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axiom of determinacy | https://ncatlab.org/nlab/source/axiom+of+determinacy |
This [[axiom]] in [[set theory]] has important role implications in [[descriptive set theory]] (since regularity properties of certain classes of subsets of [[Polish space]]s are implied by corresponding statements of determinacy). There are well-established relationships between the variants of the axiom and reflection principles/[[large cardinal]] axioms.
## Definition
+-- {: .num_defn }
###### Definition
A [[game]] is called **determined** iff some player has a winning strategy. Typically in set theory Gale-Stewart games are considered, i. e. two players pick natural numbers (which might be required to be at most some $X$). The winning condition is then given as a subset $A$ of $X^\omega$ (for $X\le\omega$) and player 1 is considered to win iff the path chosen during the game is an element of $A$.
The [[set]] providing the winning condition is also called determined in the case that the game is determined.
=--
## Statement
The axiom of determinacy (AD) states that every Gale-Stewart game is determined. In ZF this axiom contradicts the axiom of choice.
## Restrictions and relations to reflection principles
Determinacy for [[Borel set]]s can be proven in ZFC, however it cannot be proven in Zermelo set theory (without the [[axiom of replacement]] which is equivalent to [[Levy-Montague reflection]]).
The existence of infinitely many [[Woodin cardinal]]s implies the axiom of projective determinacy (PD) stating that all [[projective set]]s are determined.
The existence of infinitely many Woodin cardinals is equiconsistent to AD.
## Reference
* Wikipedia, _[Axiom of determinacy](http://en.wikipedia.org/wiki/Axiom_of_determinacy)_
[[!redirects projective determinacy]] |
axiom of extensionality | https://ncatlab.org/nlab/source/axiom+of+extensionality |
# The axiom of extensionality
* table of contents
{: toc}
## Idea
In material [[set theory]], the _axiom of extensionality_ says that the global membership relation $\in$ is an [[extensional relation]] on the class of all [[pure sets]].
Since any relation becomes extensional on its [[extensional quotient]], one can interpret this axiom as a definition of [[equality]]. However, because the extensional quotient map need not reflect the relation, there is still content to the axiom: if two sets would be identified in the extensional quotient, then they must be members of the same sets and have the same sets as members.
If one models [[pure sets]] in structural [[set theory]], then this property may be made to hold by construction.
## Statements
### Weak extensionality
#### In unsorted set theories
In any [[unsorted set theory]], the **axiom of [[weak extensionality]]** states that given a set $A$ and a set $B$, $A = B$ if and only if for all $C$, $C \in A$ if and only if $C \in B$.
If the set theory does not have [[equality]] as a primitive, we could define equality as the predicate
$$A = B \coloneqq \forall C.(C \in A) \iff (C \in B)$$
The axiom of weak extensionality is a foundational axiom in most [[material set theories]], such as Zermelo set theory and Mac Lane set theory, both which do not have the [[axiom of foundation]], as well as [[ZFC]] which does have the [[axiom of foundation]].
In [[fully formal ETCS]], where the basic objects of the theory are functions, $0$ represents both the [[empty set]] and the [[identity function]] on the empty set, and $1$ represents the [[singleton]], the [[identity function]] of the singleton, and the sole [[element]] of the singleton all at the same time, there are three possible notions of sets:
* as [[identity functions]] $\mathrm{set}(a) \coloneqq \mathrm{dom}(a) = a$ or $\mathrm{set}(a) \coloneqq \mathrm{codom}(a) = a$
* as [[functions]] into the [[singleton]] $\mathrm{set}(a) \coloneqq (\mathrm{codom}(a) = 1)$
* as functions from the [[empty set]] $\mathrm{set}(a) \coloneqq (\mathrm{dom}(a) = 0)$
The elements are functions with domain $1$, $\mathrm{element}(a) \coloneqq (\mathrm{dom}(a) = 1)$
When sets are defined as functions into the singleton, the membership relation $a \in b$ is defined by the function $a$ being an element, the function $b$ being a set, and the codomain of $a$ being $b$
$$a \in b \coloneqq \mathrm{set}(a) \wedge \mathrm{element}(b) \wedge \mathrm{codom}(a) = b$$
Weak extensionality is a theorem in this case. By the universal property of the singleton, any two sets $A$ and $B$ with the same domain are equal to each other, which means that any proposition $P$ between $A$ and $B$ implies that $A = B$, and thus that $\forall C.(C \in A) \iff (C \in B)$ implies that $A = B$. The converse follows from [[indiscernibility of identicals]].
#### In two-sorted set theories
In any [[two-sorted set theory]], the **axiom of [[weak extensionality]]** states that given a set $A:Set$ and a set $B:Set$, $A =_{Set} B$ if and only if for all $a:Element$, $a \in A$ if and only if $a \in B$.
If the set theory does not have [[equality]] of sets as a primitive, we could define equality of sets as the predicate
$$A =_{Set} B \coloneqq \forall a:Element.(a \in A) \iff (a \in B)$$
### Material strong extensionality
Let $\sim$ be a [[bisimulation]], a binary relation such that for all sets $A$ and $B$ such that $A \sim B$, the following conditions hold:
* for all sets $C$ such that $C \in A$, there exists a set $D$ such that $D \in B$ and $C \sim D$
* for all sets $D$ such that $D \in B$, there exists a set $C$ such that $C \in A$ and $C \sim D$
The **axiom of strong extensionality** states that for every bisimulation $\sim$ and for every set $A$ and $B$, $A \sim B$ implies that $A = B$.
If the set theory does not have [[equality]] as a primitive, we could define equality as the [[terminal]] [[bisimulation]], as the bisimulation $=$ such that for every bisimulation $\sim$ and for every set $A$ and $B$, $A \sim B$ implies that $A = B$.
In any set theory with the [[axiom of foundation]], the axiom of weak extensionality implies the axiom of strong extensionality.
### Structural strong extensionality
In any [[structural set theory]], the axiom of strong extensionality states that for all sets $A$ and $B$ with an [[injection]] $i:A \hookrightarrow B$, the two definitions of $i$ being a [[bijection]] are logically equivalent to each other:
* there exists a function $i^{-1}:B \to A$ such that for all elements $a \in A$ and $b \in B$, $i^{-1}(i(a)) = a$ and $i(i^{-1}(b)) = b$
* for every element $x \in B$ there exists an element $y \in A$ such that $i(y) = x$.
Similar to how in material set theory one can use the axiom of extensionality to define [[equality]] of sets, in structural set theory one can use the axiom of strong extensionality to define [[bijection]] of sets.
## In type theory
In [[dependent type theory]], the *[[axiom of extensionality]]* is a property of [[power sets]], and states that given a [[type]] $A$ and [[subtypes]] $B:\mathcal{P}(A)$ and $C:\mathcal{P}(A)$, there is an [[equivalence of types]] between the [[identity type]] $B =_{\mathcal{P}(A)} C$ and the [[dependent function type]] $\prod_{x:A} (x \in_A B) \simeq (x \in_A C)$, where $(\Omega, \mathrm{El}_\Omega)$ is the [[type of all propositions]] and $x \in_A B \coloneqq \mathrm{El}_\Omega(B(x))$ is the local membership relation between elements $x:A$ and subtypes $B:\mathcal{P}(A)$. The axiom of extensionality holds in the [[dependent type theory]] if and only if [[function extensionality]] holds.
## See also
* [[extensionality]]
* [[identity of indiscernibles]]
## References
* [[Michael Shulman]], _Comparing material and structural set theories_ ([arXiv:1808.05204](https://arxiv.org/abs/1808.05204))
* [[Håkon Robbestad Gylterud]], [[Elisabeth Bonnevier]], *Non-wellfounded sets in HoTT* ([arXiv:2001.06696](https://arxiv.org/abs/2001.06696))
category: foundational axiom
[[!redirects Axiom of extensionality]]
[[!redirects axiom of extensionality]]
[[!redirects axiom of weak extensionality]]
[[!redirects axiom of strong extensionality]] |
axiom of foundation | https://ncatlab.org/nlab/source/axiom+of+foundation |
# The axiom of foundation
* table of contents
{: toc}
In material [[set theory]], the __axiom of foundation__, also called the __axiom of regularity__, states that the membership relation $\in$ on the proper class of all [[pure set]]s is [[well-founded relation|well-founded]]. In structural set theory, accordingly, one uses well-founded relations in building structural models of well-founded pure sets.
## Statement
Given a proper class $A$ of [[pure set]]s, suppose that $A$ has the property that, given any pure set $x$,
$$ \forall t,\; t \in x \Rightarrow t \in A ,$$
then $x \in A$. Such an $A$ may be called a _membership-inductive_ class. Then the __axiom of foundation__ states that the only membership-inductive class of pure sets is the class of all pure sets. In this form, the axiom of foundation is also called *$\in$-induction*.
Although the statement here refers to proper classes, it can also be formulated as an axiom schema that makes no mention of classes.
## Alternative formulations
While the statement above follows how the axiom of foundation is generally *used* ---to prove properties of pure sets by [[transfinite induction]]---, it is complicated. Two alternative formulations are given by the following lemmas:
+-- {: .num_lemma #InfiniteDescent}
###### Lemma
The axiom of foundation holds if and only if there exists no infinite descending [[sequence]] $\cdots \in x_2 \in x_1 \in x_0$.
=--
+-- {: .proof}
###### Proof
First, suppose the axiom of foundation holds, and suppose there were such a sequence. Let $A$ be the class of all sets which are not equal to $x_i$ for any $i$. Then for any $x$, if $x$ were equal to some $x_i$, then some $t\in x$ would also be equal to some $x_j$, namely $x_{i+1}$; hence if all $t\in x$ are in $A$, so is $x$. Thus, by the axiom of foundation, all sets are in $A$, a contradiction.
Second, suppose there are no such sequences, and let $A$ be as in the statement of foundation. Suppose that $A$ does not contain all sets. Then there exists an $x_0\notin A$. By hypothesis on $A$, there must exist an $x_1\in x_0$ such that $x_1 \notin A$. And hence an $x_2\in x_1$ such that $x_2 \notin A$, and so on, producing an infinite descending $\in$-sequence in contradiction to our hypothesis; hence $A$ must contain all sets.
=--
This is essentially a version of Fermat\'s method of [[infinite descent]] modified to apply to [[transfinite induction]] instead of only to ordinary [[induction]]. It arguably provides the most direct intuitive picture of what the axiom means. Note that as a special case, it implies that no set can contain itself, since then $\cdots \in x\in x\in x$ would be an infinite chain.
+-- {: .num_lemma #MembershipMinimal}
###### Lemma
The axiom of foundation holds if and only if every [[inhabited set|inhabited]] pure set $x$ has a member $y \in x$ such that no $t \in x$ satisfies $t \in y$. (Such a $y$ is called a _membership-minimal_ element of $x$.)
=--
+-- {: .proof}
###### Proof
First, suppose the axiom of foundation holds, let $x$ be a pure set without a membership-minimal element, and let $A = \{ y | y\notin x \}$. If $z$ is a set such that all $t\in z$ are in $A$, then we cannot have $z\in x$, since then we would have some $t\in z$ with $t\in x$ and hence $t\notin A$; hence $z\in A$. So $A$ satisfies the assumptions of the foundation axiom, and hence $x$ is empty.
Second, suppose that every inhabited pure set has a membership-minimal element, and let $A$ be as in the statement of foundation. Since every pure set is contained in a [[transitive set]], it suffices to show that $A \cap x = x$ for any transitive $x$. Let $y = \{ z\in x \mid z \notin A \}$. If $y$ is inhabited, then it contains a membership-minimal element $z$, i.e. we have $z\in x$, $z\notin A$, and for every $t\in z$ we have $t\notin z$---but $t\in x$ since $x$ is transitive, hence $t\in A$. Thus this $z$ contradicts the assumption on $A$, so $y$ must be empty, as desired.
=--
This version is favoured by [[classical mathematics|classical]] set theorists as a statement of the axiom, since it uses neither higher-order reasoning (as our first definition does) or infinity (as infinite descent does).
However, neither of these is acceptable in [[constructive mathematics]], since both lemmas require at least the principle of [[excluded middle]] to prove at least one direction. In particular, the nonexistence of infinite descending sequences is too weak to allow proofs by transfinite induction (except for special forms of $A$), while the requirement that every inhabited pure set have a minimal element is unnecessarily strong and itself implies excluded middle.
More precisely, the necessary set-theoretic axioms for the above proofs are the following. (Is it known, in any case, that proofs using fewer axioms don't exist?)
* The "only if" direction of Lemma \ref{InfiniteDescent} requires only the axiom of infinity (for "infinite sequence" to make sense).
* The "if" direction of Lemma \ref{InfiniteDescent} evidently requires the principle of excluded middle, the axiom of infinity, and the axiom of [[dependent choice]]. It also appears to require the [[axiom of collection]] (since dependent choice, as usually stated, only chooses elements from a sequence of nonempty *sets*, rather than nonempty classes), and a principle of [[induction]] strong enough to recursively construct functions into a [[proper class]] (which is usually proven using the [[axiom of separation]]) in order to put together these nonempty sets into a sequence we can apply dependent choice to.
* The "only if" direction of Lemma \ref{MembershipMinimal} requires only excluded middle.
* The "if" direction of Lemma \ref{MembershipMinimal} requires excluded middle, also that every set is contained in a transitive one (the [[axiom of transitive closure]], which follows from [[axiom of replacement|replacement]]), as well as the [[axiom of separation]] in the form "the intersection of any class with a set is a set."
Another version of the axiom of foundation which *is* intuitionistically acceptable, but makes no reference to proper classes is:
+-- {: .num_lemma #WFTransitive}
###### Lemma
The axiom of foundation holds if and only if the relation $\in$ on any [[transitive set]] is a [[well-founded relation]].
=--
+-- {: .proof}
###### Proof
Suppose the axiom of foundation holds, let $x$ be a transitive set, and let $S\subseteq x$ be such that for any $y\in x$, if all $t\in y$ are in $S$, then $y\in S$. Let $A$ be the class of all sets $y$ such that if $y\in x$, then $y\in S$; then $A$ satisfies the conditions in the axiom of foundation, so it contains all sets, and hence $S=x$.
Now suppose $\in$ is well-founded on any transitive set and let $A$ satisfy the conditions in foundation. Since every set is contained in a transitive one, it suffices to show that $A\cap x = x$ for any transitive $x$, but this follows directly from the assumption.
=--
In this case:
* The "only if" direction requires no notable axioms at all, while
* The "if" direction requires [[transitive closure]], as in Lemma \ref{MembershipMinimal}, and also the axiom of separation.
Finally, another commonly cited version of foundation, equivalent to it at least over the other axioms of [[ZF]], is:
+-- {: .num_lemma #CumulativeHierarchy}
###### Lemma
The axiom of foundation holds if and only if every pure set is an element of $V_\alpha$ for some [[ordinal]] $\alpha$.
=--
Here the $V_\alpha$ are the [[cumulative hierarchy]] defined by [[transfinite recursion]] as $V_\alpha = P(\bigcup_{\beta\lt \alpha} V_\beta)$.
## Anti-foundation
Most of set theory works without the axiom of foundation, but not the deep study of well-founded pure sets. However, one might want to do material set theory without assuming that all sets are well-founded, then one would not assume this axiom.
Alternatively, one can adopt the __axiom of anti-foundation__, which says:
* Given any [[extensional relation|extensional]] [[binary relation]] $\prec$ on any [[set]] $S$, there exists a unique [[transitive set]] $U$ such that $(U,\in)$ is [[isomorphism|isomorphic]] (necessarily uniquely) to $(S,\prec)$.
Since any relation has an [[extensional quotient]], we may also phrase the axiom thus:
* Given any binary relation $\prec$ on any [[set]] $S$, there exists a unique [[transitive set]] $U$ and [[surjection]] $f : S \to U$ such that $f(s_1) \in f(s_2)$ if and only if $s_1 \in s_2$, for $s_1, s_2$ in $S$. (That is, $f$ is almost an isomorphism between $(S, \prec)$ and $(U, \in)$, but needn't be [[injection|injective]].)
Just as there are several versions of an [[extensional relation]], there are several versions of this axiom. Note that the existence part of the statement is a set-formation axiom, while the uniqueness part is a strong version of the [[axiom of extensionality]] (which is equivalent to the usual one for well-founded sets).
If you include the hypothesis that $\prec$ be [[well-founded relation|well-founded]], then the statement is a theorem ([[Mostowski's collapsing lemma]]), while the converse is the axiom of foundation.
If you adopt the axiom of anti-foundation (with the strongest notion of extensional relation) instead of foundation, then the universe of [[pure sets]] becomes the [[corecursion|corecursively]] defined ill-founded sets instead of the [[recursion|recursively]] defined well-founded sets.
## Structural meaning
Since the axiom of foundation is about pure sets, there seems little point to it in a [[structural set theory]]. However, it does have a structural consequence: every set $S$ is the underlying set of elements of a well-founded model for a pure set (in any of the ways described at [[pure set]]). If one assumes the [[axiom of choice]], however, then this statement follows from the [[well-ordering theorem]], since in that case $S$ is the underlying set of a model for a von Neumann [[ordinal number]]. But the axiom of foundation has no stronger structural consequence, since this statement already suffices to ensure that a model of structural set theory can be reconstructed from the material set theory consisting of its well-founded pure sets.
That this statement is the correct structural version of antifoundation may be justified by appeal to the [[material-structural adjunction]].
## See also
* [[Mostowski's principle]]
category: foundational axiom
[[!redirects axiom of foundation]]
[[!redirects axiom of anti-foundation]]
[[!redirects axiom of antifoundation]]
[[!redirects anti-foundation]]
[[!redirects antifoundation]]
[[!redirects axiom of regularity]]
[[!redirects membership induction]]
[[!redirects membership-induction]]
|
axiom of full comprehension | https://ncatlab.org/nlab/source/axiom+of+full+comprehension |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--
\tableofcontents
## Definition
In an [[unsorted set theory|unsorted]] [[material set theory]], the [[axiom]] or rule of **full** or **unrestricted comprehension** says that for any property $P$, there exists an [[set]] $\{ x \mid P(x) \}$ of all objects satisfying $P$.
## Resolving the inconsistency
An [[unsorted set theory]] with the unrestricted [[comprehension]] rule is called [[naive set theory]], and is inconsistent due to [[Russell's paradox]] and [[Curry's paradox]]. Here we mention several approaches to this issue.
### Separating the sets and elements
One way to resolve the inconsistency is to require the set theory to have multiple [[sorts]], one for the [[elements]] and one for the [[sets]], and not allow any reflection rules which send sets to elements or elements to sets. This means that the membership relation is only between elements and sets, and there is no notion of $x \in x$ necessary to express [[Russell's paradox]] and [[Curry's paradox]]. This is the approach taken by [[structural set theories]].
### Restricted comprehension
Standard [[unsorted set theories]] such as [[ZFC]] avoid this [[paradox]] by replacing unrestricted comprehension with the [[axiom scheme of separation]] (or "restricted comprehension"), which restricts $x$ to lie in some previously specified set $X$.
### Stratified comprehension
Set theories such as [[New Foundations]] instead replace comprehension by a rule of "stratified comprehension". This permits a "[[set of all sets]]" but still appears to avoid paradox.
### Substructural logics
It is also possible to retain full comprehension but avoid paradox by modifying the ambient logic. Passing to [[constructive logic]] doesn't help, and indeed the root issue has nothing to do with [[negation]] as such, since [[Curry's paradox]] can be stated without any [[negation]]. One might think that [[paraconsistent logic]] would help, but many paraconsistent logics are still vulnerable to Curry's paradox. Perhaps the most obvious culprit is the [[contraction rule]], and indeed [[linear logic]] (including some paraconsistent logics) can admit a full comprehension rule without explosion.
### Normal logics
Another possibility is to keep the contraction rule but restrict the use of the [[cut rule]]. It is not necessary to forbid all uses of cut, since many cuts can be normalized or eliminated. Indeed, in ordinary consistent logic, *all* cuts can be eliminated; but in the presence of full comprehension they cannot all be. Thus, another way to avoid paradox with full comprehension is to permit only proofs that can be [[normal form|normalized]].
Note that unlike a restriction on contraction, this is a "global" restriction: the proofs of two lemmas can independently be valid, but their combination may no longer be so. Similar "global" restrictions on logic were investigated by [Fitch 1953](#Fitch52), [69](#Fitch69).
## References
### In linear logic
In [[linear logic]]:
* Grishin, V. N., "Predicate and set theoretic calculi based on logic without contraction rules" (Russian), _Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya_, 45(1): 47 – 68, 1981. English translation in Math. USSR Izv., 18(1): 41 – 59, 1982. ([math-net.ru](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=1547&option_lang=eng))
* [[Jean-Yves Girard]], Light Linear Logic, _Information and Computation_, 14(3):123-137, 2003. ([pdf.gz](http://iml.univ-mrs.fr/~girard/LLL.pdf.gz))
* [[Kazushige Terui]], Light Affine Set Theory: A Naive Set Theory of Polynomial Time, _Studia Logica: An International Journal for Symbolic Logic_, Vol. 77, No. 1 (Jun., 2004), pp. 9-40. ([jstor](http://www.jstor.org/stable/20016605)) ([pdf](http://www.kurims.kyoto-u.ac.jp/~terui/lastfin.pdf)). See also Terui's slides, [Linear Logic and Naive Set Theory (Make our garden grow)](http://www.kurims.kyoto-u.ac.jp/~terui/summer3.pdf)
### Global restrictions
Several global restrictions were considered in
* {#Fitch52} [[Frederic Fitch]], *Symbolic Logic: An introduction*, Ronald Press, New York 1952
* {#Fitch69} [[Frederic Fitch]], *A method for avoiding the Curry paradox*, in *Essays in Honor of Carl G. Hempel*, Reidel, Dordrecht, Holland 1969, pp. 255--265 ([doi:10.1007/978-94-017-1466-2](https://link.springer.com/book/10.1007/978-94-017-1466-2))
The notation therein is somewhat difficult to follow for a modern reader, especially due to the somewhat confused treatment of what nowadays would be called free and bound variables. A more modern explanation of Fitch's restrictions can be found in:
* Susan Rogerson, _Natural deduction and Curry's paradox_, Journal of Philosophical Logic (2007) 36: 155--179. [pdf](https://link.springer.com/content/pdf/10.1007/s10992-006-9032-0.pdf)
The normalizability restriction is also discussed philosophically in
* Tennant, N.: *Proof and paradox*, Dialectica 36 (1982), 265-296
and other references (someone add!).
category: foundational axiom
[[!redirects full comprehension]]
[[!redirects full comprehension rule]]
[[!redirects axiom of full comprehension]]
[[!redirects axioms of full comprehension]]
[[!redirects axiom scheme of full comprehension]]
[[!redirects axiom schemes of full comprehension]]
[[!redirects axiom schema of full comprehension]]
[[!redirects axiom schemas of full comprehension]]
[[!redirects axiom schemata of full comprehension]]
[[!redirects unrestricted comprehension]]
[[!redirects unrestricted comprehension rule]]
[[!redirects axiom of unrestricted comprehension]]
[[!redirects axioms of unrestricted comprehension]]
[[!redirects axiom scheme of unrestricted comprehension]]
[[!redirects axiom schemes of unrestricted comprehension]]
[[!redirects axiom schema of unrestricted comprehension]]
[[!redirects axiom schemas of unrestricted comprehension]]
[[!redirects axiom schemata of unrestricted comprehension]]
|
axiom of infinity | https://ncatlab.org/nlab/source/axiom+of+infinity |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Foundations
+--{: .hide}
[[!include foundations - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
In the [[foundations]] of [[mathematics]], an [[axiom of infinity]] is any axiom that asserts that [[infinite set]]s exist. In [[set theory]] and [[set-level type theory]], infinite sets cannot be constructed from finite sets, so their existence must be posited as an extra axiom. Further axioms, in this vein which assert the existence of even larger sets that cannot be constructed from smaller ones are called [[large cardinal]] axioms.
## Statement
### Natural numbers
One common form of the axiom of infinity says that the particular set $N$ of [[natural number]]s exists. In material [[set theory]] this often takes the form of asserting that the von Neumann [[ordinal number]] $\omega$ exists, where $\omega$ is characterized as the smallest set such that $\emptyset\in\omega$ and whenever $a\in \omega$ then $a\cup \{a\}\in \omega$. In structural set theory the usual form of the axiom of infinity is the existence of a [[natural numbers object]].
In [[dependent type theory]], the axiom of infinity for a [[Tarski universe]] is given by the element
$$\mathrm{axinf}_U:\sum_{\mathbb{N}:U} \sum_{0:T(\mathbb{N})} \sum_{s:T(\mathbb{N}) \to T(\mathbb{N})} \prod_{C:T(\mathbb{N}) \to U} \prod_{c_0:T(C(0))} \prod_{c_s:\prod_{x:T(\mathbb{N})} T(C(x)) \to T(C(s(x)))} \sum_{c:\prod_{x:T(\mathbb{N})} T(C(x))} (c(0) =_{T(C(0))} c_0) \times \prod_{x:T(\mathbb{N})} (c(s(x)) =_{T(C(s(x)))} c_s(c(x)))$$
or
$$\mathrm{axinf}_U:\sum_{\mathbb{N}:U} \sum_{0:T(\mathbb{N})} \sum_{s:T(\mathbb{N}) \to T(\mathbb{N})} \prod_{C:U} \prod_{c_0:T(C)} \prod_{c_s:T(C) \to T(C)} \exists!c:T(\mathbb{N}) \to T(C).(f(0) =_{T(C)} c_0) \times \prod_{n:T(\mathbb{N})} c(s(n)) =_{T(C)} c_s(c(n))$$
which states that there is a [[natural numbers type]] in the universe.
There is an alternate way to express the axiom of infinity in a Tarski universe, as the axiom of resizing the [[set truncation]] of the [[type of finite types]] in $U$, since $\mathrm{isFinite}$ and set truncations are definable from the [[type of propositions]] in $U$, $\sum_{A:U} \mathrm{isProp}(A)$, but they are all usually large, and so have to be resized to be small:
$$\mathrm{axinf}_U:\sum_{\mathbb{N}:U} T(\mathbb{N}) \simeq \left[\sum_{A:U} \mathrm{isFinite}(T(A))\right]_0$$
### Integers
#### Inductive definition
Instead of a defining the natural numbers via its induction principle, one can instead define the [[integers]] via its induction principle, and then use the fact that disjoint unions are disjoint and $\mathbb{Z} \cong \mathbb{Z} \uplus \mathbb{Z}$ to construct the natural numbers.
#### Second-order definition
Alternatively, one can assume a (trichotomous) [[ordered integral domain]] $\mathbb{Z}$, such that every (trichotomous) ordered integral subdomain of $\mathbb{Z}$ is equivalent to the [[improper subset]] of $\mathbb{Z}$. This defines the [[integers]], since the integers are the [[initial object|initial]] (trichotomous) ordered integral domain and are [[strict initial object|strictly initial]]. Since the integers as defined automatically comes with a [[total order]] $\leq$ and a [[pseudo-order]] $\lt$, one can define the natural numbers as the set of non-negative integers.
The benefit of such a definition of infinity is that it allows for a [[strongly predicative mathematics|strongly predicative]] definition of the [[natural numbers]], the [[rational numbers]], and the [[real numbers]] in [[dependent type theory]], since one doesn't need arbitrary [[function sets]] or [[dependent function types]] to define the natural numbers, which one needs to characterize the natural numbers by its usual recursion or induction principle. Instead, one only needs dependent function types of a family of propositions, weak function extensionality, and the power set of the ordered integral domain $\mathbb{Z}$. Unlike the case for arbitrary ordered integral domains, being an ordered integral subdomain is only a property of an element of a power set, and then one can construct the type of ordered integral subdomains of $\mathbb{Z}$, from which one can characterize $\mathbb{Z}$ as having a contractible type of ordered integral subdomains of $\mathbb{Z}$.
### Rational numbers
If one has [[power sets]], one can assume an ([[trichotomous]]) [[ordered field]] $\mathbb{Q}$, such that every (trichotomous) ordered [[subfield]] of $\mathbb{Q}$ is equal to the [[improper subset]] of $\mathbb{Q}$. This defines the [[rational numbers]], since the rational numbers are the initial (trichotomous) [[ordered field]] and are [[strict initial object|strictly initial]]. The rational numbers are automatically infinite, and one can construct the [[integers]] $\mathbb{Z}$ as the [[intersection]] of all [[ordered integral domain|ordered integral subdomains]] of $\mathbb{Q}$, and since the integers as defined automatically comes with a [[total order]] $\leq$ and a [[pseudo-order]] $\lt$, one can define the natural numbers as the set of non-negative integers.
## Generalizations
In the form of an NNO, the axiom of infinity generalises to the existence of [[inductive type]]s or [[W-type]]s. These can be constructed from a NNO if [[power set]]s exist, but in [[predicative mathematics|predicative]] theories they can be added as additional axioms.
One could also posit the existence of the set of [[extended natural numbers]] instead of the set of natural numbers, as the set of extended natural numbers have [[countable|countably infinite]] cardinality and is the categorical [[duality|dual]] of the natural numbers in Set, a [[terminal coalgebra]] for the endofunctor $F(X) = 1 + X$ in Set. This generalises to the existence of [[coinductive types]] or [[M-types]], which can be added as additional axioms.
One could also posit the existence of [[FinSet]], the collection of [[finite sets]]. In dependent type theory this is a [[type of finite types]], a [[universe]] $\mathcal{U}$ that satisfies the axiom of finiteness (see below).
## Alternatives
Broadly speaking, [[finite mathematics]] is mathematics that does not use or need the axiom of infinity; a finitist is an extreme breed of [[constructive mathematics|constructivist]] that believes that mathematics is better without the axiom of infinity, or even that this axiom is false.
A more extreme case is to *deny* the axiom of infinity with an __axiom of finiteness__: every set is [[finite set|finite]]. There is one of these for every definition of 'finite' given on that page; here is the strongest stated directly in terms of [[set theory]] as an axiom of [[induction]]:
* Any property of sets that is invariant under [[isomorphism]] and holds for the [[empty set]] must hold for all sets if, whenever it holds for a set $X$, it holds for the [[disjoint union]] $X \uplus \{*\}$.
In [[material set theory]], this is equivalent given the [[axiom of foundation]] (which guarantees that $X$ and $\{X\}$ are [[disjoint sets|disjoint]]):
* Any property of sets that holds for the empty set must hold for all sets if, whenever it holds for a set $X$, it holds for the [[union]] $X \cup \{X\}$.
In higher categorical terms, the above axiom of finiteness could be stated as follows: [[Set]] is an initial algebra of the 2-endofunctor $F(X) \cong X \coprod 1$ in the [[(2,1)-category]] [[Grpd]].
In [[dependent type theory]], given a [[Tarski universe]] $(U, T)$ that is closed under the [[empty type]], the [[unit type]], and [[sum types]], the axiom of finiteness for the universe states that
* For all type families $A:U \vdash C(A)$ such that $T(A) \simeq T(B)$ implies that $C(A) \simeq C(B)$, elements $c_0:C(\mathbb{0})$ and dependent functions $c_s:\prod_{A:U} C(A) \to C(A + \mathbb{1})$, there exists a unique dependent function $c:\prod_{A:U} C(A)$ such that $c(\mathbb{0}) =_{C(\mathbb{0})} c_0$ and for all $A:U$, $c(A + 1) =_{C(A + 1)} c_s(c(A))$.
In [[dependent type theory]] with [[dependent product types]], [[dependent sum types]], [[identity types]], [[function extensionality]], and a [[type of all propositions]], the *axiom of finiteness* for the entire type theory is an [[axiom schema]] which states that given a type $A$, one could derive a witness that the type is a [[finite type]]:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{finWitn}_A:\mathrm{isFinite}(A)}$$
where
$$
\mathrm{isFinite}(A) \equiv
\begin{array}{c}
\prod_{S:(A \to \mathrm{Prop}) \to \mathrm{Prop}} (((\lambda x:A.\bot) \in S) \times \prod_{P:A \to \mathrm{Prop}} \prod_{Q:A \to \mathrm{Prop}} (P \in S) \\
\times (\exists!x:A.x \in Q) \times (P \cap Q =_{A \to \mathrm{Prop}} \lambda x:A.\bot) \to (P \cup Q \in S)) \to ((\lambda x:A.\top) \in S)
\end{array}
$$
The membership relation and the subtype operations used above are defined in the nLab article on [[subtypes]].
In particular, the axiom of finiteness for the entire type theory implies the [[principle of excluded middle]] for the [[type of all propositions]], since the only finite propositions are the [[decidable propositions]]. Furthermore, the axiom of finiteness implies that the type theory is a [[set-level type theory]] because every finite type is an [[h-set]].
## Related concepts
* [[finite set]], [[finite object]]
* [[Set]], [[FinSet]]
## References
In relation to [[classifying toposes]]:
* [[Andreas Blass]], *Classifying topoi and the axiom of infinity*, Algebra Universalis **26** (1989) 341-345 [[doi:10.1007/BF01211840](https://doi.org/10.1007/BF01211840)]
For constructing the [[natural numbers]] from the [[integers]]:
* {#Sattler23} [[Christian Sattler]], *Natural numbers from integers* ([pdf](https://www.cse.chalmers.se/~sattler/docs/naturals.pdf))
category: foundational axiom
[[!redirects axiom of infinity]]
[[!redirects axioms of infinity]]
[[!redirects axiom of finiteness]]
[[!redirects axioms of finiteness]]
[[!redirects axiom of finity]]
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axiom of materialization | https://ncatlab.org/nlab/source/axiom+of+materialization | # Axioms of materialization
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## Idea
An **axiom of materialization** states that every [[set]] is [[isomorphic]] to a [[pure set]] of a certain form. They can be formulated either in [[structural set theory]] or [[material set theory]], and in reasonably strong set theories they are often provable.
## Versions
* The **axiom of well-founded materialization** says that every set is isomorphic to a well-founded pure set, or equivalently (at least in a strong enough set theory) that every set can be embedded into a well-founded extensional graph. In material set theory this is also known as **Coret's Axiom B** ([Coret 64](#Coret64)).
* The **axiom of ill-founded materialization** says that every set can be embedded in an [[extensional relation|strongly extensional]] graph.
Since well-founded extensional graphs are strongly extensional, the well-founded axiom implies the ill-founded one.
## Provability
* The axiom of well-founded materialization follows from the [[axiom of choice]]: every [[well-ordered set]] is a well-founded extensional graph, so if every set is well-orderable then the axiom follows.
* In material set theory, the axiom of well-founded materialization follows from the [[axiom of foundation]], since then every set *is already* a well-founded pure set. Similarly, the axiom of ill-founded materialization follows from the [[axiom of anti-foundation]].
* In the presence of [[excluded middle]] and the [[axiom of replacement]], every strongly extensional graph can be embedded into a well-founded extensional graph. Specifically, a graph is a coalgebra for the powerset functor, hence has a cone over the [[transfinite construction of free algebras|terminal coalgebra sequence]] for that endofunctor (which exists using the axiom of replacement, even though it does not converge to a terminal coalgebra). The kernels of the maps in this cone are the approximations to the bisimilarity of the graph, and hence converge to it at some ordinal $\alpha$ (namely the [[Hartogs number]] of the lattice of binary relations; this uses excluded middle). By strong extensionality, the bisimilarity is the identity, so the $\alpha^{\mathrm{th}}$ map in the cone is an injection into a well-founded set.
Thus, given excluded middle and replacement, the axiom of ill-founded materialization implies the well-founded one, hence the two are equivalent. In particular, if the axiom of foundation in ZF is replaced by the axiom of anti-foundation, then the axiom of *well-founded* materialization is also provable.
It is unclear whether the two axioms of materialization are distinct in weaker theories lacking excluded middle or replacement, such as [[IZF]] or [[Zermelo set theory]] with the axiom of foundation removed.
## Consequences
Either axiom of materialization implies that the [[category of sets]] is [[equivalence of categories|equivalent]] to its subcategory of the relevant kind of pure sets. Thus, for instance, [[Scott's trick]] can be used to prove isomorphism-invariant properties (see for instance [this MO question](https://math.stackexchange.com/questions/305859/scotts-trick-without-the-axiom-of-regularity)).
## See also
* [[Mostowski's principle]]
* [[Mostowski set theory]]
## References
* {#Coret64} J. Coret, *Formules stratifides et axiome de fondation*, Comptes Rendus hebdomadaires des seances de l'Academie des Sciences de Paris serie A, vol. 264 (1964)
* {#Shulman19} [[Mike Shulman]], *Comparing material and structural set theories*. Annals of Pure and Applied Logic 170(4), 2019, p465–504, [arxiv](https://arxiv.org/abs/1808.05204v2)
category: foundational axiom
[[!redirects axiom of well-founded materialization]]
[[!redirects axiom of ill-founded materialization]]
[[!redirects axioms of materialization]]
[[!redirects Coret's axiom B]]
[[!redirects Coret's Axiom B]]
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axiom of multiple choice | https://ncatlab.org/nlab/source/axiom+of+multiple+choice |
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### Context
#### Foundations
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# The axiom of multiple choice
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This article is about an axiom of constructive mathematics. Some set theory literature instead uses this name for an unrelated weakening of [[axiom of choice|AC]]. For that notion, see [[(classical) axiom of multiple choice]].
## Idea
The **axiom of multiple choice** (AMC) is a weaker version of the [[axiom of choice]], which can hold in [[constructive mathematics]].
## Statement
A set-indexed family $\{D_c\}_{c\in C}$ of sets is said to be a *collection family* if for any $c\in C$ and any [[surjection]] $E\twoheadrightarrow D_c$, there exists a $c'\in C$ and a surjection $D_{c'}\twoheadrightarrow D_c$ which factors through $E$.
Depending on the author, the *axiom of multiple choice* is one of the following statements:
1. for every set $X$, there exists a collection family $\{D_c\}_{c\in C}$ such that $X\cong D_c$ for some $c$ ([[Michael Rathjen]]'s formulation, attributed to [[Peter Aczel]] and [[Alex Simpson]]), or
1. for every set $X$, there exists a collection family $\{D_c\}_{c\in C}$, with $C$ inhabited, and a family of surjections $\{D_c \to X\}_{c\in C}$ (the formulation originally given by [[Ieke Moerdijk]] and [[Erik Palmgren]]), or
1. for every set $X$, the full subcategory $(Set/X)_{surj}$ of the slice category $Set/X$ consisting of the surjections has a weakly initial set (in [[Benno van den Berg]]'s formulation; this is also called [[WISC]]).
The nLab uses the initialization AMC to cover either the first two formulations.
The third is a weaker condition, and while some may refer to as a "weak axiom of multiple choice", van den Berg obviously does not; he calls his the AMC and the Moerdijk-Palmgren formulation rather the "strong axiom of multiple choice".
## Relationships to other axioms
* Note that $P$ is a [[projective set]] if and only if the singleton family $\{P\}$ is a collection family. Therefore, since AC is equivalent to "all sets are projective," it implies AMC.
* An extension of this argument shows that [[COSHEP]] is sufficient to imply AMC.
* The [[Reflection Principle]] (RP) is equivalent to AMC (the one called strong AMC by van den Berg). RP is motivated by the [[regular extension axiom]] (REA) from constructive set theory. RP states that every map belongs to a representable class of small maps.
* However, AMC does not imply [[countable choice]] or any of the other usual consequences of AC.
* Rathjen proves that [[SVC]] also implies AMC. It follows that AMC holds in "most" models of set theory.
* AMC implies [[WISC]], and therefore also implies that the category of [[anafunctors]] between two [[small categories]] is [[essentially small category|essentially small]]. Thus WISC may be termed "weak axiom of multiple choice".
* A [[ΠW-pretopos]] satisfying the (weak) axiom of multiple choice is a _[[predicative topos]]_, or removing the word "weak", we may speak of a strong predicative topos.
## References
* [[Ieke Moerdijk]], [[Erik Palmgren]], _Type theories, toposes and constructive set theory: predicative aspects of AST_ (2000) ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.34.8934))
* Rathjen, "Choice principles in constructive and classical set theories"
In
* [[Benno van den Berg]], _Predicative toposes_ ([arXiv:1207.0959](http://arxiv.org/abs/1207.0959))
{#vdBerg}
[[WISC]] is called the "axiom of multiple choice".
* Jech, _The Axiom of Choice_ (1973), ISBN : 0444104844 (New York)
category: foundational axiom
[[!redirects axiom of multiple choice]]
[[!redirects multiple choice]]
[[!redirects AMC]]
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axiom of pairing | https://ncatlab.org/nlab/source/axiom+of+pairing |
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### Context
#### Set theory
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# The axiom of pairing
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## Idea
In [[material set theory]] as a [[foundation of mathematics]], the axiom of pairing is an important axiom needed to get the foundations off the ground (to mix metaphors). It states that [[pair sets]] exist.
## Statement
### Pairing
The __axiom of pairing__ (or __axiom of pairs__) states the following:
**Axiom of pairing**: _If $x$ and $y$ are (material) sets, then there exists a set $P$ such that $x \in P$ and $y \in P$._
Using the axiom of separation ([[bounded separation]] is enough), we can prove the existence of a particular set $P$ such that $x$ and $y$ are the *only* members of $P$. Using the [[axiom of extensionality]], we can then prove that this set $P$ is unique; it is usually denoted $\{x,y\}$ and called the __[[pair set]]__ of $x$ and $y$. Note that $\{x,x\}$ may also be denoted simply $\{x\}$.
One could also assume that the [[material set theory]] has a primitive binary operation $P$ which takes of a material set $x$ and $y$ and returns a material set $P(x, y)$. Then the axiom of pairing becomes
**Axiom of pairing**: _If $x$ and $y$ are (material) sets, then $x \in P(x, y)$ and $y \in P(x, y)$._
### Unordered pairing
The __axiom of unordered pairing__ (or __axiom of unordered pairs__) states the following:
**Axiom of unordered pairing**: If $x$ and $y$ are (material) sets, then there exists a set $P$ such that $x \in P$ and $y \in P$ and for all sets $z$, $z \in P$ implies that $z = x$ or $z = y$.
Using the [[axiom of extensionality]], we can then prove that this set $P$ is unique; it is usually denoted $\{x,y\}$ and called the __[[pair set]]__ of $x$ and $y$. Note that $\{x,x\}$ may also be denoted simply $\{x\}$.
One could also assume that the [[material set theory]] has a primitive binary operation $\{-,-\}$ which takes of a material set $x$ and $y$ and returns a material set $\{x, y\}$. Then the axiom of pairing becomes
**Axiom of unordered pairing**: If $x$ and $y$ are (material) sets, then $x \in \{x, y\}$, $y \in \{x, y\}$, and for all sets $z$, $z \in \{x, y\}$ implies that $z = x$ or $z = y$.
### Ordered pairing
Let us assume that the [[material set theory]] has a primitive binary operation $(-,-)$ which takes of a material set $x$ and $y$ and returns a material set $(x, y)$.
The __axiom of ordered pairing__ (or __axiom of ordered pairs__) states the following:
**Axiom of ordered pairing**: _If $x$ and $y$ are (material) sets, then $x \in (x, y)$, $y \in (x, y)$, and for all sets $a$ and $b$, $(a, b) = (x, y)$ if and only if $a = x$ and $b = y$._
$$\forall a.\forall b.\{a, b\} = \{x, y\} \iff (a = x \wedge b = y)$$
### With sets and elements different
In set theories where sets and elements are not the same thing, pairing becomes an operation on both the sets and the elements. One has to add a primitive ternary relation $p(X, Y, P)$ which says that $P$ is the [[Cartesian product]] of $X$ and $Y$, as well as primitive quaternary relations $\pi_1(X, P, c, a)$ and $\pi_2(Y, P, c, b)$ which says that element $a \in X$ is the left element of the pair $c \in P$ and element $b \in Y$ is the right element of the pair $c \in P$, and the following axiom:
**Axiom of ordered pairing**: _If $X$ and $Y$ are sets, then there exists a set $P$ such that $p(X, Y, P)$ and for every object $a$ and $b$, $a \in X$ and $b \in Y$ implies that there exists an object $c$ such that $c \in P$, $\pi_1(X, P, c, a)$, and $\pi_2(Y, P, c, b)$_
$P$ is usually denoted $X \times Y$ and called the __[[Cartesian product]]__ of $X$ and $Y$, while $c$ is usually denoted $(a, b)$ and called the __[[ordered pair]] of $a$ and $b$.
## Generalisation
The axiom of pairing is the binary part of a [[binary/nullary pair]] whose nullary part is the axiom stating the existence of the [[empty set]]. We can use these axioms and the [[axiom of union]] to prove every instance of the following __axiom__ (or rather theorem) __schema of finite sets__:
\begin{theorem}
If $x_1, \ldots, x_n$ are sets, then there exists a set $P$ such that $x_1, \ldots, x_n \in P$.
\end{theorem}
Again, we can prove the existence of specific $P$ such that $x_1, \ldots, x_n$ are the *only* members of $P$ and prove that this $P$ is unique; it is denoted $\{x_1, \ldots, x_n\}$ and is called the __[[finite set]]__ consisting of $x_1, \ldots, x_n$.
Note that this is a _schema_, with one instance for every (metalogical) [[natural number]]. Within axiomatic set theory, this is very different from the single statement that begins with a [[universal quantification]] over the (internal) set of natural numbers. In particular, each instance of this schema can be stated and proved without the [[axiom of infinity]]. Of course, there is one proof for each natural number.
* For $n = 0$, this is simply the axiom of the empty set.
* For $n = 1$, we use the axiom of pairing with $x \coloneqq x_1$ and $y \coloneqq x_1$ to construct $\{x_1\}$.
* For $n = 2$, we use the axiom of pairing with $x \coloneqq x_1$ and $y \coloneqq x_2$ to construct $\{x_1, x_2\}$.
* For $n = 3$, we first use the axiom of pairing twice to construct $\{x_1, x_2\}$ and $\{x_3\}$, then use pairing again to construct $\big\{\{x_1, x_2\}, \{x_3\}\big\}$, then use the axiom of union to construct $\{x_1, x_2, x_3\}$.
* In general, once we have $\{x_1, \ldots, x_{n-1}\}$, we use pairing to construct $\{x_n\}$, use pairing again to construct $\big\{\{x_1, \ldots, x_{n-1}\}, \{x_n\}\big\}$, then use the axiom of union to construct $\{x_1, \ldots, x_n\}$. (A direct proof of a single statement for $n \gt 3$ can actually go faster than this; the length of the shortest proof is [[logarithmic]] in $n$ rather than linear in $n$.)
Note that these 'finite sets' are precisely the [[Kuratowski-finite sets]] in a [[constructive mathematics|constructive]] treatment.
## Related notions
* [[pairing structure]]
In the $n$Lab, the term '[[pairing]]' usually refers to *[[ordered pair|ordered]]* pairs.
## References
For the axiom of ordered pairing see:
* [[Håkon Robbestad Gylterud]], [[Elisabeth Bonnevier]], *Non-wellfounded sets in HoTT* ([arXiv:2001.06696](https://arxiv.org/abs/2001.06696))
[[!redirects axiom of pairing]]
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category: foundational axiom
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axiom of replacement | https://ncatlab.org/nlab/source/axiom+of+replacement |
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# Axioms of replacement and collection
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## Idea
>_Zwei äquivalente Vielheiten sind entweder beide 'Mengen' oder beide inkonsistent._ [[Georg Cantor]] (1899)[^Cant]
[^Cant]: _'Two equivalent multiplicities either are both "sets" or are both inconsistent'_, letter to Dedekind from 28th July 1899 ([Cantor 1932](#Cantor99), p.444). This is suggested as an early formulation of the axiom of replacement by van Heijenoort (1967, p.113). A categorical formalization of Cantor's idea as an extension for [[ETCS]] is given in [McLarty (2004)](#McLarty).
**Axioms of collection** and replacement are [[axiom schemata]] in [[set theory]] that permit to construct new [[sets]] from other already given sets, thereby contributing substantially to the size of the set-theoretic [[universe]], and hence are seen as 'strong' [[foundational axioms]].
The most famous of these [[axiom schemata]] is the **axiom of replacement**[^name] of [[Zermelo-Fraenkel set theory]] that was suggested by [[Abraham Fraenkel|A. Fraenkel]] and formulated by [[Thoralf Skolem|T. Skolem]] in 1922. Given a unary operation $F$ and a set $x$ it permits to collect all $F(y)$ for $y\in x$ into a new set.
[^name]: The term 'replacement', or 'Ersetzungsaxiom' in German, is apparently due to [Fraenkel (1922)](#Fraenkel22) and was intended as a provisory terminology until the final formalization of Zermelo's notion of a 'definite property' which was identified with a first-order formula in the language of set theory by Skolem in the same year (and independently earlier by [[Hermann Weyl|H. Weyl]]).
The resulting expansiveness of the set-theoretic universe is somewhat peripheral to the practice of 'ordinary' mathematics and therefore a [[structural set theory]] like [[ETCS]] can omit replacement without incurring a great loss[^etcs]. Even in the context of a ZF-equivalent material set theory the axiom of replacement can be traded in for a [[reflection principle]][^bellmach].
[^bellmach]: See [Bell-Machover 1977](#BellMach77), p.495.
[^etcs]: It is possible, however, to augment a categorical set theory with a version of replacement if necessary as shown in ([Osius 1974](#Osius74), section 9) resulting in a system with the full strength of ZF. According to [McLarty (2004)](#McLarty04), Osius' ideas go back to discussions between Lawvere and the Berkeley logicians on reflection principles in 1963. McLarty's paper proposes another equivalent way to flesh out replacement categorically!
Axioms of replacement and collection become useful, however, whenever [[recursion|recursively]] constructing a set that is 'larger' than any set known before:
>what the axiom of replacement is mainly needed for in mathematical practice is to define families of sets indexed by some set $I$ carrying some inductive structure as, typically, the set $N$ of natural numbers.[^streicher]
[^streicher]: [[Thomas Streicher]] ([2005, p.79](#Streicher05)). See there for further discussion of the role of replacement for _mathematics beyond $V_{\omega +\omega}$_ and the handling of similar iterated collection processes in toposes by universes.
There are many variations on these axiom schemata, but any given system should only need one.
## Statements
In general, these axioms apply to a [[binary relation]] that relates elements of one [[set]] $A$ to arbitrary sets. However, we do *not* expect that the relation itself be an object in the theory; really, we have an axiom schema with one axiom for every binary [[predicate]] of the proper form. We will write this predicate as $\phi(x,Y)$, where $x$ stands for an elment of $A$ and $Y$ stands for any set. (Note that there may well be other free variables in the predicate.)
Generally, $\phi$ will need to be an [[entire relation]] for the axiom to apply; that is, the axiom has as a hypothesis that, for every $x \in A$, there is some $Y$ such that $\phi(x,Y)$ holds. In versions called 'replacement' instead of 'collection', $\phi$ also needs to be [[functional relation|functional]]; that is, the axiom has the hypothesis that, for every $x \in A$, there is a *unique* $Y$ such that $\phi(x,Y)$ holds. Thus, most of the 'replacement' versions only make sense if the language has a notion of [[equality]] of sets.
So much for the hypothesis of the axiom; the conclusion asserts the existence of a [[family of sets]] to which appropriate $Y$s belong. In a material set theory, we can simply state the existence of set $\mathcal{F}$ such that certain $Y \in \mathcal{F}$. In a structural set theory, we state the existence of an index set $I$, a total set $E$, and a function $f\colon E \to I$ such that each [[fibre]] $f^*(x)$ for $x \in I$ is equal to (or at least isomorphic to) certain $Y$. (Often we can take $I$ to be $A$, but that does not come into the statement of the axioms.)
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Who wants to write out some of these?
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* **Bounded replacement** or **restricted replacement** or **$\Delta_0$-replacement**: for any $\Delta_0$-formula $\phi(x, y)$, for any $a$, if for every $x \in a$ there exists a unique $y$ with $\phi(x, y)$, then the set $\{y \vert \exists x \in a.\phi(x, y)\}$ exists.
* **Full replacement**: for any formula $\phi(x, y)$, for any $a$, if for every $x \in a$ there exists a unique $y$ with $\phi(x, y)$, then the set $\{y \vert \exists x \in a.\phi(x, y)\}$ exists.
## Lawvere on replacement
>A question that has been much of a "foundational" interest, though of hardly any significance for the practice of algebra, topology, functional analysis, etc. is whether, for a given $T$, all imaginable families of sets parametrized by $T$ can be represented by $E\to T$ for some $E$ and some mapping; if "imaginable" is interpreted to mean "definable", an affirmative answer to this question is essentially equivalent (for abstract, constant sets) to the postulation of the so-called "replacement schema", whereas if $\mathcal{S}$ is considered as an object in some larger realm, an affirmative answer means that $\mathcal{S}$ itself has "inaccessible cardinality". However, in view of practice and in view of the role of $\mathcal{S}$ as a limiting case of the general notion of continuously variable sets, it seems appropriate to simply define "an internal-to-$\mathcal{S}$ $T$-parametrized family of objects of $\mathcal{S}$" to mean just a morphism of $\mathcal{S}$ with domain $T$. [Lawvere (1976, p.121)](#Lawvere76)
See also the remarks on pages 721 and 727 of ([Lawvere 2000](#Lawvere00)).
## Related discussion
* [[Thomas Streicher]], [[William Lawvere]], [[Colin McLarty]] et al, *Categorical formulations of replacement*, March 2008. ([link](http://www.mta.ca/~cat-dist/archive/2008/08-3))
* MO-discussion: _Who needs Replacement anyway ?_ ([link](http://mathoverflow.net/questions/208711/who-needs-replacement-anyway))
* Akihiro Kanamori (2012). _In Praise of Replacement_. The Bulletin of Symbolic Logic 18:1 (2012 March). [pdf](http://math.bu.edu/people/aki/20.pdf).
* $n$-category Cafe (2021), [Large Sets 12](https://golem.ph.utexas.edu/category/2021/07/large_sets_12.html) ([[Tom Leinster]]) and [Large Sets 12.5](https://golem.ph.utexas.edu/category/2021/07/large_sets_125.html) ([[Mike Shulman]]) -- includes discussion of a number of different replacement axioms for [[ETCS]] and more general [[structural set theory]].
## Related entries
* [[Zermelo-Fraenkel set theory]]
* [[reflection principle]]
* [[axiom of choice]]
* [[ETCS]], [[SEAR]]
* [[large cardinals]]
* [[type theoretic axiom of replacement]]
## References
* {#BellMach77}[[John Bell|J. L. Bell]], M. Machover, *A Course in Mathematical Logic*, North-Holland Amsterdam 1977. (ch. 10, §5)
* {#Cantor99}[[Georg Cantor]], *Brief an Dedekind vom 22. Juli 1899*, pp.443-447 in Cantor, _Gesammelte Abhandlungen_, Springer Berlin 1932. English transl. pp.113-117 of van Heijenoort (ed.), _From Frege to Gödel_ , Harvard UP 1967.
* {#Fraenkel22}A. Fraenkel, *Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre*, Math. Ann. **86** (1922) pp.230-237. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002268760))
* [[Harvey Friedman|H. J. Friedman]], *Higher Set Theory and Mathematical Practice*, Ann. Math. Logic **2** (1971) pp.326-357.
* [[André Joyal]], [[Ieke Moerdijk]], *A categorical theory of cumulative hierarchies of sets*, C. R. Math. Rep. Acad. Sci. Canada **13** (1991) pp.55-58.
* {#Lawvere76}[[F. William Lawvere]], *Variable Quantities and Variable Structures in Topoi*, pp.101-131 in Heller, Tierney (eds.), _Algebra, Topology and Category Theory: a Collection of Papers in Honor of Samuel Eilenberg_ , Academic Press New York 1976.
* {#Lawvere00}[[F. William Lawvere]], *Comments on the development of topos theory*, pp.715-734 in Pier (ed.), _Development of Mathematics 1950 - 2000_ , Birkhäuser Basel 2000. ([tac reprint](http://www.tac.mta.ca/tac/reprints/articles/24/tr24abs.html))
* {#McLarty04}[[Colin McLarty]], *Exploring Categorical Structuralism*, Phil. Math. **12** no.3 (2004) pp.37-53 doi:[10.1093/philmat/12.1.37](https://doi.org/10.1093/philmat/12.1.37).
* D. A. Martin, *Borel determinacy*, Ann. Math. **102** (1975) pp.363-371.
* {#Osius74}[[Gerhard Osius]], *Categorical Set Theory: A Characterization of the Category of Sets*, JPAA **4** (1974) pp.79-119.
* [[Thoralf Skolem]], *[[Some remarks on axiomatized set theory|Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre]]*, Mathematikerkongressen i Helsingfor 4-7 Juli 1922. English transl. pp.290-301 of van Heijenoort (ed.), _From Frege to Gödel_ , Harvard UP 1967.
* [[Thomas Streicher]], *Universes in Toposes*, pp.78-90 in Crosilla, Schuster (eds.), _From Sets and Types to Topology and Analysis_ , Oxford UP 2005. ([preprint](http://www.mathematik.tu-darmstadt.de/~streicher/NOTES/UniTop.pdf))
* [[Paul Taylor]], *[[Practical Foundations of Mathematics]]*, Cambridge UP 1999. (ch. 9)
* George Tourlakis, *Lectures in Logic and Set Theory*, Volume 2: _Set Theory_, Cambridge University Press (2003). (section III.8)
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category: foundational axiom
|
axiom of separation | https://ncatlab.org/nlab/source/axiom+of+separation |
> This is about the [[axiom of separation]] in [[set theory]]. For the axioms in [[topology]] also called "separation", see [[separation axioms]].
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### Context
#### Foundations
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[[!include foundations - contents]]
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#### $(0,1)$-Category theory
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#### Type theory
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# Axiom schemes of separation
* table of contents
{: toc}
## Idea
In [[set theory]], the [[axiom scheme]] __of separation__ aka __specification__ states that, given any [[set]] $X$ and any [[property]] $P$ of the [[elements]] of $X$, there is a set
$$ \{ X | P \} = \{ a \in X \;|\; P(a) \} $$
consisting precisely of those elements of $X$ for which $P$ holds:
$$ a \in \{ X | P \} \;\leftrightarrow\; a \in X \;\wedge\; P(a) .$$
Note that $\{X|P\}$ is a [[subset]] of $X$.
It is important to specify what language $P$ can be written in. This connects the axiom to [[logic]] and the [[foundations of mathematics]]. Arguably, [[first-order logic]] developed in order to explain the meaning of [[Ernst Zermelo]]\'s axiom of separation (although Zermelo himself disagreed with the interpretation that this gave). Separation is (usually) given as an axiom *[[axiom scheme|scheme]]* because there is one axiom for each way to state a property in the language. (We also allow parameters in $P$.)
## Statements
From weaker to stronger:
* For __bounded separation__ aka __restricted separation__ aka __$\Delta_0$-separation__, $P$ must be written in the language of first-order set theory and all [[quantifications]] must be guarded by a set: of the form $\forall x \in A$ or $\exists x \in A$ for some set $A$.
* For __limited separation__ aka __$\Delta_0^{\mathcal{P}}$-separation__, $P$ must be written in the language of first-order set theory and all quantifications must be guarded by a set or a [[power class]]: of the form $\forall x \in A$, $\exists B \subseteq A$, etc. Limited separation trivially implies bounded separation, while bounded separation implies limited separation if [[power sets]] exist.
* For __full separation__ aka simply __separation__, $P$ must be written in the language of first-order set theory, but otherwise anything goes; in a [[class theory]], $P$ must be guarded by a class. Full separation trivially implies limited separation.
* For __large separation__, $P$ must be written in the language of first-order class theory; of course, this only makes sense *in* a class theory. The difference in strength between the class theories $MK$ and $NBG$ is precisely that the former has large separation but the latter does not.
* Separation is sometimes called __restricted comprehension__; for __[[full comprehension]]__, a.k.a. simply __comprehension__, no set $X$ needs to be given ahead of time. Full comprehension was proposed by [[Gottlob Frege]], but leads to [[Russell's paradox]]. However, full comprehension can sometimes be allowed if the ambient logic is nonclassical, such as [[linear logic]] or [[paraconsistent logic]].
* For __stratified comprehension__, no set $X$ is given, but $P$ is restricted to stratified formulas, in which each [[variable]] $x$ can be given a consistent [[natural number]] $\sigma(x)$ (its stratification) such that $x \in y$ appears in the formula only if $\sigma(y) = \sigma(x) + 1$. This is used in [[Van Quine]]\'s [[New Foundations]].
## In structural set theory
Set theory is usually given in [[material set theory|material]] form, with a language based on a global membership relation $\in$, and we have implicitly followed this above. However, separation makes sense also in [[structural set theory]] (although full comprehension does not, except in a structural class theory with a [[class of all sets]], where it again leads to paradox). The conclusion of the axiom is the existence of a set
$$ \{ X | P \} = \{ a\colon X \;|\; P(a) \} $$
and an [[injection]] $i_P\colon \{X|P\} \to X$ such that
$$ \exists b\colon S,\; a = i_P(b) \;\leftrightarrow\; P(a) .$$
Note that $\{X|P\}$, equipped with $i_P$, is a [[subset]] of $X$ in the structural sense.
The structural axioms can of course be stated even in a material set theory, where they are actually weaker than the corresponding material axioms; however, the material axioms follow (as usual) from the structural axioms using [[axiom of replacement|restricted replacement]], which is quite weak (and also follows from the material form of limited separation).
If a structural set theory is given by stating axioms for the [[category of sets]], then restricted separation amounts to the property that this category is a [[Heyting category]]. If it is an [[elementary topos]], then since it satisfies the power set axiom, this implies limited separation as well. Full separation is somewhat less natural to state category-theoretically, but the combination of full separation with the structural [[axiom of collection]] is equivalent to saying that the category of sets is [[autological category|autological]] (see [here](https://arxiv.org/abs/1004.3802) for now for a definition of autological categories).
The axiom schema of restricted separation is formally expressed by the rules
$$\frac{\Gamma \vdash S \; \mathrm{set} \quad \Gamma, x \in S \vdash P(x) \; \mathrm{prop}}{\Gamma \vdash \{x \in S \vert P(x)\} \; \mathrm{set}}$$
$$\frac{\Gamma \vdash S \; \mathrm{set} \quad \Gamma, x \in S \vdash P(x) \; \mathrm{prop}}{\Gamma \vdash i:\{x \in S \vert P(x)\} \to S}$$
$$\frac{\Gamma \vdash S \; \mathrm{set} \quad \Gamma, x \in S \vdash P(x) \; \mathrm{prop}}{\Gamma, y \in \{x \in S \vert P(x)\}, z \in \{x \in S \vert P(x)\}, i(y) = i(z) \; \mathrm{true} \vdash y = z \; \mathrm{true}}$$
$$\frac{\Gamma \vdash S \; \mathrm{set} \quad \Gamma, x \in S \vdash P(x) \; \mathrm{prop}}{\Gamma, x \in R, \exists y \in \{x \in S \vert P(x)\}.x = i(y) \; \mathrm{true} \vdash P(x) \; \mathrm{true}}$$
$$\frac{\Gamma \vdash S \; \mathrm{set} \quad \Gamma, x \in S \vdash P(x) \; \mathrm{prop}}{\Gamma, x \in R, P(x) \; \mathrm{true} \vdash \exists y \in \{x \in S \vert P(x)\}.x = i(y) \; \mathrm{true}}$$
## In hyperdoctrines
### Lawvere's definition
{#LawvereDefinition}
[[Lawvere]] gives a definition ([Lawvere70, p. 12-13](#Lawvere70)) of comprehension in [[hyperdoctrines]].
+-- {: .num_defn #LawvereianComprehension}
###### Definition
Let $p \colon E \to B$ be a [[bifibration]] over the category $B$, and assume that each [[fibre]] $E_X$ of $E$ has a [[terminal object]] $T_X$. For any [[morphism]] $t \colon Y \to X$ in $B$, define the _image_ $im t$ of $t$ to be the pushforward ([[dependent sum]]) $t_! T_Y$. This gives rise to a functor $im \colon B/X \to E_X$ for each $X$. Then $E$ is said to _have comprehension_, or to satisfy the _comprehension schema_, if each image functor has a [[right adjoint]] $\{-\} \colon E_X \to B/X$.
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###### Remark
This means that for each $P \in E_X$ there is a morphism $i_P \colon \{ P \} \to X$ in $B$ such that there is a [[bijection]] between commuting triangles $t = i_P \circ t'$ in $B$ and morphisms $im t \to P$ in $E_X$. Lawvere calls the morphism $\{P\} \to X$ the _[[extension (semantics)|extension]]_ of $P$, so that one could say that $E$ satisfies the comprehension schema if the extensions of all predicates exist.
=--
### Ehrhard's reformulation
Notice that, in the above situation, morphisms $\im t \to P$ in $E_X$ are in bijection with morphisms $T_Y \to t^* P$ in $E_Y$, and that these are the same as morphisms $T_Y \to P$ in the total category $E$ that lie over $t$. This leads to an alternative formulation of the definition, originally considered by [Ehrhard 1988](#Ehrhard88) (see also [Jacobs 1993](#Jacobs93), [Jacobs 99](#Jacobs99)), which does not depend on $p$ being a bifibration.
Suppose the fibres $E_X$ have terminal objects that are preserved by the reindexing functors. This is equivalent to saying that $p:E\to B$ has a section $T:B\to E$ taking each object $X$ to a terminal object $T_X$ of $E_X$ and each morphism to a cartesian arrow; such a functor is then automatically [[fully faithful]] and [[right adjoint]] to $p:E\to B$. If $B$ has a terminal object $1$, then this is additionally equivalent to saying that the entire category $E$ has a terminal object $T_1$ preserved by $p$, since then $X^*(T_1)$ will be terminal in $E_X$ by combining the universal properties of $X^*$ and $T_1$. Note also that if $p$ is a bifibration, as in Lawvere's definition, then the reindexing functors are right adjoints and hence automatically preserve all terminal objects.
Comprehension in the sense above is equivalent to the existence of a further right adjoint to the terminal-object functor $T:B\to E$.
The implication from [Lawvere's definition](#LawvereDefinition) to Ehrhard's is clear. Conversely, if a fibration $p \colon E \to B$ satisfies the Ehrhard definition, then there is a bijection between morphisms $T_Y \to P$ in $E$ and morphisms $Y \to \{P\}$ in $B$. One must then show that the composite of this last with the canonical $\{P\} \to X$ (given by $p$ applied to the counit $T_{\{P\}} \to P$) is equal to $p$ applied to the morphism $Y \to \{P\}$, thus giving a morphism in $B/X$ of the right sort. But in fact the morphism $Y \to \{P\}$ is unique with the property that applying the functor $T_{-}$ and composing with the counit gives the morphism $T_Y \to P$, by the counit's universal property, and applying $p$ to this commuting triangle in $E$ produces the required one in $B$.
### Examples
#### Subsets
The tautological example which is useful to see what the abstract [definition by Lawvere](#LawvereDefinition) axiomatizes is the following.
Let $B = $[[Set]] be the category of sets and for $X$ a [[set]] let $E_X$ be the [[poset]] of [[subsets]] of $X$, regarded as the [[propositions]] about elements in $X$. Then comprehension exists and is given by sending a subset of $X$ regarded as an object of $E_X$ (hence regarded as a proposition) to the same subset, but now regarded as a [[monomorphism]] in [[Set]] into $X$.
More generally, the same construction works for the posets of subobjects in any [[regular category]].
#### Presheaves
There is a functor $Cat_1^{op} \to CAT_1$, from the [[large category|large]] [[Cat|1-category of categories]] and functors to the '[[very large category|very large]]' 1-category of large categories and functors, that sends a category $C$ to the [[category of presheaves]] $[C^{op}, Set]$ on $C$, and a functor $F \colon C \to D$ to the pullback functor $F^* \colon [D^{op}, Set] \to [C^{op}, Set]$. These pullback functors have left and right adjoints given by [[Kan extension]].
[Lawvere 70](#Lawvere70) shows that this fibration has comprehension, with the [[extension (semantics)|extension]] of a presheaf given by its [[category of elements]] together with the canonical projection from this to the base category.
#### In dependent linear type theory
{#ExamplesDependentLinearTypeTheory}
If the [[hyperdoctrine]] has [[linear type theories]]/[[symmetric monoidal categories]] as fibers, then it is more natural to consider in def. \ref{LawvereianComprehension} not the [[terminal object]] in some fiber (if any) but the [[tensor unit]] (which of course happens to be the terminal object in the case of [[cartesian monoidal category|cartesian monoidal]]) fibers.
In this case then the image functor in def. \ref{LawvereianComprehension} is
$$
\Sigma_X \;\colon\; (Y \stackrel{f}{\to} X) \mapsto \underset{f}{\sum} 1_Y
\,.
$$
If this has a [[right adjoint]] $R_X$, hence if we have linear comprehension according to def. \ref{LawvereianComprehension}, then the induced [[comonad]]
$$
!_X \coloneqq \Sigma_X \circ R_X
$$
is (the [[dependent type theory|dependent]] version of) the canonical [[categorical semantics|categorical interpretation]] of the [[exponential modality]] of [[linear logic]]. See at _[dependent linear type theory -- The canonical co-modality](http://ncatlab.org/nlab/show/dependent+linear+type+theory#TheCanonicalComodality)_ for more.
### The comprehensive factorization system
If the extension functors $E_X \to B/X$ are [[fully faithful]], then together they make $E$ into a 'fibrewise [[reflective subcategory|reflective]]' subfibration of the codomain fibration of $B$, which is a reflective subfibration if the image functors preserve pullbacks. (See [[stable factorization system]].)
Call a morphism of the form $\{P\} \to X$ a _subtype inclusion_. Every morphism $t \colon Y \to X$ of $B$ factors through one such by means of the unit $\eta_t$ of the adjunction $im \dashv \{-\}$:
$$
\array{
Y & & \overset{\eta_t}{\to} & & \{ im t \} \\
& \mathllap{t}\searrow & & \swarrow\mathrlap{i_t} & \\
& & X & &
}
$$
This gives rise to an [[orthogonal factorization system]] precisely when each $\eta_t$ is [[orthogonal]] to all subtype inclusions.
It is shown by [(Carboni et. al., section 2.12)](#Carboni97) that this holds if and only if subtype inclusions are closed under composition.
For the subobject fibration of a regular category, this gives the usual ([[regular epimorphism|regular epi]], [[monomorphism|mono]]) factorization system, while for the fibration of presheaves over $Cat$ it gives the factorization of a functor into a [[final functor]] followed by a [[discrete fibration]]. (See also [[michaelshulman:comprehensive factorization]] for a description of the latter as a [[factorization system in a 2-category]].)
## Axiom or axiom scheme?
Although usually presented as an axiom scheme, in many cases, all instances of separation follow from finitely many special cases (which can then be packaged into a single axiom, using [[and|conjunction]], although this is probably pointless). This is the case, for example, in [[ETCS]] (a structural set theory that satisfied bounded separation) and [[NBG]] (a material class theory that satisfies full separation). In [[type theory|type-theoretic]] foundations of mathematics, separation is usually invisible, but again some form (generally only bounded) can again be proved from a few specific axioms or constructions.
## Relation to the axiom of replacement
Full separation follows from the [[axiom of replacement]] and the principle of [[excluded middle]] (along with the axiom of the [[empty set]]). Therefore, the axiom is often left out entirely of a description of [[ZFC]] (the usually accepted foundation of mathematics). In versions of set theory for [[constructive mathematics]], however, we often have replacement but only bounded or limited separation, and in any case separation must be listed explicitly.
## Related concepts
* [[axiom of full comprehension]]
## References
### General
(...)
### In hyperdoctrines
* {#Lawvere70} [[Lawvere]], F. W. _Equality in hyperdoctrines and comprehension schema as an adjoint functor_. In A. Heller, ed., _Proc. New York Symp. on Applications of Categorical Algebra_, pp. 1--14. AMS, 1970. ([[LawvereComprehension.pdf:file]])
* {#Lawvere73} [[Lawvere]], F. W. _Metric spaces, generalized logic and closed categories_ Rend. Sem. Mat. Fis. Milano, 43:135–166 (1973). Reprints in Theory and Applications of Categories, No. 1 (2002) pp 1-37 ([tac](http://www.tac.mta.ca/tac/reprints/articles/1/tr1abs.html))
* {#Ehrhard88} [[Thomas Ehrhard]]. A Categorical Semantics of Constructions. LICS 1988. ([pdf](https://www.irif.univ-paris-diderot.fr/~ehrhard/pub/categ-sem-constr.pdf))
* {#Jacobs93} [[Bart Jacobs]]. Comprehension categories and the semantics of type dependency. _Theoretical Computer Science_ 107:2 (1993), pp. 169-207.
* {#Jacobs99} [[Bart Jacobs]]. _Categorical Logic and Type Theory_. Elsevier, 1999.
* [[Finn Lawler]], section "Tabulation and comprehension" in: _[[finnlawler:thesis outline|Fibrations of predicates and bicategories of relations]]_ (2014)
* {#Carboni97} [[Aurelio Carboni]], [[George Janelidze]], [[Max Kelly]], [[Robert Paré]], _On localization and stabilization for factorization systems_, Appl. Categ. Structures **5** (1997) 1-58 [[doi:10.1023/A:1008620404444](https://doi.org/10.1023/A:1008620404444)]
On the relation between comprehension schemes and [[factorisation systems]]:
* [[Clemens Berger]], [[Ralph M. Kaufmann]], *Comprehensive Factorization Systems*, Tbilisi Math. J. 10 (2017), 255-277 ([doi:10.1515/tmj-2017-0112](https://projecteuclid.org/journals/tbilisi-mathematical-journal/volume-10/issue-3/Comprehensive-factorisation-systems/10.1515/tmj-2017-0112.short))
Discussion in [[(infinity,1)-category theory|$(\infty,1)$-category theory]]:
* [[Raffael Stenzel]], _(∞,1)-Categorical Comprehension Schemes_ [[arXiv:2010.09663](https://arxiv.org/abs/2010.09663), talk: [YT](https://www.youtube.com/watch?v=qVhRruh_1jI)]
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[[!redirects axiom scheme of stratified comprehension]]
[[!redirects axiom schemes of stratified comprehension]]
[[!redirects axiom schema of stratified comprehension]]
[[!redirects axiom schemas of stratified comprehension]]
[[!redirects axiom schemata of stratified comprehension]]
[[!redirects comprehension rule]]
[[!redirects comprehension rules]] |
axiom of set truncation | https://ncatlab.org/nlab/source/axiom+of+set+truncation |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Foundational axiom
+-- {: .hide}
[[!include foundational axiom - contents]]
=--
=--
=--
\tableofcontents
## Idea
In [[dependent type theory]], an **axiom of set truncation** is a [[axiom]] or unjustified rule which implies the [[uniqueness of identity proofs]] which states that every type is an [[h-set]]. Adding an axiom of set truncation to a dependent type theory results in a [[set-level type theory]].
## Examples
TODO: show that each of the axioms below implies the [[uniqueness of identity proofs]].
### Uniqueness of identity proofs
#### For identification types
In [[dependent type theory]], [[uniqueness of identity proofs]] for [[identification types]] is given by the following rule:
$$\frac{\Gamma \vdash A \; type \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:\mathrm{Id}_A(a, b) \quad \Gamma \vdash q:\mathrm{Id}_A(a, b)}{\Gamma \vdash \mathrm{UIP}_A(a, b, p, q):\mathrm{Id}_{\mathrm{Id}_A(a, b)}(p, q)}$$
#### For heterogeneous identification types
There is uniqueness of identity proofs for [[heterogeneous identification types]]:
$$\frac{
\begin{array}{l}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \\
\Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:\mathrm{Id}_A(a, b) \quad \Gamma \vdash y:B \quad \Gamma \vdash z:B \\
\Gamma \vdash q:\mathrm{hId}_{B}(a, b, p, y, z) \quad \Gamma \vdash r:\mathrm{hId}_{B}(a, b, p, y, z)
\end{array}
}{\Gamma \vdash \mathrm{UIP}_{B}(a, b, p, y, z, q, r):\mathrm{Id}_{\mathrm{hId}_{B}(a, b, p, y, z)}(q, r)}$$
Uniqueness of identity proofs for identification types follows from uniqueness of identity proofs for heterogeneous identification types by taking canonical element $\mathrm{pt}:\mathbb{1}$ of the [[unit type]] $\mathbb{1}$, identification $\mathrm{refl}_{\mathbb{1}}(\mathrm{pt}):\mathrm{Id}_{\mathbb{1}}(\mathrm{pt}, \mathrm{pt})$, and elements $a:A$ and $b:A$. Since the following two types are equivalent:
$$\mathrm{Id}_{A}(a, b) \simeq \mathrm{hId}_{A}(\mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}), a, b)$$
and because $\mathrm{hId}_{A}(\mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}), a, b)$ is always a [[mere proposition]] by uniqueness of identity proofs for heterogeneous identification types, it follows that $\mathrm{Id}_{A}(a, b)$ is also always a mere proposition.
Similarly, there is also uniqueness of identity proofs for [[dependent heterogeneous identification types]]:
$$\frac{
\begin{array}{l}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \\
\Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:\mathrm{Id}_A(a, b) \quad \Gamma \vdash y:B(a) \quad \Gamma \vdash z:B(b) \\
\Gamma \vdash q:\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z) \quad \Gamma \vdash r:\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z)
\end{array}
}{\Gamma \vdash \mathrm{UIP}_{x:A.B(x)}(a, b, p, y, z, q, r):\mathrm{Id}_{\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z)}(q, r)}$$
Uniqueness of identity proofs for identification types follows from uniqueness of identity proofs for dependent heterogeneous identification types by defining the type $A$ to be the dependent type $C(\mathrm{pt})$ for type family $x:\mathbb{1} \vdash C(x)$ indexed by the [[unit type]] $\mathbb{1}$, and taking the canonical element $\mathrm{pt}:\mathbb{1}$, identification $\mathrm{refl}_{\mathbb{1}}(\mathrm{pt}):\mathrm{Id}_{\mathbb{1}}(\mathrm{pt}, \mathrm{pt})$, and elements $a:C(\mathrm{pt})$ and $b:C(\mathrm{pt})$. Since the two types are equivalent:
$$\mathrm{Id}_{A}(a, b) \equiv \mathrm{Id}_{C(\mathrm{pt})}(a, b) \simeq \mathrm{hId}_{x:\mathbb{1}.C(x)}(\mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}), a, b)$$
and because $\mathrm{hId}_{x:\mathbb{1}.C(x)}(\mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}), a, b)$ is always a [[mere proposition]] by uniqueness of identity proofs for dependent heterogeneous identification types, it follows that $\mathrm{Id}_{C(\mathrm{pt})}(a, b)$ and thus by definition $\mathrm{Id}_{A}(a, b)$ is also always a mere proposition.
### Axiom K
#### For identification types
In [[dependent type theory]], [[axiom K (type theory)|axiom K]] is given by the following rule:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma\vdash a:A \quad \Gamma \vdash p:\mathrm{Id}_A(a, a)}{\Gamma\vdash K_A(a, p):\mathrm{Id}_{\mathrm{Id}_A(a, a)}(p, refl_A(a))}$$
#### For heterogeneous identification types
There is an axiom K for [[heterogeneous identification types]]:
$$\frac{
\begin{array}{l}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \\
\Gamma \vdash f:A \to B \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:\mathrm{Id}_A(a, b) \\
\Gamma \vdash q:\mathrm{hId}_{B}(a, b, p, f(a), f(b))
\end{array}
}{\Gamma \vdash K_{B}(f, a, b, p, q):\mathrm{Id}_{\mathrm{hId}_{B}(a, b, p, f(a), f(b))}(q, \mathrm{ap}_{B}(f, a, b, p))}$$
Axiom K for identification types follows from axiom K for heterogeneous identification types by taking canonical element $\mathrm{pt}:\mathbb{1}$ of the [[unit type]] $\mathbb{1}$, identification $\mathrm{refl}_{\mathbb{1}}(\mathrm{pt}):\mathrm{Id}_{\mathbb{1}}(\mathrm{pt}, \mathrm{pt})$, and elements $a:A$ and $b:A$. Since the following two types are equivalent:
$$\mathrm{Id}_{A}(a, b) \simeq \mathrm{hId}_{A}(\mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}), a, b)$$
and because $\mathrm{hId}_{A}(\mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}), a, a)$ is always a [[contractible type]] with [[center of contraction]] $\mathrm{ap}_{A}(f, \mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}))$ for element $a:A$, and function $f:\mathbb{1} \to A$, by axiom K for heterogeneous identification types, it follows that $\mathrm{Id}_{A}(a, a)$ is also always a contractible type, with center of contraction $\mathrm{refl}_A(a)$.
Similarly, there is also an axiom K for [[dependent heterogeneous identification types]]:
$$\frac{
\begin{array}{l}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \\
\Gamma \vdash f:\prod_{x:A} B(x) \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:\mathrm{Id}_A(a, b) \\
\Gamma \vdash q:\mathrm{hId}_{x:A.B(x)}(a, b, p, f(a), f(b))
\end{array}
}{\Gamma \vdash K_{x:A.B(x)}(f, a, b, p, q):\mathrm{Id}_{\mathrm{hId}_{x:A.B(x)}(a, b, p, f(a), f(b))}(q, \mathrm{apd}_{x:A.B(x)}(f, a, b, p))}$$
Axiom K for identification types follows from axiom K for dependent heterogeneous identification types by defining the type $A$ to be the dependent type $C(\mathrm{pt})$ for type family $x:\mathbb{1} \vdash C(x)$ indexed by the [[unit type]] $\mathbb{1}$, and taking the canonical element $\mathrm{pt}:\mathbb{1}$, identification $\mathrm{refl}_{\mathbb{1}}(\mathrm{pt}):\mathrm{Id}_{\mathbb{1}}(\mathrm{pt}, \mathrm{pt})$, and elements $a:C(\mathrm{pt})$ and $b:C(\mathrm{pt})$. Since the two types are equivalent:
$$\mathrm{Id}_{A}(a, b) \equiv \mathrm{Id}_{C(\mathrm{pt})}(a, b) \simeq \mathrm{hId}_{x:\mathbb{1}.C(x)}(\mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}), a, b)$$
and because $\mathrm{hId}_{x:\mathbb{1}.C(x)}(\mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}), a, b)$ is always a [[contractible type]] with [[center of contraction]] $\mathrm{apd}_{x:\mathbb{1}.C(x)}(f, \mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}))$ for element $a:A$, and dependent function $f:\prod_{x:\mathbb{1}} C(x)$, by axiom K for heterogeneous identification types, it follows that $\mathrm{Id}_{C(\mathrm{pt}}(a, a)$ and thus by definition $\mathrm{Id}_{A}(a, a)$ is also always a contractible type, with center of contraction $\mathrm{refl}_A(a)$.
### Leibniz’s identity of indiscernibles
In [[dependent type theory]] with [[power sets]], [[Leibniz's identity of indiscernibles]] is given by the following [[inference rule]]:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, x:A, y:A \vdash \mathrm{idOfInd}_A(x, y):\mathrm{Id}_A(x, y) \simeq \left(\prod_{P:\mathcal{P}(A)} P(x) \simeq P(y)\right)}$$
This implies that all types $A$ are sets, because the type
$$\prod_{P:\mathcal{P}(A)} P(x) \simeq P(y)$$
is always a proposition, in the context of [[weak function extensionality]], and thus by the [[equivalence of types]], $\mathrm{Id}_A(x, y)$ is also a proposition for all $x:A$ and $y:A$, which means that $A$ is a set.
### Set-theoretic principle of unique choice
In [[dependent type theory]] with [[power sets]], the [[set-theoretic principle of unique choice]] is given by the following rule:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash \mathrm{puc}_{A, B}:\left(\sum_{R:A \times B \to \mathcal{P}(\mathbb{1})} \mathrm{isTotal}(R) \times \mathrm{isFunctional}(R)\right) \simeq (A \to B)}$$
Because the type $A \times B \to \mathcal{P}(\mathbb{1})$ is a [[set]], the type
$$\sum_{R:A \times B \to \mathcal{P}(\mathbb{1})} \mathrm{isTotal}(R) \times \mathrm{isFunctional}(R)$$
is also a set, and the equivalence only exists if the function type $A \to B$ is a set, which only occurs if $B$ is a set. Thus, this rule implies that all types are sets.
### Boundary separation
In [[cubical type theory]], [[boundary separation]] is given by the following rule:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash r:I \quad \Gamma, (r =_I 0) \vee (r =_I 1) \; \mathrm{true} \vdash a \equiv b:A}{\Gamma \vdash a \equiv b:A}$$
### Axiom of existence
In [[dependent type theory]], the [[type-theoretic axiom of existence]] states that every type has a [[choice operator]], a function from its [[bracket type]] to the type itself, and is given by the following rule:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \epsilon_A:[A] \to A}$$
Indeed, it suffices for the [[identity types]] to have [[choice operators]]:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, x:A, y:A \vdash \epsilon_{\mathrm{Id}_A}(x, y):[\mathrm{Id}_A(x, y)] \to \mathrm{Id}_A(x, y)}$$
### Equality reflection
In [[dependent type theory]], [[equality reflection]] is given by the following rule:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:\mathrm{Id}_A(a, b)}{\Gamma \vdash a \equiv b:A}$$
### Strong pattern matching
...
### $S^1$-localization
In [[dependent type theory]], the [[axiom of S1-localization|axiom of $S^1$-localization]] is given by the following rules:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash f:S^1 \to A}{\Gamma \vdash \mathrm{circlocal}_{A}:\mathrm{isEquiv}(\mathrm{const}_{A, S^1})}$$
### Axiom of finiteness
In [[dependent type theory]] with [[dependent product types]], [[dependent sum types]], [[identity types]], [[function extensionality]], and a [[type of all propositions]], the [[axiom of finiteness]] is given by the following rule:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{finWitn}_A:\mathrm{isFinite}(A)}$$
where
$$
\mathrm{isFinite}(A) \equiv
\begin{array}{c}
\prod_{S:(A \to \mathrm{Prop}) \to \mathrm{Prop}} (((\lambda x:A.\bot) \in S) \times \prod_{P:A \to \mathrm{Prop}} \prod_{Q:A \to \mathrm{Prop}} (P \in S) \\
\times (\exists!x:A.x \in Q) \times (P \cap Q =_{A \to \mathrm{Prop}} \lambda x:A.\bot) \to (P \cup Q \in S)) \to ((\lambda x:A.\top) \in S)
\end{array}
$$
The membership relation and the subtype operations used above are defined in the nLab article on [[subtypes]].
### Circle type axioms
Any one of the following axioms on the [[circle type]] implies [[axiom K]] or [[uniqueness of identity proofs]] for all types in the dependent type theory, and thus is an axiom of set-truncation:
* That $S^1$ is a [[set]],
$$\mathrm{trunc}_0^{S^1}:\mathrm{isSet}(S^1)$$
* That $S^1$ is a [[proposition]] or a [[contractible type]], which are the same condition as $S^1$ being a [[set]] because $S^1$ is a [[pointed connected type]]
$$\mathrm{trunc}^{S^1}:\mathrm{isProp}(S^1) \qquad \mathrm{contr}_{S^1}:\mathrm{isContr}(S^1)$$
* That there is an [[identification]] $K:\mathrm{refl}_{S^1}(\mathrm{base}) =_{S^1} \mathrm{loop}$
### Kuratowski-finite implies finite
In [[dependent type theory]], in general there are [[Kuratowski-finite]] types which are not [[h-sets]], such as the [[circle type]]. However, every [[finite type]] is an [[h-set]], and hence why they are usually called [[finite sets]].
A naive translation into dependent type theory of the traditional set-theoretic theorem in classical mathematics that "Kuratowski-finite implies finite" results in the statement that every Kuratowski-finite type is finite. This implies that the circle type is a finite type, which implies that the circle type is an h-set, which implies that every type is an h-set. Thus, that every Kuratowski-finite type is finite is an axiom of set truncaiton.
The statement "Kuratowski-finite implies finite" in classical mathematics is really a statement about h-sets, similar to how the [[axiom of choice]] is really a statement about h-sets. This is because it is still true in [[constructive mathematics]] that every Kuratowski-finite type with [[decidable equality]] is a [[finite type]], and in dependent type theory, every type with decidable equality is an [[h-set]].
## See also
* [[h-set]]
* [[set-level type theory]]
[[!redirects axioms of set truncation]]
[[!redirects set truncation axiom]]
[[!redirects set truncation axioms]]
[[!redirects rule of set truncation]]
[[!redirects rules of set truncation]]
[[!redirects set truncation rule]]
[[!redirects set truncation rules]] |
axiom of stack completions | https://ncatlab.org/nlab/source/axiom+of+stack+completions | # Axiom of stack completion
\tableofcontents
## Definition
An [[internal category]] $\mathbb{A}$ in an [[elementary topos]] $\mathcal{E}$ is called an **(intrinsic) stack** if the [[indexed category]] that it represents, $(X\in \mathcal{E}) \mapsto \mathcal{E}(X,\mathbb{A}) \in Cat$, is a [[stack]] for the [[regular topology]] of $\mathcal{E}$. An internal functor is called a **weak equivalence** if it is internally [[essentially surjective]] and [[fully faithful]]; this is strictly weaker than being an internal equivalence if the [[axiom of choice]] does not hold in $\mathcal{E}$. However, if $f:\mathbb{A} \to \mathbb{B}$ is a weak equivalence and $\mathbb{C}$ is a stack, then the induced functor $\mathrm{Cat}(\mathcal{E})(\mathbb{B},\mathbb {C})\to \mathrm{Cat}(\mathcal{E})(\mathbb{A},\mathbb{C})$ is an equivalence.
We say that $\mathcal{E}$ satisfies the **axiom of stack completions (ASC)** if every internal category admits a weak equivalence to an internal category that is a stack (see [Bunge-Hermida](#BH)).
## Examples
* If $\mathcal{E}$ satisfies the [[axiom of choice]], then every internal category is a stack, so ASC holds trivially.
* If $\mathcal{E}$ is a [[Grothendieck topos]], then internal stack completions can be constructed using the [[small object argument]], as [[fibrant replacements]] in the [[model structure]] on $\mathrm{Cat}(\mathcal{E})$ constructed by [Joyal and Tierney](#JT), so it satisfies ASC.
## References
* {#BH} Bunge and Hermida, "Pseudomonadicity and 2-stack completions"
* {#JT} [[André Joyal]], [[Myles Tierney]], *Strong stacks and classifying spaces*, Lecture Notes in Mathematics **1488**, Springer (1991) [[doi:10.1007/BFb0084222](https://doi.org/10.1007/BFb0084222)]
[[!redirects axiom of stack completion]]
[[!redirects ASC]]
|
axiom of union | https://ncatlab.org/nlab/source/axiom+of+union |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Set theory
+-- {: .hide}
[[!include set theory - contents]]
=--
=--
=--
# The axiom of union
* table of contents
{: toc}
## Idea
In [[material set theory]] as a [[foundation of mathematics]], the axiom of union is an important axiom needed to get the foundations off the ground (to mix metaphors). It states that [[unions]] exist.
## Statement
The __axiom of union__ states the following:
+-- {: .un_defn}
###### Axiom (union)
If $\mathcal{X}$ is a (material) set, then there exists a set $U$ such that $a \in U$ whenever $a \in B \in \mathcal{X}$.
=--
Using the axiom of separation ([[bounded separation]] is enough), one can prove the existence of a particular set $U$ such that the members of the members of $\mathcal{X}$ are the *only* members of $U$. Using the [[axiom of extensionality]], we can then prove that this set $U$ is unique; it is usually denoted $\bigcup\mathcal{X}$ and called the __[[union]]__ of (the elements of) $\mathcal{X}$.
A slightly different notation may be used when $\mathcal{X}$ is (Kuratowski)-[[finite set|finite]]; for example, $\bigcup\{A,B,C\}$ may be denoted $A \cup B \cup C$. If $(B_k \;|\; k \in I)$ is a [[family of sets]], then we may write $\bigcup_{k \in I} B_k$ (and the usual variations for a [[sequence]] of sets) for $\bigcup \{B_k \;|\; k \in I\}$; however, we require the [[axiom of replacement]] to prove that the latter set (the [[range]] of the family) exists in general.
## Related notions
If $\mathcal{X}$ is given as a collection of [[subsets]] of some ambient [[set]] $S$, then the axiom of union is not necessary; $S$ itself already satisfies the conclusion of the hypothesis (and then bounded separation gives us the union that we want). This is the only case when unions are taken in [[structural set theory]]. However, structural set theory makes use of _[[disjoint unions]]_, and [[predicative mathematics]] requires an axiom giving their existence. (In impredicative mathematics, we can construct disjoint unions from [[power sets]] and [[cartesian products]].)
One could also include [[union structure]] in the set theory.
category: foundational axiom
|
axiom schema | https://ncatlab.org/nlab/source/axiom+schema |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A family of [[axioms]] depending on parameters.
Axiom schemata are formalized in a [[deductive system]] by the use of [[inference rules]] which have [[hypotheses]]. For example, the [[axiom schema of replacement]] in [[set theory]] is given by the following [[inference rule]]:
$$\frac{\Gamma, x, y \vdash \phi(x, y) \; \mathrm{prop} \quad \Gamma \vdash a \quad \Gamma \vdash \forall x.(x \in a) \implies \exists! y.\phi(x, y) \; \mathrm{true}}{\Gamma \vdash \{y \vert \exists x.(x \in a) \implies \phi(x, y)\}}$$
Compare with the bare notion of [[axiom]], which are inference rules without any hypotheses.
## Examples
* [[axiom schema of separation]]
* [[axiom schema of replacement]]
* The [[set-theoretic axiom of choice]] in bare [[Martin-Löf type theory]] or [[cubical type theory]] without any [[type universes]] is an axiom schema, since without universes one cannot quantify over types as one could in in a [[type theory]] with [[universes]] or for sets in [[set theory]], and so the only way to express the axiom of choice is as an inference rule with a hypothesis; i.e. an axiom schema.
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A, y:B(x) \vdash P(x, y) \; \mathrm{type}}{\Gamma \vdash \mathrm{ac}_{A, B, P}:(\mathrm{isSet}(A) \times \Pi x:A.(\mathrm{isSet}(B(x)) \times \Pi y:B(x).\mathrm{isProp}(P(x, y)))) \to ((\exists y:B(x).P(x, y)) \to (\exists g:(\Pi x:A.B(x)).\Pi x:A.P(x, g(x))))}$$
## Related concepts
* [[inference rule]]
## References
See also:
* Wikipedia, *[Axiom schema](https://en.wikipedia.org/wiki/Axiom_schema)*
[[!redirects axiom schemas]]
[[!redirects axiom schemata]]
[[!redirects axiom scheme]]
[[!redirects axiom schemes]]
|
Axiomatic field theories and their motivation from topology | https://ncatlab.org/nlab/source/Axiomatic+field+theories+and+their+motivation+from+topology |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--
+-- {: .standout}
This is a sub-entry of [[geometric models for elliptic cohomology]] and [[A Survey of Elliptic Cohomology]]
See there for background and context.
This entry here indicates, generally, how [[FQFT]]s may be related to [[cohomology theory|cohomology theories]].
=--
> **raw material**: this are notes taken more or less verbatim in a seminar -- needs polishing
next:
* [[(1,1)-dimensional Euclidean field theories and K-theory]]
* [[(2,1)-dimensional Euclidean field theories and tmf]]
***
+-- {: .standout}
All of the following assumes that the reader is well familiar with the basic ideas indicated at [[FQFT]].
=--
In the following $d-RB$ or $R Bord_d$ and the like denotes a category of $d$-dimensional [[Riemannian cobordism]]s that are equipped with _Riemannian structure_, i.e. with [[Riemannian metric]].
Similarly $d-E B$ or $E Bord_d$ or the like denotes a category of cobordisms with _Euclidean structure_, by which is meant a _flat_ Riemannian metric.
The definition of this is discussed in detail in
* [[bordism categories following Stolz-Teichner]]
**definition**
$d$-dimensional Riemannian field theories are symmetric monoidal functors $d-RB \to TV$ from $d$-dimensional Riemannian bordisms to [[topological vector space]]s.
A field theory is very similar to a [[representation]] of a group. Only where a representation of a [[group]] $G$ is a functor from the [[delooping]] $\mathbf{B}G = {*}//G$ of $G$ to [[Vect]], an [[FQFT]] is a representation of a more complicated domain category.
**how does [[topology]] enter?**
for $X$ some [[topological space]] there is also a [[symmetric monoidal category]]
$$
d-RB(X)
$$
of Riemannian bordisms equipped with a continuous map to $X$.
Notice that $d RB(X)$ does depend covariantly on $X$. This means that $Fun^\otimes(d RB(X), TV)$ is contravariant in $X$.
When special structure is around, however, we also have a push-forward of such functors along morphisms.
**Example: push-forward to the point**: for $X$ as above and $X \to {*}$ the unique map to the [[point]] heuristically we want a map
$$
d RFT(X) \stackrel{p_*}{\to} d RFT(pt)
$$
notice that this push-forward is not an [[adjoint functor]]. Instead, it is a map that comes from integration over fibers. In particular it will change the degree of [[cohomology theory|cohomology theories]].
heuristically the pushforward
$$
d RFT(X) \stackrel{p_*}{\to} d RFT(pt)
$$
acts on field theories $E_X$ over $X$
$$
E_X \mapsto E
$$
by the assignment
$$
E(Y^{d-1}) \mapsto
\Gamma\left(
\array{
E_X(Y)
\\
\downarrow
\\
Maps(Y,X)
}
regarded as a vector bundle
\right)
$$
for instance when $E_X(Y) = \mathbb{C}$ then $E(Y) = \Gamma(Maps(Y,X))$.
For instance take $\Sigma$ to be the pair of pants with input $Y_0$ and output $Y_1$ and let $F : \Sigma \to X$ be a map.
Then $E(\Sigma)$ will take some section $\Psi \in E(Y_0)$ to
$$
E(\Sigma)(\Psi) : (f_1 : Y_1 \to X)
\mapsto
\int_{\{F : \Sigma \to X\} | F/Y_1 = f_1} E_X(F)(\Psi(f_0))
\frac{1}{Z}\exp(-S(F) dvol)
\,.
$$
Here the expression $\frac{1}{Z}\exp(-S(F) dvol)$ denotes a would-be measure which is still to be defined.
Here $f_0 = F/Y_0$ is the restriction of $F$ to $Y_0$.
Contravariant functors with push-forward also arise as part of a [[cohomology theory]]
$$
h^n : Diff^{op} \to Ab
$$
from the category [[Diff]] of smooth [[manifold]]s to the category [[Ab]] of abelian groups that satisfies the axioms of [[generalized (Eilenberg-Steenrod) cohomology]] theory.
These Eilenberg-Steenrod axioms are
1. **homotopy axiom** for $h^n$
1. **Mayer-Vietoris axiom** for $h^n$
1. **suspension isomorphism** $h^n(X) \simeq h^{n+1}_{cvs}(X \times \mathbb{R})$
here in the last axiom for [[topological space]]s we'd simply have the [[suspension]] $h^{n+1}(\Sigma X)$. Here in order to stay within [[manifold]]s we instead do as indicated, where ${}_{cvs}$ means "compact vertical support".
Concerning the **homotopy axiom**: given any [[functor]] that sends manifolds to a category of [[FQFT]]s over $X$
$$
d-FTs := h : Diff^{op} \to C
$$
in $dRFT$s we can make it a homotopy functor by defining
$\omega_0, \omega_1 \in h(X)$ to be [[concordance|concordant]] if $\exists \omega \in h(X \times \mathbb{R})$ such that $\omega/(X\times \{i\}) \simeq \omega_i$ in $h(X)$, $i = 0,1$
then under this relation
$$
X \mapsto d-RFT(X)/\simeq
$$
is a homotopy invariant functor
**Homework**: for $h(X) = \Omega^n_{closed}(X)$
we have that $\omega_0$ concordant to $\omega_1$ exactly when $\omega_0 = \omega_1 + d \alpha$ for some $\alpha \in \Omega^{n-1}(X)$
Concerning the **Mayer-Vietoris axiom**: if $X = U \cup V$ then the functor should respect this gluing.
suppose $E \in 2-RFT(X)$ is a 2d Riemmanian field theory. let $\gamma : S^1 \to X$ be a loop in $X$. then $E(\gamma)$ is a [[vector space]].
If $\gamma$ sits neither entirely in $U$ or in $V$, then there is no way that the vector space $E(\gamma)$ can be reconstructed by knowing just the restriction of $R$ to $U$ and $V$.
So this is a **problem** for the definition of field theories so far.
The **proposed solution** (from [[What is an elliptic object?]]) is to use _extended_ [[FQFT]]s instead. This introduces **locality** into [[FQFT]]s, at the expense of working with [[n-category|n-categories]].
This will however not be studied here for the moment.
Concerning the **suspension isomorphism**: for that first we need for $n \in \mathbb{Z}$ the notion of a field theory over $X$ of **degree $n$**, i.e.
$$
X \mapsto d-RFTs^n(X)
$$
such that for $n=0$ this is an ordinary Riemannian field theory, in $d-RFT(X)$.
This requires to replace [[manifold]]s by [[supermanifold]]s.
**Example** let $d = 0$ and consider 0-dimensional TFTs over $X$.
so consider
$$
Fun^\otimes(0 Bord(X), TV)
$$
in $0 Bord(X)$ there is only a single [[object]]: the [[empty set]] $\emptyset$ which has a unique map $\emptyset \to X$
the collecton of morphisms is $\{finite set \to X\}$ in there is $Hom_{Diff}(pt,X)$.
here composition of morphisms is the same as tensor product of objects: both comes from the disjoint union of these finite set domains.
$$
E : 0 Bord(X) \to TV_{\mathbb{R}}
$$
$$
\emptyset \mapsto \mathbb{R}
$$
$$
x \in X \mapsto E(X) \in \mathb{R}
$$
so a priori we have
$$
Fun^\otimes(0 Bord(X), TV) \simeq Maps(X, \mathbb{R})
\,,
$$
where on the right we have _all_ maps.
This is not quite what is intended. We want to see _smooth_ maps on the right. To get that, we need to talk about _smooth functors_ on the left. This is the topic of later discussion, which will yield
$$
SmoothFun^\otimes(0 Bord(X), TV) \simeq C^\infty(X, \mathbb{R})
\,.
$$
But one other ting goes wrong the: the corresponding homotopy functor is $C^\infty(X)/\simeq = \{0\}$. So turning this into an Eilenberg-Steenrod theory yields the trivial theory.
The way out to that will be to go to [[supermanifold]]s.
**Punchline** of this session here:
$$
(0|1)-TFTs(X)
\simeq
SmoothFun^\otimes((0|1)Bord(X), TV)
$$
and then it is a $lemma^G$ that
$$
C^\infty(SuperDiff(\mathbb{R}^{0|1}, X))^G
\simeq
\Omega^\bullet(X)^G
= (\oplus_{n = 0}^{\infty} \Omega^n(X))^G
$$
where $G$ is the [[supergroup]] $G = Diff(\mathbb{R}^{0|1}) \wim42 \mathbb{R}^{0|1} \rtimes \mathbb{R}^\times$.
The degree decomposition on forms then turns out to be the eigenvalue decompositon of the $\mathbb{R}^\times$-part, while the deRham differential is the $\mathbb{R}^{0|1}$-action (cite Kontsevich, Severa here ...)
so then from that one finds that the $G$-invariant bit is the _closed_ 0-forms
$$
\Omega^\bullet(X)^G
\simeq
\Omega^0_{closed}(X)
$$
So we get $n$-**Lemma** :
$$
(0|1)-TFT^n(X) \simeq \Omega^n_{cl}(X)
$$
and
$$
(0|1)-TFT^n(X)_{concord} \simeq H^n_{dR}(X)
$$
**push-forward**
Now with the super-directions included, there is a notion of push-forward of the TFTs that does shift the degree, and we get the following
**Theorem (Stolz-Teichner-Kreck-Hohnhold)**
when $X$ is oriented
$$
(0|1)TFT^n(X) \stackrel{push}{\to} (0|1)TFT^0
\simeq \mathbb{R}
$$
$$
\Omega^n_{cl}(X) \stackrel{\int_{X_n}}{\to}
\Omega^0_{cl}(pt)
\simeq
\mathbb{R}
$$
and similarly after dividing out concordance, the push-forward becomes the push-forward in [[deRham cohomology]].
## References
* [[Stefan Stolz]] (notes by Arlo Caine), _Supersymmetric Euclidean field theories and generalized cohomology_ Lecture notes (2009) ([pdf](http://www.nd.edu/~jcaine1/pdf/Lectures_complete.pdf))
{#Stolz}
|
axioms of plane geometry | https://ncatlab.org/nlab/source/axioms+of+plane+geometry | # Axioms of plane geometry
Under construction.
For Croatian version see [link](https://ncatlab.org/zoranskoda/show/aksiomi+planimetrije)
## Literature
* related pages include [[isometry]], [[congruence in geometry]]
* wikipedia [Euclidean geometry](https://en.wikipedia.org/wiki/Euclidean_geometry)
* Jacquelline Lelong-Ferrand, Les fondements de la géométrie, Chap. 5
* [[David Hilbert]], _Grundlagen der Geometrie_. Leipzig, Teubner (1899); _Foundations of geometry_, 1905, 1971
* Oscar Veblen, A system of axioms for geometry, Trans. Amer. Math. Soc. __5__ (1904) [jstor](https://archive.org/details/jstor-1986462)
* A. Kolmogorov
* George D. Birkhoff, A set of postulates for plane geometry, Annals of Mathematics 33: pp. 329-345 (1932)
* [[Hermann Weyl]], _Raum, Zeit, Materie_, 1918; engl. transl. Space Time Matter 1922, reprinted 1952
* Gustave Choquet, _Geometry in a modern setting_
* Friedrich Bachmann, _Aufbau der Geometrie aus dem Spiegelungsbegriff_, 1959; Russ. ed. Построение геометрии на основе понятия симметри, Nauka 1969 (transl. by Р.И.Пименов, redaction И.М.Яглом)
[[!redirects aksiomi planimetrije]] |
axion | https://ncatlab.org/nlab/source/axion |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _axion_ is a hypothetical type of [[field (physics)|field]]/[[fundamental particle]] originally hypothesized ([Weinberg 77](#Weinberg77), [Wilczek 78](#Wilczek78)) as a solution to the [[strong CP problem]] in the [[standard model of particle physics]].
After the initial model for the axion was quickly ruled out by [[experiment]] (see [Wilczek 78](#Wilczek78) for the early history) a variant model was found ([Dine-Fischler-Srednicki 81](#DineFischlerSrednicki81)), called the "invisible axion", which does not violate experimental bounds.
The "invisible axion" turns out to also be a natural candidate for [[dark matter]] [Preskill-Wise-Wilczek 83](#PreskillWiseWilczek83), hence potentially also solves one of the problems with the [[standard model of cosmology]]. More recently it is argued in [Hui-Ostriker-Tremaine-Witten 16](#HOTW16) that indeed axionic dark matter ([[fuzzy dark matter]]) potentially solves the remaining problems of standard WIMP dark matter models: WIMP dark matter models work exceedingly well on [[cosmology|cosmological]] scales but has serious experimental problems on the scale of [[galaxies]]. Due to the extreme lightness of axion particles, their [[de Broglie wavelength]] may be of the scale of galaxies and hence their quantum properties may become relevant at this scale to deviate in the right way from the WIMP models.
The Pecchei-Quinn shift symmetry of the axion and the peculiar nature of the axion [[interaction term]] which are needed to make the axion model work this way have been argued to naturally arise in [[string theory]], if the axion is identified with the [[KK-reduction]] of the [[higher gauge fields]] in [[string theory]]. This we discuss [below](#AsArisingFromStringTheory).
## As a solution to the strong CP problem
{#AsASolutionToTheStrongCPProblem}
The problem here is that the [[action functional]] for [[QCD]] ([[Yang-Mills theory]]) a priori contains an arbitrary [[theta angle]] $\theta$
$$
S
\;\colon\;
\nabla \mapsto
\frac{1}{g^2 }\int_X tr(F_\nabla \wedge \star F_\nabla) \;+\;
\theta \int_X tr(F_\nabla \wedge F_\nabla)
$$
but that -- since the term $tr(F_\nabla \wedge F_\nabla)$ causes parity violation, which is strongly constrained by [[experiment]] -- there must be some reason why $\theta$ vanishes or else is extremely small.
(Here the term $tr(F_\nabla \wedge F_\nabla)$ is the [[invariant polynomial]] for the [[second Chern class]], measuring the [[instanton number]] of the [[gauge field]] $\nabla$.)
The solution to this problem via axions is to assume that $\theta$ is not really a fundamental parameter, but instead is the [[vacuum expectation value]] of a dynamical [[field (physics)|field]] $a$ (the axion).
The idea is that a standard [[kinetic action]]
$$
S_{kin} (a) \propto \int_X a \wedge \ast a
$$
together with the axion [[interaction]] term
$$
S_{int} (a,\nabla) \propto \int_X a \, tr(F_\nabla \wedge F_\nabla)
$$
makes $a$ have vanishing [[expectation value]] $\langle a\rangle$. This would give a dynamical explanation why under the identification of the [[theta angle]] with this expectation value
$$
\theta \coloneqq \langle a\rangle
$$
the theta angle vanishes.
### The Vafa-Witten mechanism
{#VafaWittenMechanism}
That this is indeed the case is due to [Vafa-Witten 84, around (2)](#VafaWitten84). These authors argue via the [[Wick rotation|Wick rotated]] [[path integral]] as follows:
Under [[Wick rotation]], parity-violating terms in the [[Lagrangian density]] pick up an imaginary factor $i$. Therefore the [[path integral]] expression for the Wick rotated [[vacuum energy]] is
$$
\begin{aligned}
E_{vac}(a)
&=
- log \left( \underset{\nabla,a}{\int} \exp(- S(\nabla,a)) \, D \nabla \, D a \right)
\\
&=
- log
\left(
\underset{\nabla}{\int} \exp\left(
\frac{1}{g^2 }\int_X tr(F_\nabla \wedge \star F_\nabla)
\right)
\, D \nabla
\;
\underset{a}{\int}
\exp\left(
i a \int_X tr(F_\nabla \wedge F_\nabla)
\right)
\, D a
\right)
\end{aligned}
\,.
$$
Now due to the factor of $i$ in front of $\theta$ in this expression, the real part of the argument of the logarithm necessarily becomes _smaller_ with $a$. Therefore the negative logarithm becomes _larger_ with $a$. Accordingly $E_{vac}(a)$ must have a minimum at $\theta = 0$ (according to [Vafa-Witten 84, p.2](#VafaWitten84)).
## In string theory
{#AsArisingFromStringTheory}
In [[string theory]] axion fields
1. are naturally present as the [[KK-reduction]] of the [[higher gauge fields]] (e.g. [Svrcek-Witten 06](#SvrcekWitten06));
reviewed below in:
_[As KK-Reduction of higher gauge fields](#AxionsAsKKReductionOfHigherGaugeFields)_
1. are generically of positive but tiny mass ([ADDKM 09](#ADDKM09), [Acharya-Bobkov-Kumar 10](#AcharyaBobkovKumar10)) as they need to be to be a solution to the [[strong CP problem]] and be candidates for [[fuzzy dark matter|BEC/superfluid/fuzzy dark matter]];
reviewed below in:
_[Stringy axion phenomenology](#StringyAxionPhenomenology)_
This makes axion fields in string theory form a curious confluence point relating
* abstract concepts related to [[higher gauge theory]]
with
* fundamental question in [[standard model of particle physics|particle physics]]/[[standard model of cosmology|cosmology]] [[phenomenology]]:
$$
\array{
\mathbf{\text{higher gauge theory}}
&&
&&
\mathbf{\text{particle physics/cosmology phenomenology}}
\\
\\
\left.
\array{
\text{higher gauge fields}
\\
\text{higher characteristic classes}
\\
\updownarrow
\\
\text{non-perturbative QFT/string effects}
\\
\text{in HET: Green-Schwarz anomaly cancellation}
\\
\text{in IIA/B: higher WZW term for Green-Scharz D-branes}
}
\right\}
&\longrightarrow&
\array{
\text{axion fields}
\\
\text{in the string spectrum}
}
&\longrightarrow&
\left\{
\array{
\text{solve strong CP-problem as with P-Q robustly}
\\
\text{solve dark matter problem by FDM}
}
\right.
}
$$
### Axions as KK-reduction of higher gauge fields
{#AxionsAsKKReductionOfHigherGaugeFields}
In [[string theory]] axion fields _naturally_ arise as the [[KK compactification]] of the [[higher gauge fields]] ([[B-field]], [[C-field]], [[RR-fields]]). Here we review how this comes about.
Notice that this means that axions in string theory are as in the output of the original Peccei–Quinn proposal, but do not actually make use of the Peccei–Quinn mechanism.
As amplified in particular in [Svrcek-Witten 06](#SvrcekWitten06):
> An obvious question about the axion hypothesis is how natural it really is. Why introduce a global PQ "symmetry" if it is not actually a symmetry? What is the sense in constraining a theory so that the classical Lagrangian possesses a certain symmetry if the symmetry is actually anomalous? It could be argued that the best evidence that PQ "symmetries" are natural comes from string theory, which produces them without any contrivance. ... the string compactifications always generate PQ symmetries, often spontaneously broken at the string scale and producing axions, but sometimes unbroken. ([Svrcek-Witten 06, pages 3-4](#SvrcekWitten06))
Similarly from [ADDKM 09](#ADDKM09):
> However, at the [[effective field theory]] level it is hard to judge how natural it is to have such a "fake" global PQ symmetry which is explicitly broken exclusively by QCD. Note, that in order not to spoil the solution to the strong CP problem all other sources of explicit PQ symmetry breaking should be at least 10 orders of magnitude down with respect to the potential generated by the QCD anomaly, and one may be especially cautious about the viability of such a proposal, given the common lore that global symmetries always get broken by quantum gravitational effects [12, 13, 14]. This makes it natural to inquire whether axions arise naturally in the most developed quantum theory of gravity—string theory. ([ADDKM 09, p. 3](#ADDKM09))
The mechanism discussed in [Svrcek-Witten 06](#SvrcekWitten06), in all its incarnations in the various perspectives on string theory ([[heterotic string theory|heterotic strings]], [[type II string theory|type II strings]], [[11-dimensional supergravity]] with [[M-theory]] effects included, etc.) share the following properties:
1. the axion field itself is the [[double dimensional reduction]] of one of the [[higher gauge fields]] in string theory, the (twisted) [[Kalb-Ramond field]] for heterotic string, the [[RR-field]] for the type II string, or the [[supergravity C-field]] in 11-dimensional supergravity with [[M-brane]] effects;
1. accordingly the Peccei-Quinn periodic shift symmetry of the axion results arises as the $U(1)$-[[gauge symmetry]] that is the dimensional reduction of the $B^2 U(1)$ (heterotic [[B-field]]) or $B^3 U(1)$ (11d [[C-field]]) [[higher gauge symmetry]], where the $U(1)$-periodicity is ultimately due to the higher [[Dirac charge quantization]] for these fields;
1. the axion [[interaction term]] of the form $\propto a \langle F \wedge F\rangle $ arises as the [[double dimensional reduction]] of the self-interaction terms of these higher gauge fields:
* in type II string theory it is a component of the [[higher WZW term]] on coincident [[D6-branes]] with a component $\propto A_{3}^{RR} \wedge \langle F \wedge F\rangle$,
* for 11d C-fields it is the term $\propto C_{3} \wedge \langle F \wedge F\rangle$ induced by [[anomaly cancellation]] for M-theory at conical singularities (as discussed at _[[M-theory on G2-manifolds]]_)
* for heterotic string theory it arises from integrating out the [[Lagrange multiplier]] that enforces the [[Green-Schwarz anomaly cancellation]] in 4d.
We now discuss this in more detail:
* _[In type IIA string theory](#IntypeIIA)_
* _[In heterotic string theory](#InHeteroticStringTheory)_
#### In type IIA string theory
{#IntypeIIA}
Consider [[string phenomenology]] in [[type IIA string theory]] in the guise of [[intersecting D-brane models]] with the [[standard model of particle physics]] supported in intersecting [[D6-branes]] whose [[worldvolume]] fills all of 4d [[spacetime]].
The [[higher WZW term]] in the [[Green-Schwarz action functional]] for the [[D6-brane]] is of the form
$$
\propto \int_{X_7} A^RR \wedge \langle \exp(F_\nabla) \rangle
$$
where $A^RR$ denotes the collective inhomogenous [[background field|background]] [[RR-field]] and $F_\nabla$ is the [[curvature]] 2-form of the [[Chan-Paton gauge field]] on the D-brane. The $\langle -\rangle$ denotes the [[Killing form]] [[invariant polynomial]], hence the [[trace]] for the [[special unitary Lie algebra]] regarded as a [[matrix Lie algebra]]. $X_7$ denotes the [[worldvolume]] of the [[D6-brane]]
Hence one of the three non-vanishing summands in this expression is
$$
\propto \int_{X_7} A_3^{RR} \wedge \langle F_\nabla \wedge F_\nabla\rangle
\,.
$$
Now assuming a Kaluza-Klein ansatz $X_7 = X_4 \times Y_3$ with
$$
a \coloneqq \int_{Y_3} A_3^{RR}
$$
the effective axion field in 4d, then this term becomes the
axion [[interaction]] term
$$
\propto \int_{X_4} a \langle F_{\nabla} \wedge F_{\nabla}\rangle
$$
([Svrcek-Witten 06, around (6.9)](#SvrcekWitten06))
#### In heterotic string theory
{#InHeteroticStringTheory}
In [[heterotic string theory]] [[KK-compactification|KK-compactified]] to 4d, the 4d [[B-field]], dualized, serves as the axion field, called the "model independent axion" ([Svrcek-Witten 06, starting p. 15](#SvrcekWitten06)).
This works as follows: By the [[Green-Schwarz anomaly cancellation]] mechanism, then [[B-field]] in [[heterotic string theory]] is a twisted 2-form field, whose [[field strength]] $H$ locally has in addition to the exact differential $d B$ also a fundamental 3-form contribution, so that
$$
H = d B + C
$$
(locally). Moreover, the differential $d H$ is constrained to be the Pontryagin 4-form of the gauge potential $\nabla$ (minus that of the [[Riemann curvature]], but let's suppress this notationally for the present purpose):
$$
d H = tr \left(F_\nabla \wedge F_\nabla\right)
\,.
$$
Now suppose [[KK-compactification]] to 4d has been taken care of, then this constraint may be implemented in the [[equations of motion]] by adding it to the [[action functional]], multiplied with a [[Lagrange multiplier]] :
$$
S
=
\underset{
\text{kinetic action} \atop \text{for B-field}
}{
\underbrace{\int_X H \wedge \star H}
}
+
\underset{
\text{Green-Schwarz constraint}
}{
\underbrace{
\int_X a \left( d H - tr(F_\nabla \wedge F_\nabla) \right)
}
}
\,.
$$
Now by the usual argument, one says that instead of varying by $a$ and thus implementing the [[Green-Schwarz anomaly cancellation]] constraint, it is equivalent to first vary with respect to the other fields, and then insert the resulting equations in terms of $a$ into the action functional.
Now since we are dealing with a twisted [[B-field]], with free 3-form component $C$, we actually vary with respect to $C$. This yields the [[Euler-Lagrange equation]] [[equation of motion|of motion]]
$$
d a = \star H
\,,
$$
hence the usual relation or [[electro-magnetic duality]], expressing what used to be the [[Lagrange multiplier]] now as the magentic dual [[field (physics)|field]] to the twisted [[B-field]].
Plugging this algebraic [[equation of motion]] back into the above [[action functional]] for $H$ gives
$$
\tilde S
=
\underset{\text{kinetic action} \atop \text{for axion field}}{\underbrace{\int_X d a \wedge \star d a}}
+
\underset{\text{axionic} \atop \text{interaction}}{\underbrace{\int_X a \, tr(F_\nabla \wedge F_\nabla)}}
\,.
$$
This now is an action functional for an axion field $a$ of just the form required [above](#AsASolutionToTheStrongCPProblem) for the solution of the [[strong CP-problem]].
### Stringy axion phenomenology
{#StringyAxionPhenomenology}
From [Acharya-Kane-Kumar 12, p. 3](#AcharyaKaneKumar12):
> Now consider the axions. Due to the shift symmetries mentioned above, there are no [[perturbative QFT|perturbative]] contributions to their potential. [[non-perturbative effect|Non-perturbative effects]] though, such as strong gauge dynamics, gauge [[instantons]], gaugino condensation and stringy instantons will generate a potential for the axions. Because any such contribution is exponentially suppressed by couplings and/or extra-dimensional volumes, in our world with perturbative gauge couplings (at high scales), the axion masses will be exponentially small.
> Furthermore, since there are large numbers of axions in general, their masses will essentially be uniformly distributed on a logarithmic scale ([5](#ADDKM09)). See ([6](#AcharyaBobkovKumar10)) for a detailed calculation of axion masses in string/M effective theories.
> Like the moduli, the axions are also very weakly coupled to matter and therefore do not thermalize in general. Moreover, since their masses are tiny - ranging from $m_{3/2}$ to even below the Hubble scale today - many of them, including the QCD axion, start coherent oscillations during the time that the moduli are dominating the energy density, but before BBN.
Observe that ultralight axion fields is precisely what is required in HEP [[phenomenology]] for
1. the solution to the [[strong CP-problem]];
1. the solution of the [[dark matter]] problem via [[fuzzy dark matter|BEC/superfluid/wave/fuzzy dark matter]] (e.g. [Hui-Ostriker-Tremaine-Witten 16](#HOTW16)).
## Related concepts
* [[axion inflation]]
* [[fuzzy dark matter]]
* [[strong CP problem]]
* [[G2-MSSM]]
## References
### General
The axion as such was originally proposed in
* {#Weinberg77} [[Steven Weinberg]], _New light boson_, Phys. Rev. Lett. 40:223-6 (1977)
* {#Wilczek78} [[Frank Wilczek]], _Problem of strong P and T invariance in the presence of instantons_, Phys. Rev. Lett. 40:279-82 (1978)
The experimentally viable variant as the "invisible axion" is due to
* {#DineFischlerSrednicki81} [[Michael Dine]], [[Willy Fischler]], [[Mark Srednicki]], _A simple solution to the strong CP problem with a harmless axion_, Phys. Lett. B 104:199-202 (1981)
The observation that the "invisible axion" is a candidate for [[dark matter]] is due to three groups:
* {#PreskillWiseWilczek83} [[John Preskill]], M. Wise, [[Frank Wilczek]], _Cosmology of the invisible axion_, Phys. Lett. B 120:127-32 (1983)
* {#AbbottSikivie83} L.F. Abbott, P. Sikivie, _A Cosmological Bound on the Invisible Axion_, Phys.Lett. B120:133-36 (1983)
* {#DineFischler83} [[Michael Dine]], [[Willy Fischler]], _The Not So Harmless Axion_, Phys. Lett. B120 :137-141 (1983)
A historical recollection of the development until here is in
* [[Frank Wilczek]], _Birth of axions_ in _This Week's Citation Classic_ (1991) ([pdf](http://www.garfield.library.upenn.edu/classics1991/A1991FE76900001.pdf))
The argument that the [[topological Yang-Mills theory|topological]] [[interaction]] term $\propto a \langle F \wedge F\rangle$ gives the axion field $a$ a vanishing [[vacuum expectation value]] is due to
* {#VafaWitten84} [[Cumrun Vafa]], [[Edward Witten]], _Parity Conservation in Quantum Chromodynamics_, Phys. Rev. Lett. 53, 535 (1984) ([publisher](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.53.535))
A reformulation of this effect in terms of [[Chern-Simons forms]] is discussed in
* [[Gia Dvali]], _Three-Form Gauging of axion Symmetries and Gravity_ ([arXiv:hep-th/0507215](https://arxiv.org/abs/hep-th/0507215))
Review:
* {#KusterRaffeltBeltran08} Markus Kuster, Georg Raffelt, Berta Beltrán (eds.), _Axions: Theory, cosmology, and Experimental Searches_, Lect.
Notes Phys. 741 (Springer, Berlin Heidelberg 2008) (<a href="https://doi.org/10.1007/978-3-540-73518-2_2">doi:10.1007/978-3-540-73518-2_2</a>)
* Jihn E. Kim, Gianpaolo Carosi, _Axions and the Strong CP Problem_, Rev.Mod.Phys.82:557-602,2010 ([arXiv:0807.3125](https://arxiv.org/abs/0807.3125))
* Masahiro Kawasaki, Kazunori Nakayama, _Axions : Theory and Cosmological Role_, Annual Review of Nuclear and Particle Science Vol.63:1-552 ([arXiv:1301.1123](https://arxiv.org/abs/1301.1123))
* Luca Di Luzio, Maurizio Giannotti, Enrico Nardi, Luca Visinelli, _The landscape of QCD axion models_ ([arXiv:2003.01100](https://arxiv.org/abs/2003.01100))
* Igor G. Irastorza, *An introduction to axions and their detection* ([arXiv:2109.07376](https://arxiv.org/abs/2109.07376))
See also:
* Wikipedia, _[Axion](http://en.wikipedia.org/wiki/Axion)_
### In string theory
{#ReferencesInStringTheory}
Discussion of the various ways that axions naturally appear in [[string theory]] is in
* {#SvrcekWitten06} Peter Svrcek, [[Edward Witten]], _Axions In String Theory_, JHEP 0606:051,2006 ([arXiv:hep-th/0605206](http://arxiv.org/abs/hep-th/0605206))
Specifically for the [[F-theory]] sector of string theory:
* {#Grimm14} Thomas Grimm, _Axion Inflation in F-theory_ ([arXiv:1404.4268](http://arxiv.org/abs/1404.4268))
Specifically open string axions
* [[Gabriele Honecker]], section 4 of _From Type II string theory towards BSM/dark sector physics_, International Journal of Modern Physics A Vol. 31 (2016) 1630050 ([arXiv:1610.00007](https://arxiv.org/abs/1610.00007))
A textbook account of axion [[string phenomenology]] is in
* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], section 16.2 of _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge University Press 2012
Discussion of stringy axion [[cosmology]] (such as [[fuzzy dark matter]]) is in
* {#ADDKM09} Asimina Arvanitaki, Savas Dimopoulos, Sergei Dubovsky, Nemanja Kaloper, John March-Russell, _String Axiverse_, Phys.Rev.D81:123530,2010 ([arXiv:0905.4720](https://arxiv.org/abs/0905.4720))
* {#AcharyaBobkovKumar10} [[Bobby Acharya]], Konstantin Bobkov, [[Piyush Kumar]], _An M Theory Solution to the Strong CP Problem and Constraints on the Axiverse_, JHEP 1011:105, 2010 ([arXiv:1004.5138](https://arxiv.org/abs/1004.5138))
* {#AcharyaKaneKumar12} [[Bobby Acharya]], [[Gordon Kane]], [[Piyush Kumar]], _Compactified String Theories -- Generic Predictions for Particle Physics_, Int. J. Mod. Phys. A, Volume 27 (2012) 1230012 ([arXiv:1204.2795](https://arxiv.org/abs/1204.2795))
### In holographic QCD
Realization in the [[Witten-Sakai-SUgimoto model]] for [[holographic QCD]]:
* Francesco Bigazzi, Alessio Caddeo, Aldo L. Cotrone, Paolo Di Vecchia, Andrea Marzolla, _The Holographic QCD Axion_ ([arXiv:1906.12117](https://arxiv.org/abs/1906.12117))
### Experimental signature
Discussion of [[experiment|experimental]] signatures of and constraints on axion physics:
#### In particle physics
{#ReferencesExperimentalSignatureInParticlePhysics}
Discussion of [[experiment|experimental]] signatures of and constraints on axion physics from [[particle physics]]:
Specifically contribution of axions to the [[anomalous magnetic moment]] of the [[electron]] and [[muon]] in [[QED]]:
* Yannis Semertzidis, _Magnetic and Electric Dipole Moments in Storage Rings_, chapter 6 of Markus Kuster, Georg Raffelt, Berta Beltrán (eds.), _Axions: Theory, cosmology, and Experimental Searches_, Lect.
Notes Phys. 741 (Springer, Berlin Heidelberg 2008) ([Kuster-Raffelt-Beltran 08](#KusterRaffeltBeltran08), <a href="https://doi.org/10.1007/978-3-540-73518-2_2">doi:10.1007/978-3-540-73518-2_2</a>)
* Roberta Armillis, Claudio Coriano', Marco Guzzi, Simone Morelli, _Axions and Anomaly-Mediated Interactions: The Green-Schwarz and Wess-Zumino Vertices at Higher Orders and g-2 of the muon_, JHEP 0810:034,2008 ([arXiv:0808.1882](https://arxiv.org/abs/0808.1882))
* W.J. Marciano, A. Masiero, P. Paradisi, M. Passera, _Contributions of axion-like particles to lepton dipole moments_, Phys. Rev. D 94, 115033 (2016) ([arXiv:1607.01022](https://arxiv.org/abs/1607.01022))
* Martin Bauer, Matthias Neubert, Andrea Thamm, _Collider Probes of Axion-Like Particles_, J. High Energ. Phys. (2017) 2017: 44. ([arXiv:1708.00443](https://arxiv.org/abs/1708.00443), <a href="https://doi.org/10.1007/JHEP12(2017)044">doi:10.1007/JHEP12(2017)044</a>)
The basic relevant [[Feynman diagrams]] are worked out here:
* [pdf](http://www-personal.umich.edu/~jbourj/peskin/6-3.pdf)
See also
* [[Frank Wilczek]], _New Ideas in Axion Searches_, talk 2017 ([pdf](http://frankwilczek.com/2017/axion_searches_01.pdf))
For the [ABRACADABRA](http://abracadabra.mit.edu/) (A Broadband/Resonant Approach to Cosmic Axion Detection with an Amplifying B-field Ring Apparatus) approach to axion detection, see
* Jonathan Ouellet, _ABRACADABRA: A Broadband Search for Axion Dark Matter_, talk 2017 ([pdf](https://indico.fnal.gov/event/13702/session/3/contribution/45/material/slides/0.pdf))
For the ADMX experiment:
* {#Avignone18} Frank T. Avignone III, _[Homing in on Axions?](https://physics.aps.org/articles/v11/34)_, APS Physics 11, 34, 2018
#### In astrophysics
Axion signatures in [[gravitational waves]] potentially visible by [[LIGO]]-type [[experiments]] are discussed in
* {#HJSSZ18} Junwu Huang, Matthew C. Johnson, Laura Sagunski, Mairi Sakellariadou, Jun Zhang, _Prospects for axion searches with Advanced LIGO through binary mergers_ ([arXiv:1807.02133](https://arxiv.org/abs/1807.02133))
#### In cosmology
Discussion of [[experiment|experimental]] signatures of and constraints on axion physics in [[cosmology]], where the axion is a ([[fuzzy dark matter|fuzzy]]) [[dark matter]]-candidate:
* Joseph P. Conlon, M.C. David Marsh, _Searching for a 0.1-1 keV Cosmic Axion Background_ ([arXiv:1305.3603](http://arxiv.org/abs/1305.3603))
> Primordial decays of [[string theory]] [[moduli stabilization|moduli]] at $z \sim 10^{12}$ naturally generate a [[dark radiation]] Cosmic Axion Background (CAB) with $0.1 - 1 keV$ energies. This CAB can be detected through axion-[[photon]] conversion in astrophysical [[magnetic fields]] to give quasi-thermal excesses in the extreme ultraviolet and soft X-ray bands. Substantial and observable luminosities may be generated even for axion-photon couplings $\ll 10^{-11} GeV^{-1}$. We propose that axion-photon conversion may explain the observed excess emission of soft X-rays from galaxy clusters, and may also contribute to the diffuse unresolved cosmic X-ray background. We list a number of correlated predictions of the scenario.
Discussion of the axion as a candidate for [[dark matter]] ([[fuzzy dark matter]]) is in
* {#HOTW16} [[Lam Hui]], [[Jeremiah Ostriker]], [[Scott Tremaine]], [[Edward Witten]], _On the hypothesis that cosmological dark matter is composed of ultra-light bosons_, Phys. Rev. D 95, 043541 (2017) ([arXiv:1610.08297](https://arxiv.org/abs/1610.08297))
with its experimental bounds
* Nilanjan Banik, Adam J. Christopherson, Pierre Sikivie, Elisa Maria Todarello, _New astrophysical bounds on ultralight axionlike particles_, Phys. Rev. D 95, 043542 (2017) ([arXiv:1701.04573](https://arxiv.org/abs/1701.04573))
and as a candidate for [[dark energy]]:
* [[Stephon Alexander]], [[Robert Brandenberger]], [[Juerg Froehlich]], _Tracking Dark Energy from Axion-Gauge Field Couplings_ ([arXiv:1601.00057](https://arxiv.org/abs/1601.00057))
[[!redirects axions]]
[[!redirects invisible axion]]
[[!redirects invisible axions]]
|
axion inflation | https://ncatlab.org/nlab/source/axion+inflation |
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#### Physics
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[[!include physicscontents]]
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* table of contents
{:toc}
## Idea
[[cosmic inflation]] with [[axion]] [[field (physics)|fields]] playing the role of the [[inflaton field]].
## References
General:
* Enrico Pajer, Marco Peloso, _A review of Axion Inflation in the era of Planck_ ([arXiv:1305.3557](http://arxiv.org/abs/1305.3557))
Models in [[string theory]] ([[F-theory]]):
* {#Grimm14} [[Thomas Grimm]], _Axion Inflation in F-theory_ ([arXiv:1404.4268](http://arxiv.org/abs/1404.4268)) |
axionic landscape | https://ncatlab.org/nlab/source/axionic+landscape |
#Contents#
* table of contents
{:toc}
## Idea
The discussion of the [[landscape of string theory vacua]] is often done with rather broad stroke approximations to the structures that should really be classified. For instance the landscape of [[flux compactifications]] is often done with all [[fluxes]] modeled simply as [[differential forms]]. It is known however that more properly these flux fields are really [[cocycles]] in [[differential cohomology]] (For instance the [[B-field]] is really in [[ordinary differential cohomology]] while the [[RR-field]] is really in [[differential K-theory]]). If one does take this refinement into account, then after [[KK-compactification]] what previously looked like plane [[scalar fields]] now become _[[axion]]_ fields, which inherit periodicity from their origin in [[higher gauge fields]]. Taking this [[axion|axionic]] nature of some of the scalar fields in string theory compactifications into account is hence known as considering the _axionic landscape_ of string theory vacua.
## References
* [[Michael Dine]], Guido Festuccia, John Kehayias, Weitao Wu, _Axions in the Landscape and String Theory_, JHEP 1101:012,2011 ([arXiv:1010.4803](https://arxiv.org/abs/1010.4803))
* Gaoyuan Wang, Thorsten Battefeld, _Vacuum Selection on Axionic Landscapes_, JCAP 04 (2016) 025 ([arXiv:1512.04224](https://arxiv.org/abs/1512.04224))
[[!redirects axionic landscapes]]
[[!redirects axionic landscape of string theory vacua]]
[[!redirects axionic landscapes of string theory vacua]] |
Azumaya algebra | https://ncatlab.org/nlab/source/Azumaya+algebra |
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#### Algebra
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[[!include higher algebra - contents]]
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#Contents#
* table of contents
{:toc}
## Idea
An *Azumaya algebra* over a [[commutative unital ring]] $R$ is an [[associative algebra|algebra]] over $R$ that has an inverse up to [[Morita equivalence]]. That is, $A$ is an Azumaya algebra if there is an $R$-algebra $B$ such that $B \otimes_R A$ is Morita equivalent to $R$, which is the unit for the tensor product of $R$-algebras. Thus, Morita equivalence classes of Azumaya algebras over $R$ form a group, which is called the [[Brauer group]] of $R$.
## Definition
### Traditional
In what follows, $R$ is a [[commutative unital ring]] and algebras over $R$ are assumed associative and unital but not necessarily commmutative. An __Azumaya algebra__ over $R$ is an algebra $A$ over $R$ obeying any of the following equivalent conditions:
* There exists an $R$-algebra $B$ such that the [[tensor product]] of $R$-algebras $B \otimes_R A$ is [[Morita equivalence|Morita equivalent]] to $R$.
* The $R$-algebra $A^{\mathrm{op}} \otimes_R A$ is [[Morita equivalence|Morita equivalent]] to $R$, where $A^{\mathrm{op}}$ is the opposite algebra of $A$.
* The [[center]] of $A$ is $R$, and $A$ is a [[separable algebra]].
* As a left $R$-[[module]], $A$ finitely generated, faithful and [[projective object|projective]], and the canonical morphism $A\otimes_R A^{op}\to End_R(A)$ is an [[isomorphism]].
When $R$ is a field, an Azumaya algebra is the same as a [[central simple algebra]] over $R$.
For any commutative ring $R$ there is a [[monoidal bicategory]] with
* algebras over $R$ as objects,
* bimodules as morphisms,
* bimodule homomorphisms as 2-morphisms.
Given any monoidal bicategory we can take its [[core]]: that is, the sub-monoidal bicategory where we only keep objects invertible up to equivalence, morphisms invertible up to 2-isomorphism, and invertible 2-morphisms. This core is a [[3-group]], sometimes called the [[Picard 3-group]], and it has [[Azumaya algebras]] over $R$ as its objects.
### Over a scheme
More generally, [[Grothendieck]] defines an __Azumaya algebra__ over a [[scheme]] $X$ as a [[sheaf]] $\mathcal{A}$ of $\mathcal{O}_X$-algebras such that for each point $x\in X$, the corresponding [[stalk]] $\mathcal{A}_x$ is an Azumaya $\mathcal{O}_{X,x}$-algebra.
The [[Brauer group]] $Br(X)$ classifies Azumaya algebras over $X$ up to a suitably defined equivalence relation: $\mathcal{A}\sim\mathcal{B}$ if $\mathcal{A}\otimes_{\mathcal{O}_X} \mathbf{End}(\mathcal{E}) \cong \mathcal{B}\otimes_{\mathcal{O}_X}\mathbf{End}(\mathcal{F})$ for some locally free sheaves of $\mathcal{O}_X$-modules $\mathcal{E}$ and $\mathcal{F}$ of finite rank. The group operation of $Br(X)$ is induced by the tensor product. The Brauer group can be reexpressed in terms of second [[nonabelian cohomology]]; indeed a sheaf of Azumaya algebras over $X$ determines an $\mathcal{O}_X^*$-[[gerbe]] (or $U(1)$-gerbe in the [[manifold]] context).
[[Karoubi K-theory]] involves an element in a Brauer group and in the original Karoubi--Donovan paper is related to a twisting with a "local system" which involves Azumaya algebras.
### In terms of (derived) étale cohomology
{#InTermsOfEtaleCohomology}
For $R$ a [[ring]] and $H^n_{et}(-,-)$ the [[etale cohomology]], $\mathbb{G}_m$ the [[multiplicative group]] of the [[affine line]]; then
* $H^0_{et}(R, \mathbb{G}_m) = R^\times$ ([[group of units]])
* $H^1_{et}(R, \mathbb{G}_m) = Pic(R)$ ([[Picard group]]: iso classes of invertible $R$-modules)
* $H^2_{et}(R, \mathbb{G}_m)_{tor} = Br(R)$ ([[Brauer group]] Morita classes of Azumaya $R$-algebras)
More generally, this works for $R$ a (connective) [[E-infinity ring]]
(the following is due to [[Benjamin Antieau]] and [[David Gepner]]).
Let $GL_1(R)$ be its [[infinity-group of units]].
If $R$ is [[connective spectrum|connective]], then the first [[Postnikov tower|Postikov stage]] of the [[Picard group|Picard]] [[infinity-groupoid]]
$$
Pic(R) \coloneqq Mod(R)^\times
$$
is
$$
\array{
\mathbf{B}_{et} GL_1(-) &\to& Pic(-)
\\
&& \downarrow
\\
&& \mathbb{Z}
}
\,,
$$
where the top morphism is the inclusion of locally free $R$-modules.
so $H^1_{et}(R, GL_1)$ is not equal to $\pi_0 Pic(R)$, but it is off only by $H^0_{et}(R, \mathbb{Z}) = \prod_{components of R} \mathbb{Z}$.
Let $Mod_R$ be the [[(infinity,1)-category]] of $R$-[[module spectra|modules]].
There is a notion of $Mod_R$-[[enriched (infinity,1)-category]], of "$R$-linear $(\infty,1)$-categories".
$Cat_R \coloneqq Mod_R$-modiles in [[presentable (infinity,1)-categories]].
Forming module $(\infty,1)$-categories is then an [[(infinity,1)-functor]]
$$
Alg_R \stackrel{Mod}{\to} Cat_R
$$
Write $Cat'_R \hookrightarrow Car_R$ for the image of $Mod$. Then define the [[Brauer group|Brauer]] [[infinity-group]] to be
$$
Br(R) \coloneqq (Cat'_R)^\times
$$
One shows (Antieau-Gepner) that this is exactly the Azumaya $R$-algebras modulo Morita equivalence.
**Theorem** (B. Antieau, D. Gepner)
1. For $R$ a connective $E_\infty$ ring, any Azumaya $R$-algebra $A$ is étale locally trivial: there is an [[etale topology|etale cover]] $R \to S$ such that $A \wedge_R S \stackrel{Morita \simeq}{\to} S$.
(Think of this as saying that an Azumaya $R$-algebra is étale-locally a Matric algebra, hence Morita-trivial: a "bundle of compact operators" presenting a (torsion) $GL_1(R)$-2-bundle).
1. $Br : CAlg_R^{\geq 0} \to Gpd_\infty$ is a sheaf for the [[etale cohomology]].
**Corollary**
1. $Br$ is [[connected object in an (infinity,1)-topos|connected]]. Hence $Br \simeq \mathbf{B}_{et} \Omega Br $.
1. $\Omega Br \simeq Pic$, hence $Br \simeq \mathbf{B}_{et} Pic$
[[Postnikov tower]] for $GL_1(R)$:
$$
for\; n \gt 0: \pi_n GL_1(S) \simeq \pi_n
$$
hence for $R \to S$ étale
$$
\pi_n S \simeq \pi_n R \otimes_{\pi_0 R} \pi_0 S
$$
This is a [[quasi-coherent sheaf]] on $\pi_0 R$ of the form $\tilde N$ (quasicoherent sheaf associated with a module), for $N$ an $\pi_0 R$-module. By vanishing theorem of higher cohomology for quasicoherent sheaves
$$
H_{et}^1(\pi_0 R, \tilde N) = 0; for p \gt 0
$$
For every [[(infinity,1)-sheaf]] $G$ of [[infinity-groups]], there is a [[spectral sequence]]
$$
H_{et}^p(\pi_0 R; \tilde \pi_q G) \Rightarrow \pi_{q-p} G(R)
$$
(the second argument on the left denotes the $qth$ Postnikov stage). From this one gets the following.
* $\tilde \pi_0 Br \simeq *$
* $\tilde \pi_1 Br \simeq \mathbb{Z}$;
* $\tilde \pi_2 Br \simeq \tilde \pi_1 Pic \simeq \pi_0 GL_1 \simeq \mathbb{G}_m$
* $\tilde \pi_n Br$ is quasicoherent for $n \gt 2$.
there is an [[exact sequence]]
$$
0 \to
H_{et}^2(\pi_0 R, \mathbb{G}_m)
\to \pi_0 Br(R)
\to H_{et}^1(\pi_0 R, \mathbb{Z})
\to 0
$$
(notice the inclusion $Br(\pi_0 R) \hookrightarrow H_{et}^2(\pi_0 R, \mathbb{G}_m)$)
this is [[split exact sequence|split exact]] and so computes $\pi_0 Br(R)$ for connective $R$.
Now some more on the case that $R$ is not connective.
Suppose there exists $R \stackrel{\phi}{\to} S$ which is a faithful
[[Galois extension]] for $G$ a [[finite group]].
**Examples**
1. (real into complex [[K-theory spectrum]]) $KO \to KU$ (this is $\mathbb{Z}_2$)
1. [[tmf]] $\to tmf(3)$
Give $R \to S$, have a [[fiber sequence]]
$$
Gl_1(R/S) \stackrel{fib}{\to} GL_1(R) \to GL_1(S)
\to
Pic(R/S) \stackrel{fib}{\to} Pic(R) \to Pic(S)
\to
Br(R/S) \stackrel{fib}{\to} Br(R) \to Br(S) \to \cdots
$$
**Theorem** (descent theorems) (Tyler Lawson, David Gepner) Given $G$-Galois extension $R \stackrel{\simeq}{\to} S^{hG}$ ([[homotopy fixed points]])
1. $Mod_R \stackrel{\simeq}{\to} Mod_S^{hG}$
1. $Alg_R \stackrel{\simeq}{\to} Alg_S^{hG}$
it follows that there is a homotopy fixed points spectral sequence
$$
H^p(G, \pi_\bullet \Sigma^n GL_1(S))
\Rightarrow
\pi_{-n} GL_1(S)
$$
**Conjecture** The spectral sequence gives an Azumaya $KO$-algebra $Q$ which is a nontrivial element in $Br(KO)$ but becomes trivial in $Br(KU)$.
### Azumaya categories
Borceux and Vitale have noted that the [[monoidal bicategory]] of $R$-algebras, bimodules and bimodule morphisms can be generalized in the context of [[enriched category]] theory, leading to a concept of "Azumaya category". An Azumaya algebra over the commutative ring $R$ is then a one-object Azumaya category enriched over $R Mod$.
More precisely, they consider an arbitrary [[Benabou cosmos]] $V$, meaning a [[complete]] and [[cocomplete]] [[symmetric monoidal]] [[closed category|closed]] category. This gives a monoidal bicategory $V Mod$ with
* $V$-enriched categories as objects,
* $V$-enriched [[profunctors]] as morphisms, and
* $V$-natural transformations between $V$-enriched profunctors as 2-morphisms.
The [[core]] of this monoidal bicategory is a 3-group, and they call the objects of the core **Azumaya categories**.
* [[Francis Borceux]] and [[Enrico Vitale]], Azumaya categories, *Applied Categorical Structures* **10** (2002), 449-467. ([pdf](https://perso.uclouvain.be/enrico.vitale/Azumaya.pdf))
## Related concepts
* [[group of units]], [[Picard group]], [[Brauer group]], [[Picard 3-group]]
## References
* [G. Cortiñas](http://mate.dm.uba.ar/~gcorti), [[Charles Weibel]], _Homology of Azumaya algebras_, Proc. AMS __121__, 1, pp. 1994 ([jstor](http://www.jstor.org/stable/2160364))
* [[John Duskin]], _The Azumaya complex of a commutative ring_, in Categorical Algebra and its Appl., Lec. Notes in Math. 1348 (1988) [doi:10.1007/BFb0081352](http://dx.doi.org/10.1007/BFb0081352)
* {#GrothendieckBrauer} [[Alexander Grothendieck]], _Le groupe de Brauer I, II, III_, in Dix exposes sur la cohomologie des schemas (I: Algèbres d'Azumaya et interprétations diverses) North-Holland Pub. Co., Amsterdam (1969)
* [[Max Karoubi]], [[Peter Donovan]], _Graded Brauer groups and $K$-theory with local coefficients_ ([pdf](http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1970__38_/PMIHES_1970__38__5_0/PMIHES_1970__38__5_0.pdf))
* M-A. Knus, M. Ojanguren, _Théorie de la descente et algèbres d'Azumaya_, Lec. Notes in Math. __389__, Springer 1974, [doi:10.1007/BFb0057799](http://dx.doi.org/doi:10.1007/BFb0057799), MR0417149
* J. Milne, _Étale cohomology_, Princeton Univ. Press
* [[Ross Street]], _Descent_, Oberwolfach preprint (sec. 6, Brower groups) [pdf](http://www.math.mq.edu.au/~street/Descent.pdf); _Some combinatorial aspects of descent theory_, Applied categorical structures __12__ (2004) 537-576, [math.CT/0303175](http://arxiv.org/abs/math/0303175) (sec. 12, Brower groups)
* [[Enrico Vitale]], _A Picard-Brauer exact sequence of categorical groups_, [pdf](http://www.math.ucl.ac.be/membres/vitale/cat-gruppi2.pdf)
* Ana-L. Agore, [[Stefan Caenepeel]], Gigel Militaru, _Braidings on the category of bimodules, Azumaya algebras and epimorphisms of rings_, Appl. Categor. Struct. 22, 29--42 (2014) [doi](https://doi.org/10.1007/s10485-012-9294-3)
The observation that passing to [[derived algebraic geometry]] makes also the non-torsion elements in the "[[bigger Brauer group]]" $H^2_{et}(-,\mathbb{G}_m)$ be represented by (derived) Azumaya algebras is due to
* {#Toen10} [[Bertrand Toën]], _Derived Azumaya algebras and generators for twisted derived categories_ ([arXiv:1002.2599](http://arxiv.org/abs/1002.2599))
The comparison of the Artin's theorem on characterization of Azumaya algebras
and Tomiyama-Takesaki's theorem on $n$-[[homogeneous C*-algebra]]s is in chapter 9 of
* Edward Formanek, _Noncommutative invariant theory_, in: Group actions on rings (Brunswick, Maine, 1984), 87–119, Contemp. Math. 43, Amer. Math. Soc. 1985 [doi](http://dx.doi.org/10.1090/conm/043)
See also
* Wikipedia, [Azumaya algebra](http://en.wikipedia.org/wiki/Azumaya_algebra)
* Urs Schreiber, [Picard and Brauer 2-groups](http://golem.ph.utexas.edu/string/archives/000786.html), String Theory Coffee Table, 2006.
* John Baez, [The Brauer 3-group](https://golem.ph.utexas.edu/category/2020/05/the_brauer_3group.html), $n$-Category Café, 2020.
[[!redirects Azumaya algebras]]
[[!redirects central separable algebra]] |
B meson | https://ncatlab.org/nlab/source/B+meson |
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### Context
#### Fields and quanta
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[[!include fields and quanta - table]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In the [[standard model of particle physics]] a _B-meson_ is a [[meson]] which is a [[bound state]] of a [[bottom quark]] and any other [[quark]] from the first two [[generations of fermions]] ([[up quark]] or [[down quark]] or [[strange quark]] or [[charm quark]]).
Hence the class of B-mesons goes along side that of [[K-mesons]] (light mnesons with a [[strange quark]]) and [[D-mesons]] (light meson with a [[charm quark]]).
[[!include flavors of fundamental particles -- table]]
Current [[experiments]] studying B-meson [[physics]] are the \
* [[LHCb experiment]]
* [[Belle experiment]]
* [[BaBar experiment]]
All these are increasingly seeing [[flavour anomalies]] (see there) in $B$-meson decays at high [[statistical significance]], pointing to physics beyond the [[standard model of particle physics]].
## Related concepts
* [[upsilon-meson]]
* [[K-meson]], [[D-meson]]
* [[pion]], [[omega meson]], [[rho meson]]
* [[XYZ meson]]
* B-meson [[experiments]]
* [[LHCb experiment]]
* [[Belle collaboration]]
* [[BaBar experiment]]
* B-[[baryon]]: [[Lambda baryon]]
* observations
* [[flavour anomaly]]
* [[V_cb puzzle]]
* [[pentaquark]]
## References
### General
* [[Patrick Koppenburg]], [[Zdenek Dolezal]], [[Maria Smizanska]], _Rare decays of b hadrons_, Scholarpedia, 11(6):32643 ([doi:10.4249/scholarpedia.32643](http://dx.doi.org/10.4249/scholarpedia.32643))
See also
* Wikipedia, _<a href="https://en.wikipedia.org/wiki/B_meson">B meson</a>_
### Semileptonic decay
[[semileptonic decay]]:
Review:
* R. Casalbuoni, A. Deandrea, N.Di Bartolomeo, F. Feruglio, R. Gatto, G. Nardulli, _Phenomenology of Heavy Meson Chiral Lagrangians_, Phys. Rept. 281:145-238, 1997 ([arXiv:hep-ph/9605342](https://arxiv.org/abs/hep-ph/9605342))
* Jochen Dingfelder, Thomas Mannel, _Leptonic and semileptonic decays of $B$ mesons_, Rev. Mod. Phys. 88 (2016) 3, 035008 ([spire:1488176](https://inspirehep.net/literature/1488176), [doi:10.1103/RevModPhys.88.035008](https://doi.org/10.1103/RevModPhys.88.035008))
See also
* J. P. Lees et al. (BABAR Collaboration), _Measurement of the $B^+ \to \omega \ell^+ \nu$ branching fraction with semileptonically tagged B mesons_, Phys. Rev. D 88, 072006 (2013) ([doi:10.1103/PhysRevD.88.072006](https://doi.org/10.1103/PhysRevD.88.072006))
* Xian-Wei Kang, Tao Luo, Yi Zhang, Ling-Yun Dai, Chao Wang, _Semileptonic $B$ and $B_s$ decays involving scalar and axial-vector mesons_, Eur. Phys. J. C 78, 909 (2018) ([doi:10.1140/epjc/s10052-018-6385-9](https://doi.org/10.1140/epjc/s10052-018-6385-9))
[[!include Skyrme hadrodynamics with heavy mesons -- references]]
### In holographic QCD
Discussion in [[holographic light front QCD]]:
* Mohammad Ahmady, _Holographic light-front QCD in B meson phenomenology_ (<a href="https://arxiv.org/abs/2001.00266">arXiv:2001.00266</a>)
[[!redirects B mesons]]
[[!redirects B-meson]]
[[!redirects B-mesons]]
|
B-bordism | https://ncatlab.org/nlab/source/B-bordism |
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#### Manifolds and cobordisms
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#Contemts#
* table of contents
{:toc}
## Idea
A _$B$-bordism_ is a [[cobordism]] $W$ equipped with extra "topological [[structure]]" with respect to _[[(B,f)-structure]]_, i.e. in the form of a lift of the classifying map $W \to B O$ of its [[tangent bundle]]/[[stable normal bundle]] through some [[fibration]] $B \to B O$ over the [[classifying space]] of the [[orthogonal group]].
Commonly considered are lifts through the [[Whitehead tower]] of the [[orthogonal group]]), corresponding, in this order, to cobordisms with
* [[spin structure]] - [[spin bordism]]
* [[string structure]] - [[string bordism]]
* [[fivebrane structure]] - [[fivebrane bordism]]
etc.
## Examples
[[!include flavours of cobordism cohomology theories -- table]]
## References
* [[Manifold Atlas]], _[B-Bordism](http://www.map.mpim-bonn.mpg.de/B-Bordism)_
[[!redirects B-bordisms]]
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B-brane | https://ncatlab.org/nlab/source/B-brane |
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## Idea
The [[branes]] of the [[B-model]] [[topological string]], hence a [[TQFT]]-shadow of the [[D-branes]] of the [[superstring]]. Form the [[derived category]] of [[coherent sheaves]] on the [[target space|target]] [[spacetime]].
## Related concepts
* [[A-brane]], [[homological mirror symmetry]]
* [[2d TQFT]], [[TCFT]]
* [[Kapustin-Witten TQFT]]
* [[Bridgeland stability condition]]
* [[quiver gauge theory]]
[[!include table of branes]]
## References
A classical review is
* [[Paul Aspinwall]], _D-Branes on Calabi-Yau Manifolds_ ([arXiv:hep-th/0403166](http://arxiv.org/abs/hep-th/0403166))
Further surveys of the literature include
* [[Kevin H. Lin]], MO comment on _[Matrix factorization and physics](http://mathoverflow.net/a/9748/381), 2009
Formulation of the [[chain complexes]] of [[holomorphic vector bundles]] on the B-branes via [[Lie infinity-algebroid representation]] (see there) of the holomorphic [[tangent Lie algebroid]] is discussed in
* {#Bergman08} [[Aaron Bergman]], _Topological D-branes from Descent_ ([arXiv:0808.0168](http://arxiv.org/abs/0808.0168))
The definition of a [[triangulated category]] of B-branes for the [[Landau-Ginzburg model]] via [[matrix factorization]] was proposed by [[Maxim Kontsevich]] and is written out in
* [[Anton Kapustin]], Yi Li, _D-Branes in Landau-Ginzburg Models and Algebraic Geometry_ ([arXiv:hep-th/0210296](http://arxiv.org/abs/hep-th/0210296))
* {#Orlov} [[Dmitri Orlov]], _Triangulated categories of singularities and D-branes in Landau-Ginzburg models_, Proc. Steklov Inst. Math. 2004, no. 3 (246), 227--248 ([arXiv:math/0302304](http://arxiv.org/abs/math/0302304))
* [[Dmitri Orlov]]_Derived categories of coherent sheaves and triangulated categories of singularities_, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, 503–531, Progr. Math., 270, Birkhäuser Boston,
Inc., Boston, MA, 2009 ([arXiv:math.ag/0503632](http://arxiv.org/abs/math.ag/0503632))
[[!redirects B-branes]]
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B-model | https://ncatlab.org/nlab/source/B-model |
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## Idea
Witten introduced two topological twists of a supersymmetric nonlinear [[sigma model]], which is a certain N=2 superconformal field theory attached to a compact [[Calabi-Yau variety]] $X$. One of them is the _B-model_ [[topological string]]; it is a 2-dimensional [[topological conformal field theory|topological N=1 superconformal field theory]]. In Kontsevich's version, instead of SCFT with Hilbert space, one assembles all the needed data in terms of [[Calabi-Yau category|Calabi-Yau A-infinity-category]] which is the A-infinity-category of coherent sheaves on the underlying variety. In fact only the structure of a derived category is sufficient (and usually quoted): it is now known (by the results of [[Dmitri Orlov]] and [[Valery Lunts]]) that under mild assumptions on the variety, a derived category of coherent sheaves has a unique [[enhanced triangulated category|enhancement]].
The B-model arose in considerations of [[string theory|superstring]]-propagation on Calabi--Yau varieties: it may be motivated by considering the [[vertex operator algebra]] of the 2d[[CFT|SCFT]] given by the N=2 supersymmetric nonlinear [[sigma-model]] with target $X$ and then changing the fields so that one of the super-[[Virasoro algebra|Virasoro]] generators squares to 0. The resulting "topologically twisted" algebra may then be read as being the [[BRST complex]] of a [[TCFT]].
One can also define a B-model for [[Landau-Ginzburg model|Landau?Ginzburg models]]. The category of [[D-brane|D-branes]] for the string -- the [[B-branes]] -- is given by the category of [[matrix factorization|matrix factorizations]] (this was proposed by Kontsevich and elaborated by Kapustin-Li; see also papers of Orlov). For the genus 0 closed string theory, see the work of Saito.
By [[homological mirror symmetry]], the B-model is dual to the [[A-model]].
## Properties
### Second quantization / effective background field theory
The [[second quantization]] [[effective field theory]] defined by the [[perturbation series]] of the B-model is supposed to be "Kodaira-Spencer gravity" / "BVOC theory" in 6d ([BCOV 93](#BCOV93), [Costello-Li 12](#CostelloLi12), [Costello-Li 15](#CostelloLi15)).
For more on this see at _[TCFT -- Worldsheet and effective background theories](http://ncatlab.org/nlab/show/TCFT#ActionFunctionals)_.
## Related concepts
* [[schreiber:∞-Chern-Simons theory]]
* [[sigma-model]]
* [[AKSZ sigma-model]]
* [[Poisson sigma-model]]
* [[A-model]], **B-model**
* [[half-twisted model]]
* [[twistor string theory]]
* [[Courant sigma-model]]
* [[Chern-Simons theory]]
* [[topological membrane]]
* [[topologically twisted D=4 super Yang-Mills theory]]
* [[Landau-Ginzburg model]]
[[!include gauge theory from AdS-CFT -- table]]
## References
{#References}
### General
The B-model was first conceived in
* [[Edward Witten]], _Topological sigma models_, Commun. Math. Phys. __118__ (1988) 411--449, [euclid](http://projecteuclid.org/euclid.cmp/1104162092), [MR90b:81080](http://www.ams.org/mathscinet-getitem?mr=0958805)
An early review is in
* [[Edward Witten]]. _Mirror manifolds and topological field theory_, in: Essays on mirror manifolds, pp. 120–-158. Int. Press, Hong Kong, 1992. ([arXiv:hep-th/9112056](http://arxiv.org/abs/hep-th/9112056)).
The motivation from the point of view of [[string theory]] with an eye towards the appearance of the Calabi-Yau categories is reviewed for instance in
* [[Paul Aspinwall]], _D-Branes on Calabi-Yau Manifolds_ ([arXiv:hep-th/0403166](http://arxiv.org/abs/hep-th/0403166))
A summary of these two reviews is in
* H. Lee, _Review of topological field theory and homological mirror symmetry_ ([pdf](http://people.maths.ox.ac.uk/leeh/files/CYMSmini.pdf))
That the B-model [[Lagrangian]] arises in [[AKSZ theory]] by [[gauge fixing]] the [[Poisson sigma-model]] was observed in
* {#AKSZ} M. Alexandrov, [[Maxim Kontsevich|M. Kontsevich]], [[Albert Schwarz|A. Schwarz]], O. Zaboronsky, around page 28 in _The geometry of the master equation and topological quantum field theory_, Int. J. Modern Phys. A 12(7):1405--1429, 1997
For survey of the literature see also
* [[Kevin H. Lin]], MO comment on _[Matrix factorization and physics](http://mathoverflow.net/a/9748/381), 2009
The B-model on [[genus]]-0 [[cobordism]]s had been constructed in
* S. Barannikov, [[Maxim Kontsevich]], _Frobenius manifolds and formality of Lie algebras of polyvector fields_ , Internat. Math. Res. Notices 1998, no. 4, 201--215; [math.QA/9710032](http://arxiv.org/abs/alg-geom/9710032) [doi](http://dx.doi.org/10.1155/S1073792898000166)
The construction of the B-model as a [[TCFT]] on [[cobordisms]] of arbitrary [[genus]] was given in
* [[Kevin Costello]], _The Gromov-Witten potential associated to a TCFT_ ([math.QA/0509264](http://arxiv.org/abs/math/0509264))
See also the MathOverflow discussion: [higher-genus-closed-string-b-model](http://mathoverflow.net/questions/8692/higher-genus-closed-string-b-model)
### Second quantization to Kodeira-Spencer gravity
{#ReferencesBCOV}
The [[second quantization]] [[effective field theory|effective]] field theory defined by the B-model [[perturbation series]] ("Kodeira-Spencer gravity"/"BCOV theory") is discussed in
Discussion of how the [[second quantization]] of the [[B-model]] yields [[Kodeira-Spencer gravity]]/[[BCOV theory]] is in
* {#BCOV93} [[Michael Bershadsky]], [[Sergio Cecotti]], [[Hirosi Ooguri]], [[Cumrun Vafa]], *Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes*, Commun. Math. Phys. **165** (1994) 311-428 [[arXiv:hep-th/9309140](http://arxiv.org/abs/hep-th/9309140), [doi:10.1007/BF02099774](https://doi.org/10.1007/BF02099774)]
* {#CostelloLi12} [[Kevin Costello]], [[Si Li]], _Quantum BCOV theory on Calabi-Yau manifolds and the higher genus B-model_ ([arXiv:1201.4501](http://arxiv.org/abs/1201.4501))
* [[Si Li]], _BCOV theory on the elliptic curve and higher genus mirror symmetry_ ([arXiv:1112.4063](http://arxiv.org/abs/1112.4063))
* [[Si Li]], _Variation of Hodge structures, Frobenius manifolds and Gauge theory_ ([arXiv:1303.2782](http://arxiv.org/abs/1303.2782))
* {#CostelloLi15} [[Kevin Costello]], Si Li, _Quantization of open-closed BCOV theory, I_ ([arXiv:1505.06703](http://arxiv.org/abs/1505.06703))
### Computation via topological recursion
Computation via [[topological recursion]] in [[matrix models]] and all-[[genus of a surface|genus]] proofs of [[mirror symmetry]] is due to
* {#BouchardKlemmMarinoPasquetti09} [[Vincent Bouchard]], [[Albrecht Klemm]], [[Marcos Marino]], [[Sara Pasquetti]], _Remodeling the B-model_, Commun.Math.Phys.287:117-178, 2009 ([arXiv:0709.1453](https://arxiv.org/abs/0709.1453))
* [[Bertrand Eynard]], [[Amir-Kian Kashani-Poor]], Olivier Marchal, _A matrix model for the topological string I: Deriving the matrix model_ ([arXiv:1003.1737](https://arxiv.org/abs/1003.1737))
* [[Bertrand Eynard]], [[Amir-Kian Kashani-Poor]], Olivier Marchal, _A matrix model for the topological string II: The spectral curve and mirror geometry_ ([arXiv:1007.2194](https://arxiv.org/abs/1007.2194))
* {#EynardOrantin12} [[Bertrand Eynard]], [[Nicolas Orantin]], _Computation of open Gromov-Witten invariants for toric Calabi-Yau 3-folds by topological recursion, a proof of the BKMP conjecture_ ([arXiv:1205.1103](https://arxiv.org/abs/1205.1103))
* {#FangLiuZong13} Bohan Fang, Chiu-Chu Melissa Liu, Zhengyu Zong, _All Genus Open-Closed Mirror Symmetry for Affine Toric Calabi-Yau 3-Orbifolds_ ([arXiv:1310.4818](https://arxiv.org/abs/1310.4818))
[[!redirects Kodeira-Spencer gravity]]
[[!redirects BCOV theory]]
[[!redirects B-models]]
|
B. A. Blackadar > history | https://ncatlab.org/nlab/source/B.+A.+Blackadar+%3E+history | |
B. Andrei Bernevig | https://ncatlab.org/nlab/source/B.+Andrei+Bernevig |
* [webpage](https://phy.princeton.edu/people/bogdan-bernevig)
## Selected writings
Predicting a [[quantum spin Hall effect]] in [[topological insulators]] based on mercury telluride (HgTe quantum wells):
* [[Andrei Bernevig]], [[Shou-Cheng Zhang]], _Quantum Spin Hall Effect_, Phys. Rev. Lett. 96, 106802 (2006) ([arXiv:cond-mat/0504147](https://arxiv.org/abs/cond-mat/0504147), [doi:10.1103/PhysRevLett.96.106802](https://doi.org/10.1103/PhysRevLett.96.106802))
* [[Andrei Bernevig]], [[Taylor Hughes]], [[Shou-Cheng Zhang]], _Quantum spin Hall effect and topological phase transition in HgTe quantum wells_, Science 15 December 2006: __314__, n. 5806, pp. 1757-1761 ([doi:10.1126/science.1133734](http://dx.doi.org/10.1126/science.1133734), [arXiv:cond-mat/0611399](https://arxiv.org/abs/cond-mat/0611399))
On higher order [[topological insulators]] (with protected corner-modes beyond the edge-modes):
* [[Frank Schindler]], Ashley M. Cook, Maia G. Vergniory, Zhijun Wang, Stuart S. P. Parkin, [[B. Andrei Bernevig]], [[Titus Neupert]], *Higher-Order Topological Insulators*, Science Advances 01 **4** 6 (2018) eaat0346 [[arXiv:1708.03636](https://arxiv.org/abs/1708.03636), [doi:10.1126/sciadv.aat0346](https://doi.org/10.1126/sciadv.aat0346)]
category: people
[[!redirects Andrei Bernevig]]
[[!redirects Bogdan Andrei Bernevig]]
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B1-homotopy theory | https://ncatlab.org/nlab/source/B1-homotopy+theory |
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## Idea
_$\mathbb{B}^1$-homotopy theory_ is the study of the [[localization]] of an [[(∞,1)-sheaf (∞,1)-category]] over a [[site]] of [[analytic spaces]] at morphisms $\mathbb{B}^1 \to *$, where $\mathbb{B}^1 \simeq Spm(k\{t\})$ is the _Tate ball_.
## Related concepts
* [[A1-homotopy theory]]
* [[complex analytic ∞-groupoid]]
## References
* {#Ayoub06} [[Joseph Ayoub]], _Motives of rigid varieties and
the motivic nearby functor_, talk notes 2006 ([pdf](http://user.math.uzh.ch/ayoub/Other-PDF/Venise_Hodge.pdf))
* [[Joseph Ayoub]], _Motifs des variétés analytiques rigides_ ([pdf](http://user.math.uzh.ch/ayoub/PDF-Files/MotVarRig.pdf))
* [[Joseph Ayoub]], Floian Ivorra and Julien Sebag, _Motives of rigid analytic tubes and nearby motivic sheaves_ ([pdf](http://user.math.uzh.ch/ayoub/PDF-Files/Motive-of-Tube.pdf))
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b1-meson | https://ncatlab.org/nlab/source/b1-meson |
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## Idea
One of the [[light meson|light]] [[vector mesons]], with [[Wigner classification]] being a [[Lorentz group]] [[bivector]]
([DHR 11](#DHR11), [DHR 12](#DHR12)) and [[isospin]] [[vector]].
## Related concepts
* [[h1-meson]]
* [[B-field]]
## References
* J. C. R. Bloch, Yu. L. Kalinovsky, C. D. Roberts, S. M. Schmidt, _Describing $a_1$ and $b_1$ decays_, Phys. Rev. D60:111502, 1999 ([arXiv:nucl-th/9906038](https://arxiv.org/abs/nucl-th/9906038))
Discussion within [[holographic QCD]]:
* {#DHR11} Sophia K. Domokos, [[Jeffrey Harvey]], Andrew B. Royston, _Completing the framework of AdS/QCD: $h_1$/$b_1$ mesons and excited $\omega$/$\rho$'s_, JHEP 05(2011)107 ([arXiv:1101.3315](https://arxiv.org/abs/1101.3315))
* {#DHR12} Sophia K. Domokos, [[Jeffrey Harvey]], Andrew B. Royston, _Successes and Failures of a More Comprehensive Hard Wall AdS/QCD_, JHEP 04(2013)104 ([arXiv:1210.6351](https://arxiv.org/abs/1210.6351))
[[!redirects b1-mesons]]
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## Idea
The [[higher U(1)-gauge field]] which is the [[electric-magnetic duality|EM dual]] to the [[B2-field]], for instance in [[dual heterotic string theory]]. Modeled by [[circle n-bundle with connection|circle 6-bundle with connection]]. Couples to the [[NS5-brane]].
## Related concepts
[[!include table of branes]]
[[!redirects B6-fields]] |
Baas-Sullivan theory | https://ncatlab.org/nlab/source/Baas-Sullivan+theory | +-- {: .rightHandSide}
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##Idea
The Baas-Sullivan construction allows [[generalized cohomology theories]] to be seen as a [[cobordism theory]] in which the cobordisms may have singularities.
## References
Some of the original work is contained in these papers:
* Nils Andreas Baas, Bordism theories with singularities, Proceedings of the Advanced Study Institute on Algebraic Topology (Aarhus Univ., Aarhus, 1970), Vol. I, pp. 1–16. Various Publ. Ser., No. 13, Mat. Inst., Aarhus Univ., Aarhus, 1970.
* Nils Andreas Baas, On bordism theory of manifolds with singularities, Math. Scand. 33 (1973), 279–302.
* Nils Andreas Baas, On formal groups and singularities in complex cobordism theory, Math. Scand. 33 (1973), 303–313 |
Babak Haghighat | https://ncatlab.org/nlab/source/Babak+Haghighat |
* [personal page](https://ymsc-strings.github.io/people/babak.html)
* [YMSC institute page](https://ymsc.tsinghua.edu.cn/en/info/1032/1214.htm)
* [YMSC research group page](https://ymsc-strings.github.io/people-faculty.html)
* [BIMSA page](https://www.bimsa.cn/newsinfo/581931.html)
* [InSpire page](https://inspirehep.net/authors/1057034)
* [GoogleScholar page](https://scholar.google.com/citations?user=N1SYs84AAAAJ&hl=en)
## Selected writings
On the [[M5-brane elliptic genus]]:
* Murad Alim, [[Babak Haghighat]], Michael Hecht, [[Albrecht Klemm]], Marco Rauch, Thomas Wotschke, *Wall-crossing holomorphic anomaly and mock modularity of multiple M5-branes*, Comm. Math. Phys. **339** (2015) 773–814 [[arXiv:1012.1608](https://arxiv.org/abs/1012.1608), [doi:10.1007/s00220-015-2436-3](https://doi.org/10.1007/s00220-015-2436-3)]
On [[M-strings]]:
* [[Babak Haghighat]], [[Amer Iqbal]], [[Can Kozcaz]], [[Guglielmo Lockhart]], [[Cumrun Vafa]], _M-Strings_, Commun. Math. Phys. 334, 779–842 (2015) ([arXiv:1305.6322](https://arxiv.org/abs/1305.6322), [doi:10.1007/s00220-014-2139-1](https://doi.org/10.1007/s00220-014-2139-1))
* [[Babak Haghighat]], [[Can Kozcaz]], [[Guglielmo Lockhart]], [[Cumrun Vafa]], _On orbifolds of M-Strings_, Physical Review D 89.4 (2014): 046003 ([arXiv:1310.1185](https://arxiv.org/abs/1310.1185))
On [[E-strings]]:
* {#HLV14} [[Babak Haghighat]], [[Guglielmo Lockhart]], [[Cumrun Vafa]], _Fusing E-string to heterotic string: $E + E \to H$_, Phys. Rev. D 90, 126012 (2014) ([arXiv:1406.0850](https://arxiv.org/abs/1406.0850))
and emergence of [[SU(2)]] [[flavor (particle physics)|flavor]]-[[chiral symmetry|symmetry]] on [[M5-branes]] in [[heterotic M-theory]] (in the [[D=6 N=(1,0) SCFT]] on [[small instantons]] in [[heterotic string theory]]):
* {#GHKKLV15} [[Abhijit Gadde]], [[Babak Haghighat]], [[Joonho Kim]], [[Seok Kim]], [[Guglielmo Lockhart]], [[Cumrun Vafa]], _6d String Chains_, JHEP **1802** (2018) 143 [[arXiv:1504.04614](https://arxiv.org/abs/1504.04614)]
On [[quantum Seiberg-Witten curves]] in relation to [[class S-theories]] and "[[M3]]"-[[defect branes]] inside [[M5-branes]]:
* [[Jin Chen]], [[Babak Haghighat]], [[Hee-Cheol Kim]], [[Marcus Sperling]], *Elliptic Quantum Curves of Class $\mathcal{S}_k$*, J. High Energ. Phys. **2021** 28 (2021) [[arXiv:2008.05155](https://arxiv.org/abs/2008.05155), <a href="https://doi.org/10.1007/JHEP03(2021)028">doi:10.1007/JHEP03(2021)028</a>]
On [[quantum Seiberg-Witten curves]] of [[E-string]]-theories in relation to [[D6-D8-brane bound states]]:
* [[Jin Chen]], [[Babak Haghighat]], [[Hee-Cheol Kim]], [[Marcus Sperling]], [[Xin Wang]], *E-string Quantum Curve*, Nuclear Physics B **973** (2021) 115602 [[arXiv:2103.16996](https://arxiv.org/abs/2103.16996), [doi:10.1016/j.nuclphysb.2021.115602](https://doi.org/10.1016/j.nuclphysb.2021.115602)]
On [[KK-compactification]] of [[D=6 N=(2,0) SCFT]] on [[Riemann surfaces]] [[4d superconformal gauge field theory]]:
* [[Jin Chen]], [[Babak Haghighat]], [[Shuwei Liu]], [[Marcus Sperling]], _4d $\mathcal{N}=1$ from 6d D-type $\mathcal{N}=(1,0)$_, J. High Energ. Phys. **2020** 152 (2020) [[arXiv:1907.00536](https://arxiv.org/abs/1907.00536), <a href=" https://doi.org/10.1007/JHEP01(2020)152">doi:10.1007/JHEP01(2020)152</a>]
On [[braid group representations]] for [[su(2)-anyon]]-[[anyon statistics|statistics]] from the [[monodromy]] of the [[Knizhnik-Zamolodchikov connection]] of bundles of [[conformal blocks]] over [[configuration spaces of points]]:
* [[Xia Gu]], [[Babak Haghighat]], [[Yihua Liu]], *Ising- and Fibonacci-Anyons from KZ-equations*, J. High Energ. Phys. **2022** 15 (2022) [[arXiv:2112.07195](https://arxiv.org/abs/2112.07195), <a href="https://doi.org/10.1007/JHEP09(2022)015">doi:10.1007/JHEP09(2022)015</a>]
On [[QFT with defects|defects]] in the [[KK-compactification]] of the [[D=6 N=(2,0) SCFT]] on [[4-manifolds]]:
* [[Jin Chen]], [[Wei Cui]], [[Babak Haghighat]], [[Yi-Nan Wang]], *SymTFTs and Duality Defects from 6d SCFTs on 4-manifolds*, JHEP **2023** 208 (2023) [[arXiv:2305.09734](https://arxiv.org/abs/2305.09734), <a href="https://doi.org/10.1007/JHEP11(2023)208">doi:10.1007/JHEP11(2023)208</a>]
On [[conformal blocks]] for [[Liouville theory]]:
* [[Babak Haghighat]], [[Yihua Liu]], [[Nicolai Reshetikhin]], *Flat Connections from Irregular Conformal Blocks* [[arXiv:2311.07960](https://arxiv.org/abs/2311.07960)]
* [[Xia Gu]], [[Babak Haghighat]], [[Kevin Loo]], *Irregular Fibonacci Conformal Blocks* [[arXiv:2311.13358](https://arxiv.org/abs/2311.13358)]
category: people
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BaBar experiment | https://ncatlab.org/nlab/source/BaBar+experiment |
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## Idea
A [[B-meson]]-[[experiment]].
Together with the [[LHCb experiment]] and the [[Belle experiment]], BaBar is responsible for detecting the [[flavour anomalies]].
## Related experiments
* [[LHCb experiment]]
* [[Belle experiment]]
* [[XYZ particle]]
* [[precision experiment]]
* [[experiment]]
## References
* [BaBar Public webpage](https://www.slac.stanford.edu/BFROOT/)
* Wikpedia, _[BaBar experiment](https://en.wikipedia.org/wiki/BaBar_experiment)_
[[!redirects BaBar]] |
Bachir Bekka | https://ncatlab.org/nlab/source/Bachir+Bekka |
* [personal page](https://perso.univ-rennes1.fr/bachir.bekka/)
## Selected writings
On [[Kazhdan's property (T)]]:
* [[Bachir Bekka]], P. de la Harpe and A. Valette, _Kazhdan's Propert (T)_, 2007 ([pdf](https://perso.univ-rennes1.fr/bachir.bekka/KazhdanTotal.pdf), [[BekkaHarpeValetteOnKashdanPropertyT.pdf:file]])
Proof that the [[unitary group]] [[U(ℋ)]] on a [[separable Hilbert space]] in its strong [[operator topology]] satisfies [[Kazhdan's property (T)]]:
* [[Bachir Bekka]], *Kazhdan's Property (T) for the unitary group of aseparable Hilbert space*, Geom. funct. anal. 13, 509–520 (2003) ([doi:10.1007/s00039-003-0420-0](https://doi.org/10.1007/s00039-003-0420-0))
category: people |
background field | https://ncatlab.org/nlab/source/background+field | [[!redirects background gauge field]]
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#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
### General
In [[physics]], the [[field (physics)|fields]] in ([[prequantum field theory|pre]]-)[[quantum field theory]] whose [[dynamics]] is described by the [[theory (physics)|theory]] may in general depend on certain fixed [[stuff, structure, property|structure]] which also has an interpretation of a [[field (physics)|field]], but one that is fixed and not regarded as having dynamics in the given [[model (in theoretical physics)]]. These are often called **background fields**.
### For sigma-models
In particuar a [[sigma-model]] [[quantum field theory]] describes the propagation of a [[brane]] on some [[target space]] $X$. Typically the dynamics is controled by a [[gauge field]] on $X$ under which the brane is [[charge]]d. This is the **background gauge field** of the $\sigma$-model. The term "background" alludes to the fact that this background gauge field is fixed and only serves to influence the dynamics of the brane coupled to it. More generally, one could also study the quantum dynamics of the possible background fields themselves. This would be given by the corresponding [[gauge theory]] on $X$.
## Definition
A formalization of the notion is discussed at _[[field (physics)|field]]_ in _[Definition -- Physical fields](field%20%28physics%29#DefinitionPhysicalField)_.
## Examples
* The electrically charged [[relativistic particle]] is described by a [[sigma-model]] with background gauge field the [[electromagnetic field]] on [[spacetime]].
* The [[string]] of [[string theory]] may be described by a [[sigma-model]] with background gauge field the [[Kalb-Ramond field]].
* [[Chern-Simons theory]] may be understood as the [[sigma-model]] of a [[membrane]] charged under a background field that is a [[Chern-Simons circle 3-bundle]].
## Related concepts
* [[field (physics)]]
* [[sigma model]]
* [[worldvolume]]
* [[target space]]
* **background field**
* [[genus]]
[[!redirects background field]]
[[!redirects background fields]]
[[!redirects background gauge fields]] |
background field formalism | https://ncatlab.org/nlab/source/background+field+formalism |
## References
* L.F. Abbott, _Introduction to the Background Field Method_ Acta Phys.Polon. B13 (1982) 33 ([inSPIRE:166273](http://inspirehep.net/record/166273))
* L.F. Abbott, _The Background Field Method Beyond One Loop_ Nucl.Phys. B185 (1981) 189 ([inSPIRE:155719](http://inspirehep.net/record/155719))
* L.F. Abbott, Marcus T. Grisaru, Robert K. Schaefer, _The Background Field Method and the S Matrix_ Nucl.Phys. B229 (1983) 372 ([inSPIRE:13462](http://inspirehep.net/record/13462))
[[!redirects background field formalism]] |
backpropagation | https://ncatlab.org/nlab/source/backpropagation | ## Idea
Reverse mode [[automatic differentiation]] is often referred to as backpropagation, when specialized to the context of [[neural networks]].
## See also
* [[neural network]]
* [[machine learning]]
* [[equilibrium propagation]] |
backreaction | https://ncatlab.org/nlab/source/backreaction |
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### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[physics]], the _backreaction_ of an object or [[field configuration]] is its effect on other objects/fields by their mutual [[interaction]].
Typically one speaks of backreaction only in situations where it has (previously) been _neglected_, in that all other fields have been regarded as a fixed [[background field]] for the given object/field. Given such a situation of a field in a [[background field]] one may then ask what the backreaction of the former on the latter is or would have been.
## Examples
### Particle mechanics
The orbit of an [[electron]] around a [[nucleus]] or of a planet around the sun is to good approximation computed by taking into account the [[force]] that the nucleus/sun exerts on the electron/planet, but neglecting the (much smaller) force exerted by the electron/planet on the nucleus/sun. The latter force is the _backreaction_; and one will speak about taking it into account or not.
### Cosmic inhomogeneities
Not all backreaction considered is between small and large point masses.
For example, in [[inhomogeneous cosmology]] one speaks about the backreaction (or not) of inhomogeneities in the matter density filling the [[observable universe]] on the [[gravitational field]] filling the universe.
### Branes
In [[string theory]] one speaks of _[[probe branes]]_ when the backreaction of a small number of [[branes]] on some large number of background [[black branes]] is neglected.
## References
See also
* Wikipedia, _[Backreaction](https://en.wikipedia.org/wiki/Back-reaction)_
[[!redirects backreactions]]
[[!redirects back-reaction]]
[[!redirects back-reactions]] |
Badis Ydri | https://ncatlab.org/nlab/source/Badis+Ydri |
* [webpage](https://homepages.dias.ie/ydri/index1)
## Selected writings
Review of [[matrix models]] in [[string theory]]/[[M-theory]] ([[BFSS matrix model]], [[IKKT matrix model]], [[BMN matrix model]], ...) and related topics ([[light cone gauge quantization]], [[Penrose limit]] to [[pp-wave spacetimes]], [[D-brane geometry]], [[fuzzy spheres]], ...):
* {#Ydri18} [[Badis Ydri]], _Review of M(atrix)-Theory, Type IIB Matrix Model and Matrix String Theory_ ([arXiv:1708.00734](https://arxiv.org/abs/1708.00734)), published as: _Matrix Models of String Theory_, IOP 2018 ([ISBN:978-0-7503-1726-9](https://iopscience.iop.org/book/978-0-7503-1726-9))
category: people |
Baer sum | https://ncatlab.org/nlab/source/Baer+sum |
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###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _Baer sum_ is the natural addition operation on [[abelian group extensions]]
as well on the extensions of $R$-modules, for fixed ring $R$.
For $G$ a [[group]] and $A$ an [[abelian group]], the extensions of $G$ by $A$ are classified by the degree-2 [[group cohomology]]
$$
H^2_{Grp}(G,A) = H^2(\mathbf{B}G, A) = H(\mathbf{B}G, \mathbf{B}^2 A)
\,.
$$
On [[cocycles]] $\mathbf{B}G \to \mathbf{B}^2 A$ there is a canonical addition operation coming from the additive structure of $A$, and the Baer sum is the corresponding operation on the extensions that these cocycles classify.
## Definition
Below are discussed several different equivalent ways to define the Baer sum
### On concrete cocycles
A [[cocycle]] in degree-2 [[group cohomology]] $H^2_{Grp}(G,A)$ is a [[function]]
$$
c : G \times G \to A
$$
satisfying the cocycle property.
+-- {: .num_defn }
###### Definition
Given two cocycles $c_1, c_2 : G \times G \to A$ their **sum** is the composite
$$
(c_1 + c_2) : G \times G \stackrel{\Delta_{G \times G}}{\to} (G \times G) \times (G \times G) \stackrel{(c_1,c_2)}{\to} A \times A \stackrel{+}{\to} A
$$
of
* the [[diagonal]] on $G\times G$;
* the [[direct product]] $(c_1,c_2)$;
* the group operation $+ \colon A \times A \to A$.
Hence for all $g_1, g_2 \in G$ this sum is the function that sends
$$
(c_1 + c_2) : (g_1, g_2) \mapsto c_1(g_1,g_2) + c_2(g_1, g_2)
$$
=--
### On abstract cocycles
As discussed at [[group cohomology]], a cocycle $c \colon G \times G \to A$ is equivalently a morphism of [[2-groupoids]] from the [[delooping]] [[groupoid]] $\mathbf{B}G$ of $G$ to the double-delooping [[2-groupoid]] $\mathbf{B}^2 A$ of $A$:
$$
c_1,c_2 : \mathbf{B}G \to \mathbf{B}^2 A
\,.
$$
Since $A$ is an abelian group, $\mathbf{B}^2 A$ is naturally an abelian [[infinity-group|3-group]], equipped with a group operation $+ \colon (\mathbf{B}^2 A) \times (\mathbf{B}^A)\to \mathbf{B}^2 A $.
With respect to this the sum operation is
$$
c_1 + c_2 : \mathbf{B}G \stackrel{\Delta_{\mathbf{B}G}}{\to} \mathbf{B}G \times \mathbf{B}G \stackrel{(c_1,c_2)}{\to} \mathbf{B}^2 A \times \mathbf{B}^2 A \stackrel{+}{\to} \mathbf{B}^2 A
$$
### On short exact sequences
In any category with products, for any object $C$ there is a [[diagonal]] morphism
$\Delta_C:C\to C\times C$; in a category with coproducts there is a codiagonal morphism $\nabla_C: C\coprod C\to C$ (addition in the case of modules). Every additive category is, in particular, a category with finite [[biproduct]]s, so both morphisms are there. Short exact sequences in
the category of $R$-modules, or in arbitrary abelian category $\mathcal{A}$,
form an additive category (morphisms are commutative ladders of arrows) in which the biproduct
$0 \to A_i \to \hat H_{i} \to G_i \to 0$ for $i = 1,2$ is
$0\to A_1\oplus A_2 \to H_1\oplus H_2\to G_1\oplus G_2\to 0$.
Now if $0\to M\to N\to P\to 0$ is any extension, call it $E$,
and $\gamma:P_1\to P$ a morphism,
then there is a morphism $\Gamma' = (id_M,\beta_1,\gamma)$ from an
extension $E_1$ of the form $0\to M\to N_1\to P_1\to 0$ to $E$, where the pair $(E_1,\Gamma_1)$ is unique up to isomorphism of extensions,
and it is called $E\gamma$. In fact, the diagram
$$\array{
N_1&\to &P_1\\
\downarrow\beta_1 && \downarrow\gamma\\
N&\to &P
}$$
is a pullback diagram. Every morphism of abelian extensions $(\alpha,\beta,\gamma):E\to E'$ in a unique way decomposes as
$$
E\stackrel{(\alpha,\beta_a,id)}\longrightarrow E\gamma \stackrel{(id,\beta_ 1,\gamma)}\longrightarrow E'
$$
for some $\beta_a$ with $\beta_1$ as above. In short, the morphism of extensions factorizes through $E\gamma$.
Dually, for any morphism $\alpha:M\to M_2$, there is a morphism $\Gamma_2 = (\alpha,\beta_2,id_P)$ to an extension $E_2$ of the form
$0\to M_2\to N_2\to P$; the pair $(E_2,\Gamma_2)$ is unique up to isomorphism of
extensions and it is called $\alpha E$.
In fact, the diagram
$$\array{
M&\to &N\\
\downarrow\alpha && \downarrow\beta_2\\
M_2&\to &N_2
}$$
is a pushout diagram. Every morphism of abelian extensions $(\alpha,\beta,\gamma):E\to E''$ in a unique way decomposes as
$$
E\stackrel{(\alpha,\beta_a,id)}\longrightarrow \alpha E \stackrel{(id,\beta_ 2,\gamma)}\longrightarrow E''
$$
for some $\beta_a$, with $\beta_2$ as above. In short, the morphism of extensions factorizes through $\alpha E$.
There are the following isomorphisms of extensions: $(\alpha E)\gamma\cong \alpha (E\gamma)$, $id_M E \cong E$, $E id_P \cong P$,
$(\alpha'\alpha)E\cong\alpha' (\alpha E)$, $(E\gamma)\gamma' \cong E(\gamma\gamma')$.
The Baer's sum of two extensions $E_1,E_2$ of the form $0\to M\to N_i\to P\to 0$
(i.e. with the same $M$ and $P$) is given by $E_1+E_2 = \nabla_M (E_1\oplus E_2) \Delta_P$; this gives the structure of the abelian group on $Ext(P,M)$ and $Ext:\mathcal{A}^{op}\times\mathcal{A}\to Ab$ is a biadditive (bi)functor. This is also related to the isomorphisms of extensions
$\alpha (E_1+E_2)\cong \alpha E_1+\alpha E_2$, $(\alpha_1+\alpha_2) E \cong \alpha_1 E+ \alpha_2 E$, $(E_1+E_2)\gamma \cong E_1\gamma + E_2\gamma$, $E(\gamma_1+\gamma_2)\cong E\gamma_1 + E\gamma_2$.
In different notation, if $0 \to A \to \hat G_{i} \to G \to 0$ for $i = 1,2$
are two [[short exact sequences]] of [[abelian groups]], their **Baer sum** is
$$
\hat G_1 + \hat G_2
\coloneqq
+_* \Delta^* \hat G_1 \times \hat G_2
$$
The first step forms the [[pullback]] of the short exact sequence along the diagonal on $G$:
$$
\array{
A \oplus A &\to& A \oplus A
\\
\downarrow && \downarrow
\\
\Delta^* (\hat G_1 \oplus \hat G_2) &\to& \hat G_1 \oplus \hat G_2
\\
\downarrow && \downarrow
\\
G &\stackrel{\Delta_G}{\to}& G\oplus G
}
$$
The second forms the [[pushout]] along the addition map on $A$:
$$
\array{
A \oplus A &\stackrel{+}{\to}& A
\\
\downarrow && \downarrow
\\
\Delta^* (\hat G_1 \oplus \hat G_2) &\to& +_* \Delta^*(\hat G_1 \oplus \hat G_2)
\\
\downarrow && \downarrow
\\
G &\to& G
}
$$
## Related concepts
* [[cup product]]
## References
* S. MacLane, _Homology_, 1963
* Patrick Morandi, _Ext groups and Ext functors_ ([pdf](http://sierra.nmsu.edu/morandi/oldwebpages/math683fall2002/Ext.pdf))
[[!redirects Baer sums]]
[[!redirects Baer's sum]]
|
Baer's criterion | https://ncatlab.org/nlab/source/Baer%27s+criterion |
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###Context###
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
_Baer's criterion_ is a criterion for detecting [[injective objects]] in a [[category]] of [[modules]]: [[injective modules]].
## Statement
Let $R$ be a [[ring]] and $C = R $[[Mod]] the category of $R$-[[modules]].
+-- {: .num_theorem}
###### Proposition
**([[Baer's criterion]])**
{#Baer}
An [[object]] $Q \in R Mod$ is [[injective object|injective]] precisely if for $I$ any left $R$-[[ideal]] regarded as an $R$-[[module]], any [[morphism]] $g : I \to Q$ in $C$ can be extended to all of $R$ along the inclusion $I \hookrightarrow R$.
=--
+-- {: .proof}
###### Sketch of proof
Let $i \colon M \hookrightarrow N$ be a mono in $R Mod$, and let $f \colon M \to Q$ be a map. We must extend $f$ to a map $h \colon N \to Q$. Consider the poset whose elements are pairs $(M', f')$ where $M'$ is an intermediate submodule between $M$ and $N$ and $f' \colon M' \to Q$ is an extension of $f$, ordered by $(M', f') \leq (M'', f'')$ if $M''$ contains $M'$ and $f''$ extends $f'$. By an application of [[Zorn's lemma]], this poset has a maximal element, say $(M', f')$. Suppose $M'$ is not all of $N$, and let $x \in N$ be an element not in $M'$; we show that $f'$ extends to a map $M'' = \langle x \rangle + M' \to Q$, contradiction.
The set $\{r \in R: r x \in M'\}$ is an ideal $I$ of $R$, and we have a module map $g \colon I \to Q$ defined by $g(r) = f'(r x)$. By hypothesis, we may extend $g$ to a module map $k \colon R \to Q$. Writing a general element of $M''$ as $r x + y$ where $y \in M'$, it may be shown that
$$f''(r x + y) = k(r) + f'(y)$$
is well-defined and extends $f'$, as desired.
=--
## Consequences
+-- {: .num_cor}
###### Corollary
{#DirectSumInjectives}
Let $R$ be a Noetherian ring, and let $\{Q_j\}_{j \in J}$ be a collection of injective modules over $R$. Then the direct sum $Q = \bigoplus_{j \in J} Q_j$ is also injective.
=--
+-- {: .proof}
###### Proof
By [Baer's criterion](#Baer), it suffices to show that for any ideal $I$ of $R$, a module map $f \colon I \to Q$ extends to a map $R \to Q$. Since $R$ is Noetherian, $I$ is finitely generated as an $R$-module, say by elements $x_1, \ldots, x_n$. Let $p_j \colon Q \to Q_j$ be the projection, and put $f_j = p_j \circ f$. Then for each $x_i$, $f_j(x_i)$ is nonzero for only finitely many summands. Taking all of these summands together over all $i$, we see that $f$ factors through
$$\prod_{j \in J'} Q_j = \bigoplus_{j \in J'} Q_j \hookrightarrow Q$$
for some finite $J' \subset J$. But a product of injectives is injective, hence $f$ extends to a map $R \to \prod_{j \in J'} Q_j$, which completes the proof.
=--
Conversely, a result of Bass and Papp is that $R$ is Noetherian if direct sums of injective $R$-modules are injective. See [Lam](#Lam), Theorem 3.46.
## References
* {#Lam} T.-Y. Lam, _Lectures on modules and rings_, Graduate Texts in Mathematics 189, Springer Verlag (1999).
|
Baez-Crans 2-vector space | https://ncatlab.org/nlab/source/Baez-Crans+2-vector+space | These are discussed in [[2-vector space]].
[[!redirects Baez--Crans 2-vector space]]
[[!redirects Baez–Crans 2-vector space]]
[[!redirects Baez--Crans 2-vector spaces]]
[[!redirects Baez–Crans 2-vector spaces]]
[[!redirects Baez-Crans 2-vector spaces]] |
bagdomain topos | https://ncatlab.org/nlab/source/bagdomain+topos |
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###Context###
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
+ automatic table of contents goes here
{:toc}
## Idea
The **bagdomain topos construction** is the topos analogue of the powerdomain construction in domain-theoretic database theory. Roughly speaking, an object $X$ is replaced by an object $X'$ such that points of $X'$ correspond to sets (='bags') of points of $X$.
## Related entries
* [[symmetric topos]]
* [[Artin gluing]]
* [[lax-idempotent 2-monad]]
## References
* [[Marta Bunge]], [[Jonathon Funk]], _Spreads and the Symmetric Topos_ , JPAA **113** (1996) pp.1-38.
* [[Marta Bunge]], [[Jonathon Funk]], _Spreads and the Symmetric Topos II_ , JPAA **130** (1998) pp.49-84.
* [[Marta Bunge]], [[Jonathon Funk]], _Singular Coverings of Toposes_ , Springer LNM **1890** Heidelberg 2006.
* M. Jibladze, _Lower Bagdomain as a Glueing_ , Proc. A. Razmadze Math. Inst. **118** (1998) pp.33-41. ([pdf](http://www.rmi.ge/~jib/baglue.pdf))
* [[Peter Johnstone]], _Partial Products, Bagdomains and Hyperlocal Toposes_ , pp.315-339 in Fourman, Johnstone, Pitts (eds.), _Applications of Categories in Computer Science - Proceedings of the LMS Symposium Durham 1991_ , Cambridge UP 1992.
* [[Peter Johnstone]], _Variations on the Bagdomain Theme_ , Theo. Comp. Sci. **136** (1994) pp.3-20.
* [[Peter Johnstone]], _Sketches of an Elephant I_ , Oxford UP 2002. (sec. B4.4, pp.448-456) {#elephant}
* [[Steve Vickers]], _Geometric Theories and Databases_ , pp.288-314 in Fourman, Johnstone, Pitts (eds.), _Applications of Categories in Computer Science - Proceedings of the LMS Symposium Durham 1991_ , Cambridge UP 1992. ([preprint](http://www.cs.bham.ac.uk/~sjv/GeoTh+DBs.pdf)) |
Bai-Ling Wang | https://ncatlab.org/nlab/source/Bai-Ling+Wang |
* [website](http://wwwmaths.anu.edu.au/~wangb/)
## Selected writings
On [[twisted K-theory]]:
* [[Alan Carey]], [[Bai-Ling Wang]], p. 5 of: *Thom isomorphism and Push-forward map in twisted K-theory*, Journal of K-Theory **1** 2 (2008) 357-393 ([arXiv:math/0507414](https://arxiv.org/abs/math/0507414), [doi:10.1017/is007011015jkt011](https://doi.org/10.1017/is007011015jkt011))
On [[twisted differential K-theory]]:
* [[Alan Carey]], [[Jouko Mickelsson]], [[Bai-Ling Wang]], _Differential Twisted K-theory and Applications_, Journal of Geometry and Physics, Volume 59, Issue 5, May 2009, Pages 632-653 ([arXiv:0708.3114](https://arxiv.org/abs/0708.3114), [doi:10.1016/j.geomphys.2009.02.002](https://doi.org/10.1016/j.geomphys.2009.02.002))
On [[orbifold K-theory]]:
* Jianxun Hu, [[Bai-Ling Wang]], _Delocalized Chern character for stringy orbifold K-theory_, Trans. Amer. Math. Soc. 365 (2013), 6309-6341 ([arXiv:1110.0953](https://arxiv.org/abs/1110.0953), [doi:10.1090/S0002-9947-2013-05834-5](https://doi.org/10.1090/S0002-9947-2013-05834-5))
category: people |
Baire category theorem | https://ncatlab.org/nlab/source/Baire+category+theorem |
## Idea
The _Baire category theorem_ states sufficient conditions for a [[topological space]] to be a _[[Baire space]]_.
## References
* Wikipedia, _[Baire category theorem](https://en.wikipedia.org/wiki/Baire_category_theorem)_
[[!redirects Baire category theorems]] |
Baire lattice | https://ncatlab.org/nlab/source/Baire+lattice |
+-- {: .rightHandSide}
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### Context
#### $(0,1)$-Category theory
+-- {: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Baire lattices are [[lattice|lattice-theoretic]] abstraction of the [[Baire space|Baire property]] among [[topological spaces]]. In parallel to the fact that every [[complete metric space]] is a Baire space every [[continuous poset|continuous lattice]] is a Baire lattice.
## Definitions
Recall that a [[complete lattice]] is a [[partial order|poset]] which has all small [[join|joins]] and [[meet|meets]]. For $a, b \in L$, we say that $a$ is __way below__ $b$ and write $a \ll b$ if whenever $S \subseteq L$ is a [[directed subset]] and $b \leq \bigvee S$ (where $\bigvee S$ denotes the [[join]] of $S$), then there exists $s \in S$ with $a \leq s$. Further we say that $L$ is __[[continuous lattice|continuous]]__ if for every $a\in L$, the subset
$$ \Downarrow (a) \coloneqq \{ b \in L | b \ll a \} $$
is directed and has join $a$.
For example the [[category of open subsets|lattice of open subsets]] of a topological space is a continuous lattice if and only if the [[sobrification]] of the topological space is [[locally compact]] (i.e. the topology has a basis of compact neighborhoods).
\begin{defn}
An element $p \in L$ is called __irreducible__ if $a \vee b = p$ implies $p = a$ or $p = b$.
\end{defn}
To illustrate this definition think of an [[irreducible topological space|irreducible subset]] of a topological space.
\begin{defn}
An element $d\in L$ is __dense__ if for all $a\in L$ the relation $ a \neq \bot $ implies that $ d \wedge a \neq \bot $.
\end{defn}
\begin{defn}
A [[complete lattice]] $L$ is called a __Baire lattice__ if for any countable family of dense elements $N \subset L$ and each nonzero element $u \in L$ there is an irreducible element $p \in L$ such that $ d \wedge u \nleq p $ for all $d \in N$.
\end{defn}
## Theorem
\begin{theorem}
Every continuous lattice is a Baire lattice.
\end{theorem}
## Example
Let $X$ be a [[sober space|sober]] [[topological space]]. Then the lattice of opens of $X$ is Baire if and only if the topological space $X$ has the [[Baire space|Baire property]].
## Related pages
* [[complete lattice]]
* [[Baire category theorem]]
* [[Baire space]]
## References
The concept appeared in:
* [[Karl Heinrich Hofmann]], _A note on Baire spaces and continuous lattices_ 1980. Bulletin of the Australian Mathematical Society, 21(2), pp. 265-279. ([doi:10.1017/S0004972700006080]( https://doi.org/10.1017/S0004972700006080))
Textbook accounts:
* G. Gierz, [[Karl Heinrich Hofmann]], K. Keimel, J. D. Lawson, [[Michael Mislove]], [[Dana Scott]], _Continuous Lattices and Domains_ 2003, Vol. 93 of _Encyclopedia of Mathematics and its Applications_ ([doi:10.1017/CBO9780511542725](https://doi.org/10.1017/CBO9780511542725))
[[!redirects Baire lattices]]
|
Baire space | https://ncatlab.org/nlab/source/Baire+space |
> This entry is about the class of [[topological spaces]] satisfying the Baire category theorem. For *the* Baire space used in computable analysis, descriptive set theory, etc, see instead at _[[Baire space of sequences]]_.
# Baire spaces
* table of contents
{: toc}
## Idea
A Baire space is a [[topological space]] that satisfies the conclusion of the [[Baire category theorem]].
It should not be confused with the [[Baire space of sequences]] (which is an example of a Baire space in our sense but not a prominent one). Nor should it be confused with a [[Baire set]] (a [[subset]] somewhat analogous to a [[measurable set]] but defined by a topological property).
## Definition
A __Baire space__ is a [[topological space]] such that the [[intersection]] of any [[countable family]] of [[dense subspace|dense]] [[open subspace|open]] subspaces is also dense. Equivalently: a space such that a countable union of closed sets each with [[empty set|empty]] [[interior]] also has empty interior.
## Examples
The classical [[Baire category theorem]] states that:
* Any [[complete metric space]] (or rather its [[forgetful functor|underlying]] topological space, hence any completely metrizable topological space) is a Baire space.
A second theorem, sometimes dubbed "BCT2" (as in Wikipedia):
* Any [[locally compact Hausdorff space]], or indeed any locally compact [[sober space]] according to Wikipedia, is a Baire space. Moreover, any [[G-delta set]] of a locally compact Hausdorff space is a Baire space under the [[subspace]] topology.
Furthermore:
* Any [[open subspace]] of a Baire space is also a Baire space.
* Given a Baire space $X$, a _dense_ $G_\delta$ set in $X$ (i.e. a countable intersection of dense opens) is a Baire space under the subspace topology. See Dan Ma's blog, specifically Theorem 3 [here](https://dantopology.wordpress.com/2012/06/02/a-question-about-the-rational-numbers/).
* As mentioned above, the space of [[infinite sequences]] of [[natural numbers]], or equivalently (up to topology) the space [[irrational numbers]], is also known as '[[Baire space of sequences|Baire space]]'. It is a Baire space in the present sense (since it admits a complete metric), but the coincidence of names appears to be just a coincidence. (It is much more important that Baire space is a [[Polish space]] than that Baire space is a Baire space. Of course, every Polish space is a Baire space too.)
[[!redirects Baire space]]
[[!redirects Baire spaces]]
|
Baire space of sequences | https://ncatlab.org/nlab/source/Baire+space+of+sequences |
> This entry is about *the* Baire space of sequences of natural numbers. For another concept of the same name in [[topology]] proper, see at _[[Baire space]]_.
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### Context
#### Constructivism, Realizability, Computability
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[[!include constructivism - contents]]
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=--
# Contents
* table of contents
{: toc}
## Idea
In the study of [[computability]], [[descriptive set theory]], etc, by _Baire space_ is meant the [[topological space]] $\mathbb{N}^{\mathbb{N}}$ of infinite sequences of [[natural numbers]] equipped with the [[product topology]].
The [[continuous functions]] from Baire space to itself serve the role of _computable functions_ in [[computable analysis]]. See at _[[computable function (analysis)]]_.
## Related concepts
* [[irrational number]]
[[!include computable mathematics -- table]]
## References
Lecture notes include
* {#Bauer05} [[Andrej Bauer]], page 5 and section 5.3.1 of _Realizability as connection between constructive and computable mathematics_, in T. Grubba, P. Hertling, H. Tsuiki, and [[Klaus Weihrauch]], (eds.) _CCA 2005 - Second International Conference on Computability and Complexity in Analysis_, August 25-29,2005, Kyoto, Japan, ser. Informatik Berichte, , vol. 326-7/2005. FernUniversität Hagen, Germany, 2005, pp. 378–379. ([pdf](http://math.andrej.com/data/c2c.pdf))
Textbook accounts include
* {#Weihrauch00} [[Klaus Weihrauch]], section 2.3 of _Computable Analysis_ Berlin: Springer, 2000
[[!redirects Baire space (computability)]]
[[!redirects Baire space (set theory)]]
[[!redirects Baire space (descriptive set theory)]]
[[!redirects Baire space of sequences]]
[[!redirects Baire space of irrational numbers]]
[[!redirects set-theoretic Baire space]]
|
Baire subset | https://ncatlab.org/nlab/source/Baire+subset |
# Contents
* table of contents
{: toc}
## Idea
Baire sets are certain [[subsets]] of a [[topological space]]. They form the Baire $\sigma$-[[sigma-algebra|algebra]] of the space, and they play an important role in [[measure theory]].
## Definition
Let $X$ be a [[topological space]]. Then there is a $\sigma$-[[sigma-algebra|algebra]] $\mathcal{B}$ on $X$ generated by the [[open subsets]] of $X$ that are preimages of $(0,\infty)$ under some continuous map $X\to\mathbf{R}$. Elements of $\mathcal{B}$ are called the __Baire sets__ (or __Baire subsets__, or __Baire-measurable sets__, etc) of $X$, and $\mathcal{B}$ itself is called the __Baire $\sigma$-algebra__ on $X$.
## Properties
The Baire $\sigma$-algebra is a $\sigma$-subalgebra of the [[Borel sigma-algebra|Borel σ-algebra]] since every continuous map is [[measurable function|measurable]]. Often both σ-algebras even coincide.
This holds for [[perfectly normal spaces]],
such as [[metric spaces]] or [[regular]] [[hereditary property|hereditary]] [[Lindelöf topological space|Lindelöf]] spaces.
When working with [[locally compact topological space|locally compact spaces]] one can often instead use the following fact ([Dudley, Theorem 7.3.1](#Dudley02)):
\begin{theorem}
Let $ K $ be a compact Hausdorff space and $ \mu $ any finite Baire measure thereon. Then $ \mu $ has a unique extension to a regular Borel measure on $ K $.
\end{theorem}
## Related concepts
* [[Borel set]]
* [[Baire measure]]
## References
* {#Dudley02} Dudley, _Real analysis and probability_, 2002.
[[!redirects Baire subset]]
[[!redirects Baire subsets]]
[[!redirects Baire set]]
[[!redirects Baire sets]]
[[!redirects Baire-measurable subset]]
[[!redirects Baire-measurable subsets]]
[[!redirects Baire-measurable set]]
[[!redirects Baire-measurable sets]]
[[!redirects Baire sigma-algebra]]
[[!redirects Baire sigma-algebras]]
[[!redirects Baire ∞-algebra]]
[[!redirects Baire ∞-algebras]]
|
balanced category | https://ncatlab.org/nlab/source/balanced+category | +-- {: .rightHandSide}
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###Context###
#### Category theory
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[[!include category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
\begin{definition}\label{BalancedCategory}
A category is **balanced** if every [[morphism]] which is both a [[monomorphism]] and an [[epimorphism]] is already an [[isomorphism]].
\end{definition}
\begin{remark}
The possibility of monic epics that are not isomorphisms does not survive any strengthening of "monic" or "epic." Any monic [[extremal epimorphism]] is necessarily an isomorphism, and therefore so is any monic [[strong epimorphism]] or [[regular epimorphism]] (and dually). It follows that if all epics, or all monos, are extremal, then the category is automatically balanced.
In an "unbalanced" category it is frequently the case that the monomorphisms, the epimorphisms, or both, are not the "right" notion to consider and should be replaced by their [[extremal epimorphism|extremal]], [[strong epimorphism|strong]], or [[regular epimorphism|regular]] counterparts.
\end{remark}
## Examples and non-examples
{#Examples}
\begin{example}\label{SetIsBalanced}
The category [[Set]] is balanced (Def. \ref{BalancedCategory}).
\end{example}
More generally:
\begin{example}
Any [[topos]] and in fact any [[pretopos]] is balanced.
\end{example}
(eg. [Johnstone 1977, Cor. 1.22](#Johnstone77))
\begin{remark}
Beware the [[counterexample]]:
A [[quasitopos]], need *not* be balanced.
\end{remark}
\begin{example}\label{AbelianCategoriesAreBalanced}
Any [[abelian category]] is balanced.
In particular categories [[Mod]] of [[modules]] and [[Vect]] of [[vector spaces]] are balanced.
\end{example}
\begin{remark}
An [[additive category]] need not be balanced: A [[counterexample]] is the category of [[torsion subgroup]]-free abelian groups, each nonzero homomorphism $\mathbb{Z} \to \mathbb{Z}$ is both monic and epic.
\end{remark}
\begin{example}
The [[Grp|the category of groups]] is balanced (see [here](regular+monomorphism#Examples) and [here](Grp#eq)).
\end{example}
\begin{remark}
Not all categories of [[algebra|algebraic]] [[structures]] are balanced.
As a [[counterexample]],
the [[category of rings]] is not balanced: $\mathbb{Z}\hookrightarrow \mathbb{Q}$ is monic and epic but not an isomorphism.
On similar grounds, the [[category of commutative monoids]] is not balanced, as the inclusion $\mathbb{N} \hookrightarrow \mathbb{Z}$ is both monic and epic.
\end{remark}
\begin{remark}
Topological categories are rarely balanced. In [[Top]], for example, the monic epimorphisms are the continuous bijections.
\end{remark}
However:
\begin{example}
The category of [[compact Hausdorff spaces]] is balanced.
\end{example}
\begin{remark}
In a [[free category]] on a [[directed graph]], and also in any
[[partial order|poset]] and generally in any _[[thin category]]_, _every_ morphism is both monic and epic while only the [[identity morphisms]] are invertible; thus such categories are "as far as possible from being balanced."
\end{remark}
## References
* {#Johnstone77} [[Peter Johnstone]], Cor. 1.22 in: _Topos theory_, London Math. Soc. Monographs __10__, Acad. Press (1977), Dover (2014)
* [[Roy L. Crole]], p. 115 of: *Categories for types*, Cambridge University Press (1994) [[doi:10.1017/CBO9781139172707](https://doi.org/10.1017/CBO9781139172707)]
[[!redirects balanced categories]] |
balanced monoidal category | https://ncatlab.org/nlab/source/balanced+monoidal+category |
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###Context###
#### Monoidal categories
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[[!include monoidal categories - contents]]
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# Balanced monoidal categories#
* table of contents
{:toc}
## Definition
A __twist__, or __balance__, in a [[braided monoidal category]] $\mathscr{C}$ is a [[natural transformation | natural isomorphism]] $\theta$ from the [[identity functor]] on $\mathscr{C}$ to itself satisfying the following compatibility condition with the [[braiding]] $\beta$:
$$
\theta_{A\otimes B}
\;=\;
\beta_{B,A}
\circ
\beta_{A,B}
\circ
(\theta_A \otimes \theta_B),
\,\, \forall A, B \in \mathscr{C}
$$
A __balanced monoidal category__ is a braided monoidal category equipped with such a balance.
Beware that there is an un-related notion of *[[balanced categories]]*.
## Properties
In the language of [[string diagrams]], the balancing is represented by a 360-degree twist:
\begin{imagefromfile}
"file_name": "graphical-twist.png",
"width": 300,
"unit": "px"
\end{imagefromfile}
Every [[symmetric monoidal category]] is balanced in a canonical way. In fact, the [[identity natural transformation]] on the identity functor of $\mathscr{C}$ is a balance on $\mathscr{C}$. In this way, the twist can be seen as a way of "controlling" the non-symmetric behavior of the braiding.
A braided [[rigid monoidal category]] is balanced if and only if it is a [[pivotal category]], but a balanced monoidal category need not be rigid (cf. [Selinger 2011, Lem. 4.20](#Selinger11)).
## References
The original definition:
* [[André Joyal]], [[Ross Street]], *The geometry of tensor calculus I*, Adv. Math. __88__ 1 (1991) 55--112, [<a href="https://doi.org/10.1016/0001-8708(91)90003-P">doi:10.1016/0001-8708(91)90003-P</a>]
The above definition follows:
* [[Jeff Egger]], Appendix C in: *Of Operator Algebras and Operator Spaces* (2006) [[pdf](http://www.mscs.dal.ca/~jegger/4micah.pdf)]
See also:
* {#Selinger11} [[Peter Selinger]], *A survey of graphical languages for monoidal categories*,
Springer Lecture Notes in Physics **813** (2011) 289-355 [[arXiv:0908.3347](https://arxiv.org/abs/0908.3347), [doi:10.1007/978-3-642-12821-9_4](https://doi.org/10.1007/978-3-642-12821-9_4)]
[[!redirects balancing]]
[[!redirects balance]]
[[!redirects balanced monoidal category]]
[[!redirects balanced monoidal categories]]
|
balanced set | https://ncatlab.org/nlab/source/balanced+set |
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### Context
#### Functional analysis
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[[!include functional analysis - contents]]
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#Contents#
* table of contents
{:toc}
## Definition
A subset $B$ of a [[vector space]] $V$ over $\mathbb{R}$ or $\mathbb{C}$ is **balanced** if $\lambda x \in B$ whenever $x \in B$ and ${|\lambda|} \le 1$.
## Properties
The intersection of two balanced sets is balanced. The [[closure]] of a balanced set is again balanced.
## Examples
* The [[unit ball]] of a [[seminorm|seminormed]] space is balanced
* Any subspace is balanced
# Related concepts
* [[convex set]]
* [[absolutely convex set]]
* [[norm]]
* [[absorbing set]]
[[!redirects balanced sets]] |
ball | https://ncatlab.org/nlab/source/ball |
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### Context
#### Topology
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[[!include topology - contents]]
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#### Differential geometry
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[[!include synthetic differential geometry - contents]]
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# Contents
* table of contents
{: toc}
## Definition
### Geometric
For $n \in \mathbb{N}$ a [[natural number]], the $n$-[[dimension|dimensional]] **ball** or **$n$-disk** in $\mathbb{R}^n$ is the [[topological space]]
$$
D^n := \{ \vec x \in \mathbb{R}^n | \sum_{i} (x^i)^2 \leq 1\}
\subset \mathbb{R}^n
$$
equipped with the [[induced topology]] as a subspace of the [[Cartesian space]] $\mathbb{R}^n$.
Its [[interior]] is the **open $n$-ball**
$$
\mathbb{B}^n := \{ \vec x \in \mathbb{R}^n | \sum_{i} (x^i)^2 \lt 1 \}
\subset \mathbb{R}^n
\,.
$$
Its [[boundary]] is the $(n-1)$-[[sphere]].
More generally, for $(X,d)$ a [[metric space]] then an open ball in $X$ is a subset of the form
$$
B(x,r) \coloneqq \{y \in X \;|\; d(x,y) \lt r \}
$$
for $x \in X$ and $r \in (0,\infty) \subset \mathbb{R}$.
(The collection of all open balls in $X$ form the [[basis of a topology|basis]] of the [[metric topology]] on $X$.)
### Combinatorial
There are also combinatorial notions of _disks_. For instance that due to ([Joyal](#Joyal)), as entering the definition of the [[Theta-category]]. See for instance ([Makkai-Zawadowski](#MakkaiZawadowski)).
## Properties
{#Properties}
### Closed balls
A simple result on the _homeomorphism_ type of _closed_ balls is the following:
+-- {: .num_theorem}
###### Theorem
A [[compact space|compact]] [[convex subset|convex]] [[subset]] $D$ in $\mathbb{R}^n$ with [[nonempty set|nonempty]] [[interior]] is [[homeomorphic]] to $D^n$.
=--
+-- {: .proof}
###### Proof
Without loss of generality we may suppose the origin is an interior point of $D$. We claim that the map $\phi: v \mapsto v/{\|v\|}$ maps the boundary $\partial D$ homeomorphically onto $S^{n-1}$. By convexity, $D$ is homeomorphic to the cone on $\partial D$, and therefore to the cone on $S^{n-1}$ which is $D^n$.
The claim reduces to the following three steps.
1. The restricted map $\phi: \partial D \to S^{n-1}$ is continuous.
1. It's surjective: $D$ contains a ball $B = B_{\varepsilon}(0)$ in its interior, and for each $x \in B$, the positive ray through $x$ intersects $D$ in a bounded half-open line segment. For the extreme point $v$ on this line segment, $\phi(v) = \phi(x)$. Thus every unit vector $u \in S^{n-1}$ is of the form $\phi(v)$ for some extreme point $v \in D$, and such extreme points lie in $\partial D$.
1. It's injective: for this we need to show that if $v, w \in \partial D$ are distinct points, then neither is a positive multiple of the other. Supposing otherwise, we have $w = t v$ for $t \gt 1$, say. Let $B$ be a ball inside $D$ containing $0$; then the convex hull of $\{w\} \cup B$ is contained in $D$ and contains $v$ as an interior point, contradiction.
So the unit vector map, being a continuous bijection $\partial D \to S^{n-1}$ between [[compact Hausdorff space]]s, is a homeomorphism.
=--
+-- {: .num_cor}
###### Corollary
Any compact convex set $D$ of $\mathbb{R}^n$ is homeomorphic to a disk.
=--
+-- {: .proof}
###### Proof
$D$ has nonempty interior relative to its affine span which is some $k$-plane, and then $D$ is homeomorphic to $D^k$ by the theorem.
=--
### Open Balls
Open balls are a little less rigid than closed balls, in that one can more easily manipulate them within the _smooth_ category:
+-- {: .num_lemma}
###### Observation
The open $n$-ball is [[homeomorphic]] and even [[diffeomorphic]] to the [[Cartesian space]] $\mathbb{R}^n$
$$
\mathbb{B}^n \simeq \mathbb{R}^n
\,.
$$
=--
+-- {: .proof}
###### Proof
For instance, the smooth map
$$
x\mapsto \frac{x}{\sqrt{1+|x|^2}} : \mathbb{R}^n \to \mathbb{B}^n
$$
has smooth inverse
$$
y\mapsto \frac{y}{\sqrt{1-|y|^2}} : \mathbb{B}^n \to \mathbb{R}^n.
$$
=--
This probe from ${\mathbb{R}}^n$ witnesses the property that the open $n$-ball is a ([[smooth manifold|smooth]]) [[manifold]]. Hence, each (smooth) $n$-dimensional manifold is locally isomorphic to both ${\mathbb{R}}^n$ and $\mathbb{B}^n$.
From general existence results about [[smooth structure]]s on [[Cartesian space]]s we have that
+-- {: .num_theorem}
###### Theorem
In [[dimension]] $d \in \mathbb{N}$ for $d \neq 4$ we have:
every open subset of $\mathbb{R}^d$ which is [[homeomorphic]] to $\mathbb{B}^d$ is also [[diffeomorphic]] to it.
=--
See the first page of ([Ozols](#Ozols)) for a list of references.
+-- {: .num_remark}
###### Remark
In dimension 4 the analog statement fails due to the existence of [[exotic smooth structure]]s on $\mathbb{R}^4$. See [De Michelis-Freedman](#DeMFreed).
=--
+-- {: .num_theorem #StarShapedOpenDiffeomorphicToOpenBall}
###### Theorem
**(star-shaped domains are diffeomorphic to open balls)**
Let $C \subset \mathbb{R}^n$ be a [[star-shaped]] [[open subset]] of a [[Cartesian space]]. Then $C$ is [[diffeomorphic]] to $\mathbb{R}^n$.
=--
+-- {: .num_remark #LiteratureOnStarShapedOpenDiffeoToOpenBall}
###### Remark
Theorem \ref{StarShapedOpenDiffeomorphicToOpenBall} is a [[folk theorem]], but explicit **proofs** in the literature are hard to find. See the discussion in the References-section *[here](#ReferencesStarShapedReasonDiffeomorphicToOpenBall)*. An explicit proof has been written out by Stefan Born, and this appears as the proof of [theorem 237](http://www.math.tu-berlin.de/~ferus/ANA/Ana3.pdf#page=154) in ([Ferus 07](#Ferus07)). A simpler proof is given in [Gonnord-Tosel 98](#GonnordTosel98) reproduced [here](http://mathoverflow.net/a/212595/381).
=--
Here is another proof:
+-- {: .proof}
###### Proof
Suppose $T$ is a star-shaped open subset of ${\mathbb {R}}^n$ centered at the origin. Theorem 2.29 in [Lee 2009](#Lee09) proves that there is a function $f$ on ${\mathbb{R}}^n$ such that $f\gt 0$ on $T$ and $f$ vanishes on the complement of $T$. By applying [[bump functions]] we can assume that $f\le 1$ everywhere and $f=1$ in an open $\epsilon$-neighborhood of the origin; by rescaling the ambient space we can assume $\epsilon=2$.
The smooth vector field $V\colon x\mapsto f(x)\cdot x/{\|x\|}$ is defined on the complement of the origin in $T$. Multiply $V$ by a smooth bump function $0\le b\le 1$ such that $b=1$ for ${\|x\|} \gt 1/2$ and $b=0$ in a neighborhood of 0.
The new vector field $V$ extends smoothly to the origin and defines a smooth global flow $F\colon \mathbb{R} \times T\to T$. (The parameter of the flow is all of $\mathbb{R}$ and not just some interval $(-\infty,A)$ because the norm of $V$ is bounded by 1.) Observe that for $1/2\lt {\|x\|} \lt 2$ the vector field $V$ equals $x\mapsto x/{\|x\|}$. Also, all flow lines of $V$ are radial rays.
Now define the flow map $p\colon{\mathbb{R}}^n_{\gt 1/2}\to T_{\gt 1/2}$ as $x\mapsto F({\|x\|}-1, \frac{x}{{\|x\|}})$ for ${\|x\|} \gt 1/2$. (The subscript $\gt 1/2$ removes the closed ball of radius $1/2$.) The flow map is the composition of two diffeomorphisms,
$${\mathbb{R}}^n_{\gt 1/2}\to(-1/2,\infty)\times S^{n-1} \to T_{\gt 1/2},$$
hence itself is a diffeomorphism. (Note particularly that the latter map is surjective. In detail: a flow line is a smooth map of the form $L: (A,B) \to T$, where $A$ and $B$ can be finite or infinite. If $B$ is finite and the limit of $L(t)$ as $t \to B$ exists, then the vector field $V$ vanishes at $B$. In our case $V$ can only vanish at the boundary of $T$, which is precisely what we want for surjectivity.)
Finally, define the desired diffeomorphism $d\colon{\mathbb{R}}^n\to T$ as the gluing of the identity map for ${\|x\|} \lt 2$ and as $p$ for ${\|x\|}\gt 1/2$.
The map $g$ is smooth because for $1/2\lt {\|x\|} \lt 2$ both definitions give the same value.
=--
And here is another proof, due to Gonnord and Tosel, translated into English by Erwann Aubry and available on MathOverflow:
\begin{theorem}
Every open star-shaped set $\Omega$ in $\mathbb{R}^n$ is $C^\infty$-diffeomorphic to $\mathbb{R}^n$.
\end{theorem}
\begin{proof}
For convenience assume that $\Omega$ is star-shaped at $0$.
Let $F=\mathbf{R}^n\setminus\Omega$ and $\phi:\mathbf{R}^n\rightarrow\mathbb{R}_+$ (here $\mathbf{R}_+=[0,\infty)$) be a $C^\infty$-function such that $F=\phi^{-1}(\{0\})$.
(Such $\phi$ exists by the [[Whitney extension theorem]].)
Now we define $f:\Omega\rightarrow\mathbb{R}^n$ via the formula:
$$f(x)=\overbrace{\left[1+\left(\int_0^1\frac{dv}{\phi(vx)}\right)^2\|x\|^2\right]}^{\lambda(x)}\cdot x=\left[1+\left(\int_0^{\|x\|}\frac{dt}{\phi(t\frac{x}{\|x\|})}\right)^2\right]\cdot x.$$
Clearly $f$ is smooth on $\Omega$.
We set $A(x)=\sup\{t\gt0\mid t\frac{x}{\|x\|}\in\Omega\}$.
$f$ sends injectively the segment (or ray) $[0,A(x))\frac{x}{\|x\|}$ to the ray $\mathbf{R}_+\frac{x}{\|x\|}$.
Moreover, $f(0\frac{x}{\|X\|})=0$ and
$$\lim_{r\rightarrow A(x)}\left\|f(r\frac{x}{\|x\|})\right\|=\lim_{r\to A(x)}\left[1+\left(\int_0^{r}\frac{dt}{\phi\left(t\cdot\frac{rx}{\|x\|}\cdot\left\|\frac{\|x\|}{rx}\right\|\right)}\right)^2\right]\cdot r= \left[1+\left(\int_0^{A(x)}\frac{dt}{\phi(t\frac{x}{\|x\|})}\right)^2\right]\cdot A(x)=+\infty.$$
Indeed, if $A(x)=+\infty$, then it holds for obvious reason.
If $A(x)\lt+\infty$, then by definitions of $\phi$ and $A(x)$ we get that $\phi(A(x)\frac{x}{\|x\|})=0$.
Hence by the [[mean value theorem]] and the fact that $\phi$ is $C^1$ due to
$$\phi\left(r\frac{x}{\|x\|}\right)\le M(A(x)-r)$$
for some constant $M$ and every $r$.
As a result,
$$\int_0^{A(x)}\frac{dt}{\phi\left(t\frac{x}{\|x\|}\right)}$$
diverges.
Hence we infer that $f([0,A(x))\frac{x}{\|x\|})=\mathbf{R}_+\frac{x}{\|x\|}$ and so $f(\Omega)=\mathbf{R}^n$.
To end the proof we need to show that $f$ has a $C^\infty$-inverse.
But as a corollary from the [[inverse function theorem]] we get that it is sufficient to show that $df$ vanishes nowhere.
Suppose that $d_x f(h)=0$ for some $x\in\Omega$ and $h\neq 0$.
From definition of $f$ we get that
$$d_x f(h)=\lambda(x)h+d_x \lambda(h)x.$$
Hence $h=\mu x$ for some $\mu\neq 0$ and from that $x\neq 0$.
As a result $\lambda(x)+d_x \lambda(x)=0$.
But we have that $\lambda(x)\ge1$ and function $g(t):=\lambda(tx)$ is increasing, so $g'(1)=d_x \lambda(x)\gt0$, which gives a contradiction.
\end{proof}
+-- {: .num_example}
###### Example
Let $I(\Delta^n) \subset \mathbb{R}^n$ be the [[interior]] of the standard $n$-[[simplex]]. Then there is a diffeomorphism to $\mathbb{B}^n$ defined as follows:
Parameterize the $n$-simplex as
$$
I(\Delta^n)
=
\left\{
(x^1, \cdots, x^n) \in \mathbb{R}
|
(\forall i : x^i \gt 0)\; and \; ( \sum_{i=1}^n x^i \lt 1)
\right\}
\,.
$$
Then define the map $f : I(\Delta^n) \to \mathbb{R}^n$ by
$$
(x^1, \ldots, x^n)
\mapsto
(\log(\frac{x^1}{1 - x^1 - \ldots -x^n}),
\ldots,
\log(\frac{x^n}{1 - x^1 - \ldots - x^n}))
\,.
$$
=--
(Thanks to [[Todd Trimble]].) One way to think about it is that $I(\Delta^n)$ is the positive orthant of an open $n$-ball in $l^1$ norm, so that in the opposite direction we have a chain of invertible maps
$$\array{
\mathbb{R}^n & \stackrel{\exp^n}{\to} & \mathbb{R}_+^n & \to & I(\Delta^n) \\
& & \vec{x} & \mapsto & \vec{x}/(1 + {\|\vec{x}\|}_1)
}$$
which we simply invert to get the map $f$ above.
### Good covers by balls
One central application of balls is as building blocks for [[covering]]s. See [[good open cover]] for some statements.
## Related concepts
* [[unit ball]]
## References
{#References}
### Geometric
* {#Ozols} V. Ozols, _Largest normal neighbourhoods_ ,
Proceedings of the American Mathematical Society
Vol. 61, No. 1 (Nov., 1976), pp. 99-101 ([jstor](http://www.jstor.org/stable/2041672))
That an open subset $U \subseteq \mathbb{R}^4$ homeomorphic to $\mathbb{R}^4$ equipped with the smooth structure inherited as an open submanifold of $\mathbb{R}^4$ might nevertheless be non-diffeomorphic to $\mathbb{R}^4$, see
* De Michelis, Stefano; Freedman, Michael H. (1992) "Uncountably many exotic $\mathbb{R}^4$'s in standard 4-space", J. Diff. Geom. 35, pp. 219-254.
{#DeMFreed}
### Star-shaped regions diffeomorphic to open ball
{#ReferencesStarShapedReasonDiffeomorphicToOpenBall}
The proof that open star-shaped regions are diffeomorphic to a ball appears as
* {#Ferus07} [[Dirk Ferus]], [theorem 237](http://www.math.tu-berlin.de/~ferus/ANA/Ana3.pdf#page=154) in: _Analysis III_ (2007) ([pdf](http://www.math.tu-berlin.de/~ferus/ANA/Ana3.pdf), [[Ferus_AnalysisIII.pdf:file]])
It is a lengthy proof, due to Stefan Born.
A simpler version of the proof appears in
* {#GonnordTosel98} Stéphane Gonnord, Nicolas Tosel, page 60 of: _Calcul Différentiel_, ellipses (1998) (English translation: [MO:a/212595](http://mathoverflow.net/a/212595), [[Aubry_reproducing_GonnordAndTosel.pdf:file]])
These proofs had remained obscure (see also [this Remark](good+open+cover#LiteratureOnExistenceOfDifferentiablyGoodOpenCovers) at [[good open cover]]):
For instance in a remark below lemma 10.5.5 of
* {#Conlon08} Lawrence Conlon, _Differentiable manifolds_, Birkhäuser 2001/2008 ([doi:10.1007/978-0-8176-4767-4](https://link.springer.com/book/10.1007/978-0-8176-4767-4))
it says:
> It seems that open star shaped sets $U \subset M$ are always diffeomorphic to $\mathbb{R}^n$, but this is extremely difficult to prove.
And in
* {#Lee09} [[Jeffrey Lee]], _Manifolds and differential geometry_, Graduate Studies in Mathematics **107** (2009) ([ISBN: 978-0-8218-4815-9](https://bookstore.ams.org/gsm-107), [doi:10.1090/gsm/107](https://doi.org/10.1090/gsm/107))
one finds the statement:
> Actually, the assertion that an open geodesically convex set in a Riemannian manifold is diffeomorphic to $\mathbb{R}^n$ is common in literature, but it is a more subtle issue than it may seem, and references to a complete proof are hard to find (but see [Grom]).
Here "Grom" refers to
* [[Mikhail Gromov]], _Convex sets and Kähler manifolds_, Advances in differential geometry and topology. F. Tricerri ed., World Sci., Singapore,
(1990), 1-38. ([pdf](http://www.ihes.fr/~gromov/PDF/%5B68%5D.pdf))
where the relevant statement is 1.4.C1 on page 8. Note however that the diffeomorphism considered there is only of $C^1$ class, not $C^\infty$, so this is not a proof either.
A texbook account finally appears in
* {#GuilleminHaine19} [[Victor Guillemin]], [[Peter Haine]], Thm. 5.3.2 and Appendix C of: _Differential Forms_, World Scientific (2019) ([doi:10.1142/11058](https://doi.org/10.1142/11058))
See also the Math Overflow discussion [here](http://mathoverflow.net/questions/41853/explicit-diffeomorphim-between-open-simplex-and-open-ball).
### Combinatorial
* {#Joyal} [[Andre Joyal]], _Disks, duality and Theta-categories_ ([[JoyalThetaCategories.pdf:file]])
* {#MakkaiZawadowski} [[Mihaly Makkai]], Marek Zawadowski, _Duality for Simple $\omega$-Categories and Disks_ ([TAC](http://www.emis.de/journals/TAC/volumes/8/n7/8-07abs.html))
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|
Balázs Szendrői | https://ncatlab.org/nlab/source/Bal%C3%A1zs+Szendr%C5%91i |
* [webpage](https://people.maths.ox.ac.uk/szendroi/)
category: people
[[!redirects Balazs Szendroi]] |
Ban > history | https://ncatlab.org/nlab/source/Ban+%3E+history | < [[Ban]]
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Banach algebra | https://ncatlab.org/nlab/source/Banach+algebra |
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### Context
#### Analysis
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[[!include analysis - contents]]
=--
#### Analytic geometry
+--{: .hide}
[[!include analytic geometry -- contents]]
=--
#### Higher algebra
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[[!include higher algebra - contents]]
=--
=--
=--
# Banach algebras
* table of contents
{: toc}
## Definitions
An associative unital Banach algebra is [[monoid object]] in the [[closed monoidal category]] of [[Banach spaces]] (with [[short linear operators]] as [[morphisms]], and the usual [[internal hom]], or equivalently the [[projective tensor product]]). However, Banach algebras are not usually assumed to be unital, making them [[semigroup]] objects (or even [[magma]] objects if not assumed to be associative).
Explicitly, this means a [[Banach space]] $A$ equipped with a [[bilinear map|bilinear]] _multiplication_ map
$$ m\colon A \times A \to A ,$$
which again is usually taken to be [[associativity|associative]] (and may even be unital), such that
$$ {\|a \cdot b\|} \leq {\|a\|} \cdot {\|b\|} ,$$
where $a \cdot b$ (or just $a b$) means $m(a, b)$.
In the unital case, we should also require ${\|1\|} \leq 1$, although some authors leave this out. Other authors require ${\|1\|} = 1$, which is too strong, since it rules out the [[trivial ring|trivial algebra]]. (However, ${\|1\|} = 1$ follows from ${\|1\|} \leq 1$ and the existence of any element $a \ne 0$). One can of course always formally adjoin a unit $e$ with ${\|e\|} = 1$, forming the Banach algebra $A \oplus \langle{e}\rangle$ (using the $l^1$-[[l-1-direct sum|direct sum]]).
The explicit description in terms of $m$ is of course earlier; but the abstract description as an internal monoid makes clear the most natural definition of [[Banach coalgebra]]: a [[comonoid]] in the same monoidal category.
+-- {: .query}
YC: An earlier version of this entry said that "the correct" definition of Banach coalgebra is as a comonoid in the usual monoidal category of Banach spaces and [[short linear operators]]. I would prefer that this be amended, with similar wording as what I've chosen, since experience has shown that the most fruitful candidates for "Banach spaces with coalgebraic structure" are NOT comonoids in this sense. One should instead only require a comultiplication which takes values in something like the injective tensor product, or if you are working with Cstar objects, in something like the spatial tensor product. Does anyone object to my rewording?
=--
## Examples
* A standard associative example is $L^1(\mathbb{R})$ with [[Lebesgue measure]], where the multiplication is taken to be [[convolution]]. (This lacks a unit for the multiplication, since there is no $L^1$ function $e$ that represents the [[Dirac functional]] $f \mapsto f(0)$, via
$$ f(0) = \int_{-\infty}^{\infty} e(x) f(x) \,\mathrm{d}x ,$$
on [[continuous functions]] $f\colon X \to \mathbb{C}$.) One can generalize this example in straightforward fashion, replacing $\mathbb{R}$ by any [[locally compact space|locally compact]] [[Hausdorff space|Hausdorff]] [[topological group]] $G$, and Lebesgue measure by a [[Haar measure]] on $G$; the algebra is unital if and only if $G$ is [[discrete space|discrete]].
* For *any* [[measure space]] $X$, $L^{\infty}(X)$ is a [[commutative ring|commutative]] unital associative Banach algebra (in fact a unital $C^*$-[[C-star-algebra|algebra]], in fact a [[von Neumann algebra]] if $X$ is [[localizable measure space|localizable]]) with respect to pointwise multiplication.
* If $A$ is a Banach space, the [[internal hom]] $hom(A, A)$ is a unital Banach algebra (by [[general abstract nonsense]]).
* Any $C^*$-[[C-star algebra|algebra]] (and thus every [[von Neumann algebra]]) is in particular a Banach algebra.
* The [[normed division algebras]] are (possibly nonassociative) Banach [[division algebras]] over $\mathbb{R}$.
* The only Banach division algebra over $\mathbb{C}$ is $\mathbb{C}$ itself, by the [[Gel'fand–Mazur theorem]].
* A $JB$-[[JB-algebra|algebra]] (or more generally a [[Jordan–Banach algebra]]) is a nonassociative (but commutative) kind of Banach algebra. (The commutative associative Banach algebras also count as Jordan--Banach algebras.)
### An example of a 'nonunital' Banach algebra that has an identity element
Let $C_n$ be a cyclic group of order $n\geq 2$ and look at the Banach algebra (in the "strict" sense of a monoid object in $Ban$) that is obtained by equipping the Banach space $\ell^1(C_n)$ with the natural convolution product: $\delta_x * \delta_y = \delta_{x+y}$. There is a "short" homomorphism from $\ell^1(C_n)$ into the ground field which is just the unique linear extension of the group homomorphism $C_n \to \{1\}$ (by the free property of the $\ell^1$-functor) and we let $J$ be the kernel of this homomorphism. ($J$ is the so-called "augmentation ideal".) Now $J$ is a semigroup object in $Ban$ and as an algebra it has an identity element $p$, but a calculation/hindsight shows that $\delta_e-p$ must be the constant function $C_n \to \{1/n\}$, so that $p$ has norm $(1-1/n)+(n-1)/n = 2-2/n$.
## Arens products
If $A$ is a Banach algebra, its bidual $A^{**}$ has two naturally induced Banach algebra structures on it: these are the so-called Arens products on the second dual. These correspond to the left and right [[tensorial strength]]s for the bidual monad on the category of Banach spaces (whether with [[short linear operators]] as [[morphisms]], or all bounded linear operators). In different language, the two Arens multiplications arise from natural transformations
$$\alpha_{A B}: A^{\ast\ast} \otimes B^{\ast\ast} \to (A \otimes B)^{\ast\ast}$$
$$\,$$
$$\beta_{A B}: A^{\ast\ast} \otimes B^{\ast\ast} \to (A \otimes B)^{\ast\ast}$$
described at [[monoidal monad]]; putting $A = B$ and post-composing with $m^{\ast\ast}: (A \otimes A)^{\ast\ast} \to A^{\ast\ast}$ produces the two Arens products. They are named for Richard Arens, who has a 1955 paper which studies this construction in a more general setting . One can see Arens's "phyla" -- with hindsight and Whig history -- as a precursor of symmetric closed monoidal categories.
[Link to talk by F. E. J. Linton on Arens products](http://tlvp.net/~fej.math.wes/CMS-June2010/Arens01.htm)
### Arens regularity
Algebras where the two Arens products coincide are said to be Arens regular: since $B(H)$, the algebra of bounded linear operators on a Hilbert space $H$, has this property, so do all its closed subalgebras, in particular all $C^*$-[[C-star algebra|algebra]]s.
In contrast, if $G$ is an infinite locally compact group, a result of N. J. Young shows that $L^1(G)$ is not Arens regular. Consequently, it is not isomorphic as a topological algebra to any closed subalgebra of $B(H)$. Note that this consequence is significantly deeper than the observation that the norm on $L^1(G)$ does not satisfy the C-star identity with respect to the usual involution on $L^1(G)$. See also comments on [this MO question](http://mathoverflow.net/questions/134074/banach-algebra-counterexample).
In general, a Banach algebra $A$ is Arens regular if and only if, for each $\mu\in A^*$, the orbit maps $a\mapsto a\cdot\mu$ and $a\mapsto \mu\cdot a$ are [[weakly compact]] as linear maps $A\to A^*$. (Going from memory here, this result is due to J. S. Pym.) By using known results for factorization of [[weakly compact]] maps, it follows that if $A$ is Arens regular, one can construct a [[reflexive Banach space]] $E$ and an injective homomorphism of topological algebras $A\to B(E)$ which has closed range; this appears to have first been done explicitly by S. Kaijser.
+-- {: .query}
I suspect that here I mean something like a strict monomorphism in the category of monoid objects in TVS, but I am not sure of the details.
=--
However, this is not a characterization of Arens regularity; Young also observed that for any locally compact group $G$ one may build a reflexive Banach space $E$ and construct an injective homomorphism of topological algebras $L^1(G)\to B(E)$ which has closed range.
## Related concepts
* [[Banach module]]
[[!include analytic geometry ingredients -- table]]
## References
Named after [[Stefan Banach]].
* Zbigniew Semadeni, _Banach spaces of continuous functions_, vol. I, [gBooks](http://books.google.com/books/about/Banach_spaces_of_continuous_functions.html?id=vCDvAAAAMAAJ)
* N. Landsman, _Mathematical topics between classical and quantum mechanics_, Springer
* Walter Rudin, _Functional analysis_
* Richard V. Kadison, John R. Ringrose, _Fundamentals of the theory of operator algebras_
* F. F. Bonsall, J. Duncan, _Complete normed algebras_
* T. W. Palmer, _Banach Algebras and the General Theory of \* -Algebras_
Discussion over [[non-archimedean fields]] is in
* {#BoschGuntzerRemmert84} [[Siegfried Bosch]], [[Ulrich Güntzer]], [[Reinhold Remmert]], section 3.7 of _[[Non-Archimedean Analysis]] -- A systematic approach to rigid analytic geometry_, 1984 ([pdf](http://math.arizona.edu/~cais/scans/BGR-Non_Archimedean_Analysis.pdf))
Discussion in the context of ([[global analytic geometry]]) [[analytic geometry]] is in the following articles.
Lecture notes on the concept of the [[analytic spectrum]] of a Banach ring include
* {#Brodsky12} Sarah Brodsky, _Non-archimedean geometry_, 2012 ([pdf](http://math.berkeley.edu/~sstich/MAT_274/Math_274_3_Feb_2012.pdf))
A more [[topos theory|topos-theoretic]] perspective:
* {#BassatKremnitzer13} [[Oren Ben-Bassat]], [[Kobi Kremnizer]], section 6.5 of _Non-Archimedean analytic geometry as relative algebraic geometry_ ([arXiv:1312.0338](http://arxiv.org/abs/1312.0338))
category: analysis
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Banach analytic space | https://ncatlab.org/nlab/source/Banach+analytic+space |
#Contents#
* table of contents
{:toc}
## Idea
An infinite-dimensional [[analytic space]] equipped with [[Banach space]] structure.
## References
* Springer Encyclopedia of Mathematics, _[Banach analytic space](http://www.encyclopediaofmath.org/index.php/Banach_analytic_space)_
[[!redirects Banach analytic spaces]]
|
Banach bundle | https://ncatlab.org/nlab/source/Banach+bundle |
# Banach bundles
* table of contents
{: toc}
## Idea
A Banach bundle is a [[bundle]] in which every [[fibre]] is a [[Banach space]]. Certain other conditions apply.
## Definitions
A __Banach bundle__ is an [[open map|open]] (necessarily [[surjection|surjective]]) [[continuous map]] of [[Hausdorff topological spaces]] $p\colon Y\to B$, each of whose [[fibers]] carries a structure of a [[complex number|complex]] [[Banach space]], this structure being continuous in the base point (in other words, the global operations $\mathbb{C} \times Y \to Y$ of multiplication by a scalar, $Y \times_B Y \to Y$ of addition and $Y \to \mathbb{R}$ of taking the norm are continuous) and such that for every [[net]] $\{y_\alpha\}_{\alpha\in A}$, if ${\|y_{\alpha}\|} \to 0$ and $p(y_\alpha) \to b$, then $y_\alpha \to 0 = 0_b \in p^{-1}(b)$.
We distinguish a different concept of __Banach algebraic bundle__, where the base space $B$ is also a [[Banach algebra]] and the multiplication is defined as a map $\cdot\colon Y\times Y \to Y$ (not only $Y \times_B Y \to Y$), that is we can multiply the points in different fibers, and $p(a \cdot b) = p(a) \cdot p(b)$.
A Banach bundle is a __Hilbert bundle__ if each fiber is a [[separable space|separable]] [[Hilbert space]]. As usual, the inner product can be obtained by the polarization formula $(x,y) \coloneqq \frac{1}{4}({\|x+y\|^2} - {\|x-y\|^2})$ from the norm of a Banach space if the norm satisfies the parallelogram identity. From this, we infer that for Hilbert bundles, the inner product is continuous as a map $Y \times_B Y \to \mathbb{C}$. Hilbert bundles are important in the study of [[induced representations]] of [[locally compact group]]s, and [[Mackey theory]] in particular; more recently their study is connected to the study of [[Hilbert module]]s.
A __morphism of Banach bundles__ $(p\colon Y \to B)\to (p'\colon Y' \to B)$ over the same base is a morphism of total spaces commuting with the projections, $\mathbb{C}$-linear in each fiber, and preserving the norm. A Banach bundle is sometimes said to be Hilbertizable if it is isomorphic to the underlying Banach bundle of a Hilbert bundle; structurally, there is no difference between a Hilbert bundle and a Hilbertizable Banach bundle (again using the polarisation formula to prove that being a Hilbert space is a [[property-like structure]]).
One also considers Banach $*$-[[star-algebra|algebraic]] bundles, where an antilinear [[involution]] $*$ preserving the norm is involved, is continuous as a global map $Y \to Y$ and is an [[antihomomorphism]] of algebras satisfying $p(y^\ast) = p(y^{-1})$.
## References
* Wikipedia: [Banach bundle](http://en.wikipedia.org/wiki/Banach_bundle_%28non-commutative_geometry%29)
For Banach bundles see ch. 13 in vol. 1 (from page 125; def. 13.4 on p. 127) and for Banach algebraic bundles see from 783 on in vol. 2 of
* J. M. G. Fell, R. S. Doran, _Representations of $*$-algebras, locally compact groups, and Banach $*$-algebraic bundles_, Vol. 1. Basic representation theory of groups and algebras. Pure and Applied Mathematics, __125__, Academic Press 1988. xviii+746 pp. [MR90c:46001](http://www.ams.org/mathscinet-getitem?mr=936628) Vol. 2, Banach $*$-algebraic bundles, induced representations, and the generalized Mackey analysis. Pure and Applied Mathematics __126__, Acad. Press 1988. pp. i--viii and 747--1486, [MR90c:46002](http://www.ams.org/mathscinet-getitem?mr=936629)
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Banach coalgebra | https://ncatlab.org/nlab/source/Banach+coalgebra |
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### Context
#### Functional analysis
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[[!include functional analysis - contents]]
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# Banach coalgebras
* table of contents
{: toc}
## Idea
Banach coalgebras (or cogebras) are like [[Banach algebras]], but [[coalgebras]]. The [[dual vector space|dual]] of a Banach coalgebra is a Banach algebra (but not conversely). We can also consider Banach [[bialgebras]] (or bigebras).
## Definitions
A __Banach coalgebra__, or __Banach cogebra__, is a [[comonoid object]] in the [[monoidal category]] $Ban$ of [[Banach spaces]] with [[short linear maps]] and the [[projective tensor product]]. (Recall that a [[Banach algebra]] is a [[monoid object]] in $Ban$.)
Explicitly, we have:
1. a [[Banach space]] $A$
1. a [[short linear map]], the __comultiplication__:
$$ \Delta\colon A \to A {\displaystyle\hat{\otimes}_\pi} A $$
to the [[projective tensor product]];
1. a short [[linear functional]], the __counit__:
$$ \epsilon\colon A \to K ,$$
where $K$ is the [[ground field]];
1. an [[equation]], the __coassociativity__:
$$ (\Delta {\displaystyle\hat{\otimes}_\pi} \id_A) \Delta x = (\id_A {\displaystyle\hat{\otimes}_\pi} \Delta) \Delta x \in (A {\displaystyle\hat{\otimes}_\pi} A) {\displaystyle\hat{\otimes}_\pi} A \cong A {\displaystyle\hat{\otimes}_\pi} (A {\displaystyle\hat{\otimes}_\pi} A)$$
for each $x\colon A$;
1. an equation, the __left coidentity__:
$$ (\epsilon {\displaystyle\hat{\otimes}_\pi} \id_A) \Delta x = x \in K {\displaystyle\hat{\otimes}_\pi} A \cong A $$
for each $x\colon A$;
1. and an equation, the __right coidentity__:
$$ (\id_A {\displaystyle\hat{\otimes}_\pi} \epsilon) \Delta x = x \in A {\displaystyle\hat{\otimes}_\pi} K \cong K $$
for each $x\colon A$.
Technically, we\'ve defined a __counital coassociative Banach coalgebra__. We can leave out (3,5,6) to get a __non-counital Banach coalgebra__, and (also) leave out (4) to get a __non-coassociative Banach coalgebra__. Warning: these terms are examples of the [[red herring principle]]. Note that (3) is a [[property-like structure]] (and 4--6 are obviously just [[properties]]).
On the other hand, we can *add* the property of __cocommutativity__:
* $\tau \Delta x = \Delta x$ for each $x\colon A$,
where the [[braiding]] $\tau\colon A {\displaystyle\hat{\otimes}_\pi} A \to A {\displaystyle\hat{\otimes}_\pi} A$ is generated by $\tau (u \otimes v) = v \otimes u$. Then we have a __cocommutative Banach coalgebra__.
To __freely adjoin a counit__ to a non-counital Banach coalgebra $A$, take the Banach space $A \oplus_1 K$ (using the $l^1$-[[l-1-direct sum|direct sum]]), let $\Delta_{A \oplus K} (x,c)$ be $(\Delta_A x, 0, 0, c) \in (A {\displaystyle\hat{\otimes}_\pi} A) \oplus_1 A \oplus_1 A \oplus_1 K \cong (A \oplus_1 K) {\displaystyle\hat{\otimes}_\pi} (A \oplus_1 K)$, and let $\epsilon_{A \oplus_1 K} (x,c)$ be $c$. Then $A \oplus_1 K$ is a counital Banach coalgebra. (Freely forcing coassociativity or cocommutativity ---or even freely adjoining $\Delta$ in the first place--- is harder.)
The [[category]] __$Ban Coalg$__ of Banach coalgebras has, as [[objects]], Banach coalgebras and, as [[morphisms]], [[short linear maps]] $f\colon A \to B$ with equations
$$ \Delta_B f x = (f {\displaystyle\hat{\otimes}_\pi} f) \Delta_A x $$
and (unless we are allowing non-counital coalgebras)
$$ \epsilon_B f x = \epsilon_A x $$
for all $x\colon A$. Warning: the term 'homomorphism' is used more generally; see below.
If $A$ and $B$ are Banach coalgebras, then their __[[projective tensor product]]__ $A {\displaystyle\hat{\otimes}_\pi} B$ is a Banach coalgebra, generated by
$$ \Delta_{A {\displaystyle\hat{\otimes}_\pi} B} (x \otimes y) = \Delta_A x \otimes \Delta_B y \in (A {\displaystyle\hat{\otimes}_\pi} A) {\displaystyle\hat{\otimes}_\pi} (B {\displaystyle\hat{\otimes}_\pi} B) \cong (A {\displaystyle\hat{\otimes}_\pi} B) {\displaystyle\hat{\otimes}_\pi} (A {\displaystyle\hat{\otimes}_\pi} B) $$
and
$$ \epsilon_{A {\displaystyle\hat{\otimes}_\pi} B} (x \otimes y) = (\epsilon_A x) (\epsilon_B y) \in K .$$
Similarly (but more simply), the [[ground field]] $K$ is itself a Banach coalgebra, with $\Delta$ and $\epsilon$ both essentially the [[identity map]]. In this way, $Ban Coalg$ becomes a [[symmetric monoidal category]].
The [[full subcategory]] $Cocomm Ban Coalg$ of cocommutative Banach coalgebras becomes a [[cartesian monoidal category]] under the projective tensor product. Actually, $K$ is the [[terminal object]] even in $Ban Coalg$ (with the unique coalgebra morphism to $K$ being $\epsilon$ itself), but the [[pairing]]
$$ (f,g)(x) \coloneqq (f {\displaystyle\hat{\otimes}_\pi} g) \Delta x \in A {\displaystyle\hat{\otimes}_\pi} B $$
(given $f\colon \Gamma \to A$, $g\colon \Gamma \to B$, and $x\colon \Gamma$) is a morphism of $Ban Coalg$ only when $A$ and $B$ are cocommutative. (I believe that $Ban Coalg$ *does* have a [[product]], but it must be more complicated.)
## The dual algebras of a coalgebra
If $A$ is a Banach coalgebra, then the [[dual vector space]] $A^*$ is a [[Banach algebra]]. Actually, this is more general than $A^* = [A,K]$; if $B$ is any Banach algebra, then so is $[A,B]$ (the Banach space of [[bounded linear maps]] from $A$ to $B$).
This result is nothing special about Banach (co)algebras; it holds in any [[closed monoidal category]]. The multiplication operation in $[A,B]$ is given by
$$ (\lambda \mu) x = m (\lambda {\displaystyle\hat{\otimes}_\pi} \mu) \Delta x ,$$
where $m\colon B {\displaystyle\hat{\otimes}_\pi} B \to B$ is (generated by) the multiplication operation on $B$. $[A,B]$ is associative, unital, or commutative if $A$ and $B$ are (with 'co‑' in the names of $A$\'s properties). In particular, $A^*$ has one of these properties iff $A$ has the corresponding property.
Note that $[B,A]$ (or even $B^*$) is *not*, in general, a Banach coalgebra. (That\'s because $Ban$ is closed, not [[coclosed category|coclosed]].)
## Operators between coalgebras
Let $A$ and $B$ be Banach coalgebras.
Of course, $A$ and $B$ are [[Banach spaces]], so we may consider the whole panoply of [[linear operators]] from $A$ to $B$. In general, a linear operator is only a [[partial function]], defined on a [[linear subspace]] of $A$ (and otherwise only required to be a [[linear map]]); but in particular we consider the [[densely-defined operator]]s (each defined on a [[dense subspace|dense]] subspace of $A$), the [[linear mappings]] (each defined on all of $A$), the [[bounded operators]] (each defined on all of $A$ and [[bounded map|bounded]] or equivalently [[continuous map|continuous]]), and the [[short operators]] (each bounded with a [[norm]] at most $1$).
A __comultiplicative linear operator__ from $A$ to $B$ is a linear operator $T\colon A \to B$ such that the following hold for all $x \in \dom T$:
* $\Delta x \in (\dom T) {\displaystyle\hat{\otimes}_\pi} (\dom T)$,
* $\Delta_B T x = (T {\displaystyle\hat{\otimes}_\pi} T) \Delta_A x$ (which exists by the previous line), and
* $\epsilon_B T x = \epsilon_A x$ (which always exists).
We can also consider __densely-defined comultplicative linear operators__. A __coalgebra homomorphism__, or __cohomomorphism__, is a comultiplicative linear mapping; we can also consider __bounded homomorphisms__ and __short homomorphisms__. The last of these are, as above, the [[morphisms]] in $Ban Coalg$; of course, any of these classes of operators (except the densely-defined ones, which are not closed under [[composition]]) could be taken to be morphisms of a different category with the same objects, but then we would have [[isomorphisms]] that are not [[isometries]]. (See also [[isomorphism of Banach spaces]].)
## Beyond coalgebras
A __Banach [[bialgebra]]__, or __Banach bigebra__, is a [[bimonoid]] in $Ban$: a Banach space $A$ equipped with the structures of both a [[Banach algebra]] and a Banach coalgebra, such that $\Delta$ and $\epsilon$ are both morphisms of Banach algebras, or equivalently such that the multiplication and unit of the Banach algebra are both morphisms of Banach coalgebras. Explicitly, this requirement is:
* $\Delta (x y) = (\Delta x) (\Delta y)$ (with the induced multiplication on $A {\displaystyle\hat{\otimes}_\pi} A$),
* $\Delta 1 = 1 \otimes 1$ (which is the identity in $A {\displaystyle\hat{\otimes}_\pi} A$),
* $\epsilon (x y) = (\epsilon x) (\epsilon y)$ (with the multiplication on the right in $K$), and
* $\epsilon 1 = 1$ (with the $1$ on the right in $K$).
The [[category]] __$Ban Bialg$__ of Banach bialgebras has, as objects, Banach bialgebras and, as morphisms, short linear maps that are morphisms of both Banach algebras and Banach coalgebras.
A __Banach [[Hopf algebra]]__ is a [[Hopf object]] in $Ban$: a Banach bialgebra $A$ with a (necessarily unique) short linear map (the __antipode__) $S\colon A \to A$ such that
$$ m (S {\displaystyle\hat{\otimes}_\pi} \id) \Delta x, m (\id {\displaystyle\hat{\otimes}_\pi} S) \Delta x = 1 \epsilon x $$
(where $m$ is the multiplication with identity $1$) for all $x\colon A$.
The [[category]] __$Ban Hopf Alg$__ of Banach Hopf algebras has, as objects, Banach Hopf algebras and, as morphisms, short linear maps $f\colon A \to B$ that are morphisms of Banach bialgebras and preserve antipodes:
$$ f (S_A x) = S_B (f x) $$
for all $x\colon A$.
A __Banach $*$-[[star-algebra|coalgebra]]__ is a $*$-[[star-monoid object|monoid object]] in $Ban$: a Banach coalgebra $A$ equipped with an [[antilinear map]] (the __adjoint__) $x \mapsto x^*\colon A \to A$ such that
$$ \Delta x^* = (\tau \Delta x)^* $$
(where $\tau$ again is the [[braiding]] and $*$ on $A {\displaystyle\hat{\otimes}_\pi} A$ is generated by $(x \otimes y)^* = x^* \otimes y^*$) for all $x\colon A$ and
$$ \epsilon x^* = \overline {\epsilon x} $$
(where a bar indicates [[complex conjugation]]) for all $x\colon A$.
The [[category]] __$Ban {*} Coalg$__ of Banach $*$-coalgebras has, as objects, Banach $*$-coalgebras and, as morphisms, short linear maps $f\colon A \to B$ that are morphisms of Banach bialgebras and preserve adjoints:
$$ f(x^*) = f(x)^* $$
for all $x\colon A$.
There are also $C^*$-[[C-star-coalgebra|coalgebras]], which have their own page.
## Examples
It's well known that the [[sequence space]] $l^1$ of absolutely summable [[infinite sequences]], thought of as $l^1(\mathbb{Z})$ (where $\mathbb{Z}$ is the [[abelian group]] of [[integers]] under addition), is a [[Banach algebra]] under [[convolution]]; however, it is also a Banach coalgebra, and these structures together make it a Banach bialgebra, in fact a Banach Hopf $*$-algebra. Since $l^1$ is a Banach coalgebra, its [[dual vector space|dual space]] $l^\infty$ (the sequence space of absolutely bounded sequences) is a Banach algebra (which is also well known); and although there is no guarantee that it should work, in this case $l^\infty$ is also a Banach coalgebra, and indeed a Banach Hopf $*$-algebra too.
Explicitly: The projective tensor square $l^1 {\displaystyle\hat{\otimes}_\pi} l^1$ is the space of absolutely summable infinite [[matrix|matrices]]; convolution takes the matrix $(a_{i,j})_{i,j}$ to the sequence
$$ (\sum_{i + j = k} a_{i,j})_k $$
(summing along antidiagonals); comultiplication takes $(a_k)_k$ to the [[diagonal matrix]]
$$ \Delta a = (\sum_{i = j = k} a_k)_{i,j} $$
(which is not quite the origin of the symbol '$\Delta$' but might as well be). The tensor square $l^\infty {\displaystyle\hat{\otimes}_\pi} l^\infty$ is the space of infinite matrices with absolutely bounded entries;
+-- {: .query}
(comment added 26-08-2012 by YC: I am not convinced; Grothendieck's inequality, anyone?)
=--
the dual multiplication on $l^\infty$ takes the matrix $(a_{i,j})_{i,j}$ to the sequence
$$ (\sum_{i = j = k} a_{i,j})_k = (a_{k,k})_k $$
of its diagonal entries; the dual comultiplication (which part of me wants to call 'nvolution', but let\'s say coconvolution instead) takes $(a_k)_k$ to
$$ (\sum_{i + j = k} a_k)_{i,j} = (a_{i + j})_{i,j} $$
(so each antidiagonal is constant).
We are lucky that coconvolution exists, since the dual of a Banach algebra need not be a Banach coalgebra; but arguably coconvolution is easier to describe than convolution, so let us shift perspective and take coconvolution as basic. Then convolution necessarily exists on the dual of $l^\infty$, but (at least in [[classical mathematics]]) $l^1$ is only a subspace of that. So from this perspective, what's lucky is that $l^1$ is closed under convolution. (In [[dream mathematics]], $l^1$ is the entire dual of $l^\infty$, so no luck is required.) Of course, $l^1$ *is* the dual of $c_0$ (the space of sequences with limit $0$), but $c_0$ is *not* closed (coclosed?) under coconvolution (try any non-zero example), so we are still lucky.
[[!redirects Banach coalgebra]]
[[!redirects Banach coalgebras]]
[[!redirects Banach cogebra]]
[[!redirects Banach cogebras]]
[[!redirects BanCoalg]]
[[!redirects Ban Coalg]]
[[!redirects Banach bialgebra]]
[[!redirects Banach bialgebras]]
[[!redirects Banach bigebra]]
[[!redirects Banach bigebras]]
[[!redirects Banach Hopf algebra]]
[[!redirects Banach Hopf algebras]]
[[!redirects Banach-Hopf algebra]]
[[!redirects Banach-Hopf algebras]]
[[!redirects Banach–Hopf algebra]]
[[!redirects Banach–Hopf algebras]]
[[!redirects Banach--Hopf algebra]]
[[!redirects Banach--Hopf algebras]]
[[!redirects Banach *coalgebra]]
[[!redirects Banach *coalgebras]]
[[!redirects Banach *cogebra]]
[[!redirects Banach *cogebras]]
[[!redirects Banach *-coalgebra]]
[[!redirects Banach *-coalgebras]]
[[!redirects Banach *-cogebra]]
[[!redirects Banach *-cogebras]]
[[!redirects Banach * coalgebra]]
[[!redirects Banach * coalgebras]]
[[!redirects Banach * cogebra]]
[[!redirects Banach * cogebras]]
[[!redirects Banach star-coalgebra]]
[[!redirects Banach star-coalgebras]]
[[!redirects Banach star-cogebra]]
[[!redirects Banach star-cogebras]]
[[!redirects Banach star coalgebra]]
[[!redirects Banach star coalgebras]]
[[!redirects Banach star cogebra]]
[[!redirects Banach star cogebras]]
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Banach fixed-point theorem | https://ncatlab.org/nlab/source/Banach+fixed-point+theorem |
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#### Analysis
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#### Differential geometry
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#Contents#
* table of contents
{:toc}
## Statement
The **Banach fixed-point theorem** or **contraction mapping theorem** states:
Let $(X, \rho)$ be a [[sequentially Cauchy complete]] [[metric space]] with a point $x_0:X$ and a [[rational number]] $C : \mathbb{Q}$ such that for all $x:X$ and $y:X$, $\rho(x, y) \leq C$. Let $T : X \to X$ be an [[endomap]] with a [[rational function|rational]] [[Lipschitz constant]] $0 \lt c \lt 1$. Then $X$ has a unique [[fixed point]], a point $x$ with $\rho(T (x), x) = 0$,
such that for any $y : X$ with $\rho(T (y), y) = 0$, $x = y$.
## Related concepts
* [[inverse function theorem]]
* [[ordinary differential equation]]
## References
* Auke B. Booij, Analysis in univalent type theory ([pdf](https://etheses.bham.ac.uk/id/eprint/10411/7/Booij2020PhD.pdf))
See also:
* Wikipedia, _[Banach fixed-point theorem](https://en.wikipedia.org/wiki/Banach_fixed-point_theorem)_
[[!redirects Banach fixed point theorem]]
[[!redirects contraction mapping theorem]] |
Banach lattice | https://ncatlab.org/nlab/source/Banach+lattice | Banach lattice
(From https://www.encyclopediaofmath.org/index.php/Banach_lattice)
With minor editing:
A vector lattice that is at the same time a Banach space with a norm which satisfies the monotonicity condition:
|x|</=|y|=>||x||</=||y||
A Banach lattice is also called a KB-lineal, whereas an arbitrary normed lattice, i.e. a vector lattice with a monotone norm, is called a KN-lineal. When completing a normed lattice in norm, the order relations may be extended to the resulting Banach space so that it becomes a Banach lattice. If it is possible to introduce in a lattice a Banach topology which converts it to a Banach lattice, such a topology is unique. The simplest example of a Banach lattice is the space C(Q) of continuous functions on an arbitrary compact topological space Q with the natural (pointwise) order and with the ordinary (uniform) norm. Other examples of Banach lattices include Lp spaces and [[Orlicz spaces]] (cf. Orlicz space). In Banach lattices convergence in norm is (o)-convergence for convergence with a regulator. This is not true of normed lattices.
An important special case is a Banach lattice of bounded elements. If a lattice X contains a strong unit 1, i.e. xEX if for each there exists a /\ such that x|</=/\1 , then the smallest /\ for which this inequality is valid is taken as |x|. The normed lattice thus obtained is called a normed lattice of bounded elements; if it is complete in norm, it is called a Banach lattice of bounded elements. In a Banach lattice (and even in a normed lattice) of bounded elements convergence in norm is identical with convergence with a regulator, while the boundedness of a set of elements in norm is identical with order boundedness. If a normed lattice of bounded elements is conditionally o^-complete, it is complete in norm.
The space C(Q) is a Banach lattice of bounded elements in which the function x(q)E1 is taken as the unit. For any Banach lattice of bounded elements there exists a compact Hausdorff space such that is algebraically and lattice isomorphic to the space . This is an abstract characterization of the Banach lattice of continuous functions on a compact Hausdorff space.
In any normed lattice an additive functional that is continuous in norm is regular and, moreover, is representable as the difference of two additive functionals which are continuous in norm. In a Banach lattice each positive additive functional is continuous in norm, which means that the classes of regular functionals and additive functionals which are continuous in norm coincide. The space X' which is dual in the sense of Banach to the normed lattice X is a conditionally complete Banach lattice. In a normed lattice the Hahn–Banach theorem may be strengthened as follows: For any x0>0 there exists a positive additive functional f , which is continuous in norm, such that f(x0)=||x0||,||f||=1 .
References
[1] B.Z. Vulikh, "Introduction to the theory of partially ordered spaces" , Wolters-Noordhoff (1967) (Translated from Russian)
[2] M.M. Day, "Normed linear spaces" , Springer (1958)
|
Banach manifold | https://ncatlab.org/nlab/source/Banach+manifold |
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#### Differential geometry
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#### Manifolds and cobordisms
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[[!include manifolds and cobordisms - contents]]
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#Contents#
* table of contents
{:toc}
## Idea
A notion of [[infinite-dimensional manifold]].
A __Banach manifold__ is a [[manifold]] modelled on [[Banach spaces]]. By default, transition maps are taken to be [[smooth map|smooth]].
## Properties
### As absolute neighbourhood retracts
\begin{example}\label{ParacompactBanachManifoldsAreANRs}
Every [[paracompact topological space|paracompact]] Banach manifold is an [[absolute neighbourhood retract]].
\end{example}
By [Palais 1966, Cor. to Thm. 5 on p. 3](#Palais66).
### Embedding into the category of diffeological spaces
The category of smooth Banach manifolds has a [[full and faithful functor]] into the category of [[diffeological spaces]]. In terms of Chen smooth spaces this was observed in ([Hain](#Hain)). For more see at _[Fréchet manifold -- Relation to diffeological spaces](#Fréchet+manifold#RelationBetweenDeffeologicalAndFrechetStructure)_.
## Related concepts
* [[Hilbert manifold]]
* [[ILH manifold]]
* [[Frechet manifold]]
* [[convenient manifold]]
## References
For general references see at _[[infinite-dimensional manifold]]_.
Aspects of the [[homotopy theory]] of Banach manifolds:
* {#Palais66} [[Richard S. Palais]], *Homotopy theory of infinite dimensional manifolds*, Topology **5** 1 (1966) 1-16 (<a href="https://doi.org/10.1016/0040-9383(66)90002-4">doi:10.1016/0040-9383(66)90002-4</a>)
The [[full subcategory]] embedding into the category of [[diffeological spaces]]:
* {#Hain} [[Richard Hain]], _A characterization of smooth functions defined on a Banach space_, Proc. Amer. Math. Soc. 77 (1979), 63-67 ([web](http://www.ams.org/journals/proc/1979-077-01/S0002-9939-1979-0539632-8/home.html), [pdf](http://www.ams.org/journals/proc/1979-077-01/S0002-9939-1979-0539632-8/S0002-9939-1979-0539632-8.pdf))
[[!redirects Banach manifolds]]
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Banach module | https://ncatlab.org/nlab/source/Banach+module |
#Contents#
* table of contents
{:toc}
## Definition
For $A$ a [[Banach algebra]], then a _Banach module_ is a [[module]] over $A$ in the [[category]] of [[Banach spaces]], hence a [[Banach space]] $V$ equipped with a [[continuous function|continuous]] [[bilinear map]] $V\times A\to V$ satisfying the [[action]] property.
## Properties
* [[Tate's acyclicity theorem]]
[[!redirects Banach modules]]
|
Banach ring | https://ncatlab.org/nlab/source/Banach+ring |
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#### Analysis
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#### Higher algebra
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[[!include higher algebra - contents]]
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# Contents
* table of contents
{: toc}
## Idea
A Banach ring is a [[complete space|complete]] [[normed ring]], hence a [[commutative monoid]] in the [[monoidal category]] of complete [[normed groups]] (with [[short map|short]] group homomorphisms and the [[projective tensor product]]).
If not just in complete normed groups but in complete [[normed vector spaces]] ([[Banach spaces]]), then this is a [[Banach algebra]].
The [[Berkovich spectrum]] of a Banach ring $R$ is the [[topological space]] of multiplicative [[seminorms]] on $R$ that are bounded by the norm on $R$.
## Examples
* The [[integers]] $\mathbb{Z}$ equipped with their [[absolute value]] [[norm]] ${\vert- \vert_\infty}$ are a Banach ring.
* The integers with the $p$-adic norm ${|-|_p}$ are an incomplete normed ring whose completion is the Banach ring $\mathbb{Z}_p$ of $p$-[[adic integers]].
## Related concepts
[[!include analytic geometry ingredients -- table]]
## References
A quick review is in
* [[Sarah Brodsky]], _Non-archimedean geometry_ ([pdf](http://math.berkeley.edu/~sstich/MAT_274/Math_274_3_Feb_2012.pdf))
A standard textbook account in the context of [[rigid analytic geometry]] is
* {#BoschGuntzerRemmert84} [[Siegfried Bosch]], [[Ulrich Güntzer]], [[Reinhold Remmert]], section 1.2.4 of _[[Non-Archimedean Analysis]] -- A systematic approach to rigid analytic geometry_, 1984 ([pdf](http://math.arizona.edu/~cais/scans/BGR-Non_Archimedean_Analysis.pdf))
A set of lecture notes in the context of [[Berkovich spaces]] is
* [[Vladimir Berkovich]], section 1.2 of _Non-archimedean analytic spaces_, lectures at the _Advanced School on $p$-adic Analysis and Applications_, ICTP, Trieste, 31 August - 11 September 2009 ([pdf](http://www.wisdom.weizmann.ac.il/~vova/Trieste_2009.pdf))
Discussion from a more [[topos theory|topos-theoretic]] point of view is in
* {#BassatKremnitzer13} [[Oren Ben-Bassat]], [[Kobi Kremnizer]], section 6.5 of _Non-Archimedean analytic geometry as relative algebraic geometry_ ([arXiv:1312.0338](http://arxiv.org/abs/1312.0338))
category: analysis
[[!redirects Banach ring]]
[[!redirects Banach rings]]
[[!redirects complete normed ring]]
[[!redirects complete normed rings]]
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Banach space | https://ncatlab.org/nlab/source/Banach+space |
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### Context
#### Functional analysis
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# Banach spaces
* table of contents
{: toc}
## Idea
A Banach space $\mathcal{B}$ is both a [[vector space]] (over a [[normed field]] such as $\mathbb{R}$) and a [[complete space|complete]] [[metric space]], in a compatible way. Hence a complete [[normed vector space]].
A source of simple Banach spaces comes from considering a [[Cartesian space]] $\mathbb{R}^n$ (or $K^n$ where $K$ is the normed field) with the norm:
$$ {\|(x_1,\ldots,x_n)\|_p} \coloneqq \root p {\sum_{i = 1}^n {|x_i|^p}} $$
where $1 \leq p \leq \infty$ (this doesn't strictly make sense for $p = \infty$, but taking the limit as $p \to \infty$ and reading $\mathbb{R}^\infty = \underset{\longrightarrow}{\lim}_n \mathbb{R}^n$ as the [[direct limit]] (as opposed to the [[inverse limit]]) we arrive at the formula ${\|(x_1,\ldots,x_n)\|_\infty} \coloneqq \max_i {|x_i|}$).
However, the theory of these spaces is not much more complicated than that of finite-dimensional vector spaces because they all have the same underlying topology. When we look at infinite-dimensional examples, however, things become trickier. Common examples are [[Lebesgue spaces]], [[Hilbert spaces]], and [[sequence spaces]].
In the literature, one most often sees Banach spaces over the field $\mathbb{R}$ of [[real numbers]]; Banach spaces over the field $\mathbb{C}$ of [[complex numbers]] are not much different, since they are also over $\mathbb{R}$. But people do study them over [[p-adic numbers]] too. _Unless otherwise stated, we assume $\mathbb{R}$ below._
## Definitions
Let $V$ be a [[vector space]] over the field of [[real number]]s. (One can generalise the choice of [[field]] somewhat.) A **pseudonorm** (or **[[seminorm]]**) on $V$ is a function
$$ {\| - \|}\colon V \to \mathbb{R} $$
such that:
1. $ {\|0\|} \geq 0 $;
2. $ {\|r v\|} = {|r|} {\|v\|} $ (for $r$ a [[scalar]] and $v$ a vector);
3. $ {\|v + w\|} \leq {\|v\|} + {\|w\|} $.
It follows from the above that ${\|v\|} \geq 0$; in particular, ${\|0\|} = 0$. A **[[norm]]** is a pseudonorm that satisfies a converse to this: $v = 0$ if ${\|v\|} = 0$.
A [[norm]] on $V$ is **complete** if, given any infinite [[sequence]] $(v_1, v_2, \ldots)$ such that
\[ \label{Cauchy} \lim_{m,n\to\infty} {\left\| \sum_{i=m}^{m+n} v_i \right\|} = 0 ,\]
there exists a (necessarily unique) **sum** $S$ such that
\[ \label{converge} \lim_{n\to\infty} {\left\| S - \sum_{i=1}^n v_i \right\|} = 0 ;\]
we write
$$ S = \sum_{i=1}^\infty v_i $$
(with the right-hand side undefined if no such sum exists).
Then a **Banach space** is simply a vector space equipped with a complete norm. As in the real line, we have in a Banach space that
$$ {\left\| \sum_{i=1}^\infty v_i \right\|} \leq \sum_{i=1}^\infty {\|v_i\|} ,$$
with the left-hand side guaranteed to exist if the right-hand side exists as a finite real number (but the left-hand side may exist even if the right-hand side diverges, the usual distinction between [[absolute convergence]] and [[conditional convergence]]).
If we do not insist on the space being complete, we call it a **normed (vector) space**. If we have a [[topological vector space]] such that the topology comes from a norm, but we do not make an actual choice of such a norm, then we talk of a **normable space**.
### Banach spaces as metric spaces
The three axioms for a pseudonorm are very similar to the three axioms for a [[pseudometric]].
Indeed, in any pseudonormed vector space, let the **distance** $d(v,w)$ be
$$ d(v,w) = {\|w - v\|} .$$
Then $d$ is a pseudometric, which is __translation-invariant__ in that
$$ d(v+x,w+x) = d(v,w) $$
always holds. Conversely, given any translation-invariant pseudometric $d$ on a vector space $V$, let ${\|v\|}$ be
$$ {\|v\|} = d(0,v) .$$
Then ${\|-\|}$ satisfies the axioms (1--3) for a pseudonorm, except that it may satisfy (2) only for $r = 0, \pm 1$. (In other words, it is only a [[G-pseudonorm]].) It will actually be a pseudonorm iff the pseudometric satisfies a homogeneity rule:
$$ d(r v,r w) = {|r|} d(v,w) .$$
Thus pseudonorms correspond precisely to homogeneous translation-invariant pseudometrics.
Similarly, norms correspond to homogenous translation-invariant metrics and complete norms correspond to complete homogeneous translation-invariant metrics. Indeed, (eq:Cauchy) says that the sequence of partial sums is a [[Cauchy sequence]], while (eq:converge) says that the sequence of partial sums converges to $S$.
Thus a Banach space may equivalently be defined as a vector space equipped with a complete homogeneous translation-invariant metric. Actually, one usually sees a sort of hybrid approach: a Banach space is a normed vector space whose corresponding metric is complete.
### Maps between Banach spaces
{#morphisms}
If $V$ and $W$ are pseudonormed vector spaces, then the **norm** of a linear function $f\colon V \to W$ may be defined in either of these equivalent ways:
* $ {\|f\|} = \sup \{ {\|f v\|} \;|\; {\|v\|} \leq 1 \} $;
* $ {\|f\|} = \inf \{ r \;|\; \forall{v},\; {\|f v\|} \leq r {\|v\|} \} $.
(Some other forms are sometimes seen, but these may break down in degenerate cases.)
For finite-dimensional spaces, any linear map has a well-defined finite norm. In general, the following are equivalent:
* $f$ is [[continuous map|continuous]] (as measured by the pseudometrics on $V$ and $W$) at $0$;
* $f$ is continuous (everywhere);
* $f$ is [[uniformly continuous map|uniformly continuous]];
* $f$ is [[Lipschitz map|Lipschitz continuous]];
* ${\|f\|}$ is finite (and, in [[constructive mathematics]], [[located real number|located]]);
* $f$ is [[bounded map|bounded]] (as measured by the [[bornologies]] given by the pseudometrics on $V$ and $W$).
In this case, we say that $f$ is **bounded**. If $f\colon V \to W$ is not assumed to be linear, then the above conditions are no longer equivalent.
The bounded linear maps from $V$ to $W$ themselves form a pseudonormed vector space $\mathcal{B}(V,W)$. This will be a Banach space if (and, except for degenerate cases of $V$, only if) $W$ is a Banach space. In this way, the category $Ban$ of Banach spaces is a [[closed category]] with $\mathbb{R}$ as the unit.
The clever reader will note that we have not yet defined $\mathbf{Ban}$ as a category! (surprisingly in the _nLab_) There are many (nonequivalent) ways to do so.
In [[functional analysis]], the usual notion of '[[isomorphism]]' for Banach spaces is a bounded bijective linear map $f\colon V \to W$ such that the [[inverse function]] $f^{-1}\colon W \to V$ (which is necessarily linear) is also bounded. In this case one can accept all bounded linear maps between Banach spaces as morphisms. Analysts sometimes refer to this as the "isomorphic category".
Another natural notion of isomorphism is a surjective linear isometry. In this case, we take a morphism to be a **[[short map|short]]** [[linear map]], or [[short linear map|linear contraction]]: a linear map $f$ such that ${\|f\|} \leq 1$. This category, which is what category theorists generally refer to as $\mathbf{Ban}$, is sometimes referred to as the "isometric category" by analysts. Note that this makes the 'underlying set' (in the sense of $\mathbf{Ban}$ as a [[concrete category]] like any closed category) of a Banach space its (closed) **unit ball**
$$ Hom_Ban(\mathbb{R},V) \cong \{ v \;|\; {\|v\|} \leq 1 \} $$
rather than the set of all vectors in $V$ (the underlying set of $V$ as a vector space).
+-- {: .query}
Yemon Choi: This is really here to remind myself how to make query boxes. But while I'm at it, is it really OK to refer to the "unit ball functor" as "taking the underlying set"? I notice that on the discussion about internal homs at [[internal hom]] it is claimed that "Every closed category is a [[concrete category]] (represented by $I$), and the underlying set of the internal hom is the external hom" which seems to require "underlying set" to be interpreted in this looser sense.
_Toby_: Sure, but the point of putting 'underlying set' in scare quotes is precisely to point out that the category-theoretic underlying set is not what one would normally expect.
=--
+-- {: .query}
Mark Meckes: I've expanded this section in part to be consistent with analysts' terminology. I've made some assumptions about category theorists' conventions which might not be correct. (If I find time I might write about other categories of Banach spaces that analysts think about.)
_Toby_: Looks good to me!
=--
From a category-theorist\'s perspective, the isomorphic category is really the [[full image]] of the [[inclusion functor]] from $Ban$ to $TVS$ (the category of [[topological vector spaces]]), which may be denoted $Ban_{TVS}$. If you\'re working in $Ban_{TVS}$, then you only care about the topological linear structure of your space (although you do also care that it can be derived from some metric); if you\'re working in $Ban$, then you care about all of the structure on the space.
## Examples
Many examples of Banach spaces are parametrised by an exponent $1 \leq p \leq \infty$. (Sometimes one can also try $0 \leq p \lt 1$, but these generally don\'t give Banach spaces.)
* The [[Cartesian space]] $\mathbb{R}^n$ is a Banach space with
$$ {\|(x_1,\ldots,x_n)\|_p} = \root p {\sum_i {|x_i|^p}} .$$
(We can allow $p = \infty$ by taking a limit; the result is that ${\|x\|_\infty} = \max_i {|x_i|}$.) Every finite-dimensional Banach space is isomorphic to this for some $n$ and $p$; in fact, once you fix $n$, the value of $p$ is irrelevant up to isomorphism.
* The [[sequence space]] $l^p$ is the set of infinite [[sequence]]s $(x_1,x_2,\ldots)$ of real numbers such that
$$ {\|(x_1,x_2,\ldots)\|_p} = \root p {\sum_i {|x_i|^p}} $$
exists as a finite real number. (The only question is whether the sum converges. Again $p = \infty$ is a limit, with the result that ${\|x\|_\infty} = \sup_i {|x_i|}$.) Then $l^p$ is a Banach space with that norm. These are all versions of $\mathbb{R}^\infty$, but they are no longer isomorphic for different values of $p$. (See [[isomorphism classes of Banach spaces]].)
* More generally, let $A$ be any [[set]] and let $l^p(A)$ be the set of [[function]]s $f$ from $A$ to $\mathbb{R}$ such that
$$ {\|f\|_p} = \root p {\sum_{x: A} {|f(x)|^p}} $$
exists as a finite real number. (Again, ${\|f\|_\infty} = \sup_{x\colon A} {|f(x)|}$.) Then $l^p(A)$ is a Banach space. (This example includes the previous examples, for $A$ a countable set.)
* On any [[measure space]] $X$, the [[Lebesgue space]] $\mathcal{L}^p(X)$ is the set of measurable almost-everywhere-defined real-valued functions on $X$ such that
$$ {\|f\|_p} = \root p {\int {|f|^p}} $$
exists as a finite real number. (Again, the only question is whether the integral converges. And again $p = \infty$ is a limit, with the result that ${\|f\|_\infty}$ is the [[essential supremum]] of ${|f|}$.) As such, $\mathcal{L}^p(X)$ is a complete pseudonormed vector space; but we identify functions that are equal almost everywhere to make it into a Banach space. (This example includes the previous examples, for $X$ a set with counting measure.)
* Any [[Hilbert space]] is Banach space; this includes all of the above examples for $p = 2$.
## Operations on Banach spaces
The category $Ban$ of Banach spaces is [[complete category|small complete]], [[cocomplete category|small cocomplete]], and [[symmetric monoidal closed category|symmetric monoidal closed]] with respect to its standard internal hom (described at [[internal hom]]). Some details follow.
* The category of Banach spaces admits small [[product]]s. Given a small family of Banach spaces $\{X_\alpha\}_{\alpha \in A}$, its product in $Ban$ is the subspace of the vector-space product
$$\prod_{\alpha \in A} X_\alpha$$
consisting of $A$-tuples $\langle x_\alpha \rangle$ which are _uniformly_ bounded (i.e., there exists $C$ such that $\forall \alpha \in A: {\|x_\alpha\|} \leq C$), taking the least such upper bound as the norm of $\langle x_\alpha \rangle$. This norm is called the $\infty$-norm; in particular, the product of an $A$-indexed family of copies of $\mathbb{R}$ or $\mathbb{C}$ is what is normally denoted as $l^{\infty}(A)$.
* The category of Banach spaces admits [[equalizer]]s. Indeed, the equalizer of a pair of maps $f, g: X \rightrightarrows Y$ in $Ban$ is the [[kernel]] of $f-g$ under the norm inherited from $X$ (the kernel is closed since $f-g$ is continuous, and is therefore complete). In fact every equalizer is even a [[section]] by the [[Hahn-Banach theorem]]. Every [[extremal monomorphism]] is even already an equalizer (and a section): Let $f\colon X \to Y$ be an extremal monomorphism, $\iota\colon \Im(f) \to Y$ the embedding of $Im(f)$ into the codomain of $f$ and $f\prime \colon X \to Im(f)$ $f$ with restricted codomain. Since $f\prime$ is an epimorphism, $f=\iota f\prime$, and $f$ extremal, $f\prime$ is an isomorphism, thus $f$ is an embedding.
* The category of Banach spaces admits small [[coproduct]]s. Given a small family of Banach spaces $\{X_\alpha\}_{\alpha \in A}$, its coproduct in $Ban$ is the completion of the vector space coproduct
$$\bigoplus_{\alpha \in A} X_\alpha$$
with respect to the norm given by
$$ {\left\| \bigoplus_{s \in S} x_s \right\|} = \sum_{s \in S} {\|x_s\|} ,$$
where $S \subseteq A$ is finite and ${\|x_s\|}$ denotes the norm of an element in $X_s$. This norm is called the $1$-norm; in particular, the coproduct of an $A$-indexed family of copies of $\mathbb{R}$ or $\mathbb{C}$ is what is normally denoted as $l^1(A)$.
* {#Coequalizers} The category of Banach spaces admits [[coequalizers]]. Though one may expect the coequalizer of a pair of maps $f,g: X \rightrightarrows Y$ to be the [[cokernel]] of $f-g$ under the quotient norm (in which the norm of a coset $y + C$ is the minimum norm attained by elements of $y + C$; here $C$ is the [[image]] $(f-g)(X)$), these spaces are not Banach spaces in general, as the image of a map $f$ need not be closed (indeed, the inclusion $i: \ell_1 \hookrightarrow c_0$ has dense image in $c_0$), and so the quotient space may not be complete. However, the quotient by the closure of $(f-g)(X)$ suffices.
* To describe the tensor product $X \otimes_{Ban} Y$ of two Banach spaces (making $Ban$ symmetric monoidal closed with respect to its usual internal hom), let $F(X \times Y)$ be the free vector space generated by the set $X \times Y$, with norm on a typical element defined by
$$ {\left\| \sum_{1 \leq i \leq n} a_i (x_i \otimes y_i) \right\|} = \sum_{1 \leq i \leq n} {|a_i|} {\|x_i\|} \cdot {\|y_i\|}.$$
Let $\overline{F}(X \times Y)$ denote its completion with respect to this norm. Then take the cokernel of $\overline{F}(X \times Y)$ by the closure of the subspace spanned by the obvious bilinear relations. This quotient is $X \otimes_{Ban} Y$.
In the literature on Banach spaces, tensor product above is usually called the __projective tensor product__ of Banach spaces; see other [[tensor product of Banach spaces]]. The product and coproduct are considered __direct sums__; see other [[direct sums of Banach spaces]].
To be described:
* duals ($p + q = p q$);
* completion ($Ban$ is a [[reflective subcategory]] of $PsNVect$ (pseudo-normed vector spaces)).
* $Ban$ as a (somewhat larger) category with duals.
## Integration in Banach spaces
This paragraph describes some aspects of integration theory in Banach spaces that are relevant to understand the literature about [[AQFT]]. In the given context, elements of a Banach space $\mathcal{B}$ are sometimes called vectors, a function or measure taking values in $\mathcal{B}$ are therefore called vector functions and vector measures. Functions and measures taking values in the [[field]] that the Banach space is defined upon as a vector space are called scalar functions and scalar measures.
We will consider two types of integrals:
* integrals of vector functions with respect to a scalar measure, specifically the Bochner integral,
* integrals of scalar functions with respect to a vector measure, specifically the spectral integral of a normal operator on a Hilbert space.
### Bochner integral
The Bochner integral is a direct generalization of the Lesbegue integral to functions that take values in a Banach space. Whenever you encounter an integral of a function taking values in a Banach space in the [[AQFT]] literature, it is safe to assume that it is meant to be a Bochner integral. Two points already explained by Wikipedia are of interest:
1. A version of the dominated convergence theorem is true for the Bochner integral.
2. There are theorems that are not valid for the Bochner integral, notably the Radon-Nikodym theorem does not hold in general.
* [Wikipedia] (http://en.wikipedia.org/wiki/Bochner_integral)
_reference_: Joseph Diestel: "Sequences and Series in Banach Spaces" ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0542.46007&format=complete)), chapter IV.
### The spectral integral
The integral with respect to the spectral measure of a bounded normal operator on a Hilbert space is an example of a Banach space integral with respect to a vector measure.
In this paragraph we present a well known, but somewhat less often cited result, that is of use in some proofs in some approaches to [[AQFT]], it is the version of the dominated convergence theorem for the given setting.
Let A be a bounded normal operator on a Hilbert space and E be it's spectral measure (the "resolution of identity" in the terms of Dunford and Schwartz). Let $\sigma(A)$ be the spectrum of A. For a bounded complex Borel function f we then have
$$
f(A) \coloneqq \int_{\sigma(A)} f(\lambda) E(d\lambda)
$$
+-- {: .un_theorem}
###### Theorem (dominated convergence)
If the uniformly bounded sequence $\{f_n\}$ of complex Borel functions converges at each point of $\sigma(A)$ to the function $f$, then $f_n(A) \to f(A)$ in the strong operator topology.
=--
+-- {: .proof}
See Dunford, Schwartz II, chapter X, corollary 8.
=--
## Properties
### Relation to bornological spaces
Every [[inductive limit]] of [[Banach spaces]] is a [[bornological vector space]]. ([Alpay-Salomon 13, prop. 2.3](#AlpaySalomon13))
Conversely, every [[bornological vector space]] is an inductive limit of [[normed spaces]], and of [[Banach spaces]] if it is quasi-complete ([Schaefer-Wolff 99](#SchaeferWolff99))
## Related concepts
* [[reflexive Banach space]]
* [[projective Banach space]]
* [[Banach analytic space]]
[[!include analytic geometry ingredients -- table]]
## References
Named after [[Stefan Banach]].
* Walter Rudin, _Functional analysis_
* Dunford, Nelson; Schwartz, Jacob T.: "Linear operators. Part I: General theory." ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0635.47001&format=complete)), "Linear operators. Part II: Spectral theory, self adjoint operators in Hilbert space." ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0635.47002&format=complete))
* Z. Semadeni, _Banach spaces of continuous functions_, vol. I, Polish scientific publishers. Warszawa 1971
* {#AlpaySalomon13} Daniel Alpay, Guy Salomon, _On algebras which are inductive limits of Banach spaces_ ([arXiv:1302.3372](http://arxiv.org/abs/1302.3372))
* {#SchaeferWolff99} H. H. Schaefer with M. P. Wolff, _Topological vector spaces_, Springer 1999
* [[Jiří Rosický]], _Are Banach spaces monadic?_, ([arXiv:2011.07543](https://arxiv.org/abs/2011.07543))
category: analysis
[[!redirects Banach space]]
[[!redirects Banach spaces]]
[[!redirects Banach vector space]]
[[!redirects Banach vector spaces]]
[[!redirects Ban]]
[[!redirects norm isomorphism]]
[[!redirects norm-isomorphic]]
[[!redirects norm isomorphic]]
|
Banach-Alaoglu theorem | https://ncatlab.org/nlab/source/Banach-Alaoglu+theorem |
# The Banach--Alaoglu Theorem
* table of contents
{: toc}
## Idea
The closed unit ball of the double dual of a [[Banach space]] is weak* compact. This theorem is equivalent to the [[Tychonoff theorem]] for [[compact space|compact]] [[Hausdorff topological space|Hausdorff]] [[topological space|spaces]] (eg [Rossi 2009](#Rossi09)), and as such is not constructive in general. However, the restriction of the statement to _separable_ Banach spaces _is_ constructive, and is used in PDE theory to find solutions. A constructive proof can be found e.g. in [BridgesVita](#BridgesVita).
## Localic version
As usual, when formulating the theorem using [[locales]] instead of [[topological spaces]] the theorem holds constructively; see [Pelletier](#Pelletier) and [Henry](#Henry).
## Related entries
The following is a generalisation to [[locally convex topological vector space|locally convex]] spaces:
* [[Bourbaki-Alaoglu theorem|Bourbaki–Alaoglu theorem]]
## References
* Wikipedia, [Banach–Alaoglu theorem](https://en.wikipedia.org/wiki/Banach%E2%80%93Alaoglu_theorem)
* {#Rossi09} Stefano Rossi, _The Banach-Alaoglu theorem is equivalent to the Tychonoff theorem for compact Hausdorff spaces_, arXiv:[0911.0332](http://arxiv.org/abs/0911.0332)
* {#Pelletier} Joan Wick Pelletier, _Locales in functional analysis_ [DOI](http://dx.doi.org/10.1016/0022-4049%2891%2990013-R)
* {#Henry} Simon Henry, _Localic Metric spaces and the localic Gelfand duality_ [arxiv](https://arxiv.org/pdf/1411.0898)
* {#BridgesVita} Bridges and Vita, _Techniques of Constructive Analysis_ [link](http://www.springer.com/in/book/9780387336466)
[[!redirects Banach-Alaoglu theorem]]
[[!redirects Banach–Alaoglu theorem]]
[[!redirects Banach--Alaoglu theorem]]
[[!redirects Banach-Alaoglu Theorem]]
[[!redirects Banach–Alaoglu Theorem]]
[[!redirects Banach--Alaoglu Theorem]]
|
Banach-Steinhaus theorem | https://ncatlab.org/nlab/source/Banach-Steinhaus+theorem | * [uniform boundedness principle](https://en.wikipedia.org/wiki/Uniform_Boundedness_Principle) - article on Banach–Steinhaus theorem in WIkipedia |
Banach-Tarski paradox | https://ncatlab.org/nlab/source/Banach-Tarski+paradox |
+-- {: .rightHandSide}
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###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
What is known as the _Banach-Tarski paradox_ is the [[theorem]] ([Banach-Tarski 24](#BanachTarski24)) that the [[axiom of choice]] implies that any two [[bounded subsets]] in [[Euclidean space]] of [[dimension]] $d \geq 3$ may be partitioned by a [[finite number]] of pairwise congruent [[subsets]].
This is perceived as a [[paradox]] due to its counter-intuitive interpretation, which becomes particularly vivid if one takes one of the two bounded subsets to be the [[disjoint union]] of two copied of the other: In this case the theorem says, intuitively, that it is possible to break up any shape in 3d [[Euclidean space]] into a [[finite number]] of pieces, such that re-assembling these pieces suitably yields not just the original shape, but that and an entire other copy of it.
It has been pointed out that it is not just the use of the [[axiom of choice]] that is responsible for this perceived [[paradox]], but also the point-based concept of [[topological spaces]] as such, see the discussion _In point-free topology_ [below](#InPointFreeTopology).
## In point-free topology
{#InPointFreeTopology}
It is argued in ([Simpson 12](#Simpson12)) that the Banach-Tarski paradox disappears if one works in [[point-free topology]], hence with [[locales]] instead of just [[topological spaces]]:
> We view spaces of interest as [[locales]], and the notion of "part" is given by the standard notion of [[sublocale]], $[\cdots]$. Every [[topological space]] determines a [[locale]] $[\cdots]$. However, when a space is viewed as a locale, the notion of [[sublocale]] gives rise to new "parts" of spaces that are not merely subsets, and need not be determined by their points.
> The usual contradictions are avoided $[$this way$]$. The different pieces in the partitions defined by Vitali and by Banach and Tarski are deeply intertangled with each other. According to our notion of "part", two such intertangled pieces are not disjoint from each other, so additivity does not apply. An intuitive explanation for the failure of disjointness is that, although two such pieces share no point in common, they nevertheless overlap on the topological “glue” that bonds neighbouring points together.
## References
The original article:
* {#BanachTarski24} [[Stefan Banach]], [[Alfred Tarski]], _Sur la décomposition des ensembles de points en parties respectivement congruentes_ Fundamenta Mathematicae (in French). 6: 244–277, 1924 ([pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm6/fm6127.pdf))
Textbook account:
* [[Cornelia Druţu]], [[Michael Kapovich]], Chapter 17 of: *Geometric group theory*, Colloquium Publications **63**, AMS 2018 ([ISBN:978-1-4704-1104-6](https://bookstore.ams.org/coll-63/), [pdf](https://courses.maths.ox.ac.uk/node/view_material/35649))
See also:
* Wikipedia, _[Banach-Tarski paradox](https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox)_
Discussion in [[point-free topology]]:
* {#Simpson12} [[Alex Simpson]], _Measure, randomness and sublocales_, Annals of Pure and Applied Logic Volume 163, Issue 11, November 2012, Pages 1642-1659 ([pdf](http://homepages.inf.ed.ac.uk/als/Research/Sources/mrs.pdf), [doi:10.1016/j.apal.2011.12.014](https://doi.org/10.1016/j.apal.2011.12.014))
|
bananaspace | https://ncatlab.org/nlab/source/bananaspace |
*[bananaspace](https://www.bananaspace.org/wiki/%E9%A6%96%E9%A1%B5)* (香蕉空间) is a wiki on mathematics in Chinese created by Chenjing Bu, somewhat akin in style to (and apparently inspired by) the [[nLab]].
The name "bananaspace" is an inside-joke on a typo for *[[Banach space]]* that occurred in the presentation of a friend of the founders ([here](https://twitter.com/jasonchen0325/status/1526583860191825920), cf. [Miller 2011, p. 2](http://www-personal.umich.edu/~asnowden/teaching/2011/18.904/lec/lec-03-02a.pdf#page=2)).
## See also
* [[Stacks Project]]
* [[Kerodon]]
category: reference |
Banch-Tarski theorem | https://ncatlab.org/nlab/source/Banch-Tarski+theorem | |
Baptiste Morin | https://ncatlab.org/nlab/source/Baptiste+Morin |
* [webpage](https://www.math.u-bordeaux.fr/~bmorin/)
##Publications
* _Utilisation d'une cohomologie étale équivariante en topologie arithmétique_
Compositio Math. 144 (2008), no. 1, 32-60.
category: people
|
bar > history | https://ncatlab.org/nlab/source/bar+%3E+history | < [[bar]] |
bar and cobar construction | https://ncatlab.org/nlab/source/bar+and+cobar+construction |
+-- {: .rightHandSide}
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###Context###
#### Differential-graded objects
+--{: .hide}
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=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--
#Contents#
* automatic table of contents goes here
{:toc}
## Bar and cobar constructions
* There is a brief entry at [[bar construction]] together with a blog link
* [[Todd Trimble]], _On the Bar Construction_ ([blog](http://golem.ph.utexas.edu/category/2007/05/on_the_bar_construction.html))
* There is some discussion of the bar-cobar adjointness as it relates to [[twisting cochain|twisting cochains]], at that entry.
* Here we will concentrate on the bar-cobar adjointness itself and start exploring the links with other parts of differential algebra.
One of the earliest examples of a pair of [[adjoint functor|adjoint functors]] studied in [[algebraic topology]] was that giving the relationship between the functors for [[reduced suspension]] and [[loop space object|based loop space]]. If we consider a [[pointed topological space|pointed]] [[connected topological space]] $(X,x_0)$, then its [[reduced suspension]] $\Sigma X$ is obtained by taking the cylinder $I\times X$ and identifying the subspace $\{0,1\}\times X\cup I\times \{x_0\}$ to a point. (Think of crushing the two ends of the cylinder and the line through the base point to a point.) This can also be thought of as forming $S^1\wedge X$ the [[smash product]] of the circle with $X$.
Adjoint to $\Sigma$ is the [[based loop space]] functor: $\Omega Y$ is the space of pointed maps from $S^1$ to $Y$. This has a monoid structure (up to homotopy) given by concatenation of loops. (Back in $S^1$, we have a comonoid structure with respect to the pointed coproduct $S^1\to S^1\vee S^1$ as described at [[interval object]]. This in some sense is _'subdivision as an inverse for composition'_.)
(perhaps: Picture to go here?)
Using ordinary (co)homology to study spaces such as CW-complexes, we naturally use the complexes of (cellular) chains on spaces. The structure of chains on the suspension is easy to work out using the obvious cellular structure, but that on the loop space is much harder as $\Omega X$ is given the [[compact-open topology]] and only has the homotopy type of a CW-complex, so no nice cellular structure is given us 'on a plate'. The idea is thus to start with a chain complex model, $C_*(X)$, for a CW-complex, $X$, (usually the complex of cellular chains on $X$), and we try to construct from $C_*(X)$ a 'model' for the chain complex of the loop space $\Omega X$ of $X$. Adams' cobar construction was such a method (see below). This was adjoint to a bar construction defined by Eilenberg and MacLane.
Both directions use an abstract algebraic model of concatenation of paths and so their construction is linked to that of free monoids, and through those to [[monads]], [[operads]] and related abstract machinery to handle concatenation and its higher categorical analogues in categorical contexts.
The chain complex $C_*(X)$ has a rich coalgebraic structure induced by a cellular diagonal approximation on $X$ so the *cobar* construction will start with a [[differential graded coalgebra|dg-coalgebra]] as 'input' and as output we will hope for both a coalgebra structure (reflecting the chain coalgebra idea) and an algebra structure (coming from modelling the concatenation of loops). We therefore might hope for, and in fact do get, a [[differential graded Hopf algebra]].
Going the other way, we start with a [[differential graded algebra]] and use 'coconcatenation' or 'subdivision' to get a coalgebra structure. In fact, once again, this is a Hopf algebra.
These topologically motivated constructions can be applied in much greater generality as we will see both here and elsewhere:
## Definitions
### The bar construction
(due originally Eilenberg-MacLane) Remember this goes from 'algebras' to Hopf algebras in general.
\[B :pre \varepsilon CDGA \to pre CDGHA\]
Let $(A,d,\varepsilon) $ be a commutative, augmented differential $\mathbb{Z}$-graded algebra, $d(A_n)\subseteq A_{n-1}$, $\overline{A} = Ker \varepsilon$.
The **bar construction** $B(A,d,\varepsilon)$ is given by
$$B(A,d,\varepsilon) = (T(s\overline{A}), D),$$
where
* $T(s\overline{A})$ is the commutative [[differential graded Hopf algebra]] generated by $s\overline{A}$, $s$ being the _suspension_ (shift, translation, etc) operator discussed in [[graded vector space|graded vector spaces]],
* $D = d_I + d_E$, where
$$d_I(s a_1\otimes \ldots\otimes s a_n) = -\sum_{i = 1} ^n\eta(i-1)s a_1\otimes \ldots \otimes s a_{i-2}\otimes s d a_{i-1}\otimes\ldots s a_n,$$
and
$$d_E(s a_1\otimes \ldots\otimes s a_n) = -\sum_{i = 1} ^n\eta(i-1)s a_1\otimes \ldots \otimes s a_{i-2}\otimes s a_{i-1}.a_i\otimes \ldots s a_n,$$
with $\eta(i) = (-1)^{\sum_{k=1}^i |s a_k|}$.
**Note**
1. _the image of a 1-connected cdga is a connected commutative Hopf algebra_.
1. The construction uses the _suspension_ operator on the graded vector spaces. This mirrors the reduced suspension at the cell complex level.
* The construction uses a tensor algebra construction. This from one point of view handles the formal concatenation aspect, but has also a rich structure of a coalgebraic structure with reduced diagonal, given by
$$\bar{\Delta}(v_1\otimes \ldots \otimes v_n) = \sum_{p=1}^{n-1} (v_1\otimes \ldots \otimes v_p)\otimes(v_{p+1}\otimes \ldots \otimes v_n),$$
(see [[differential graded coalgebra]]).
This can be interpreted as looking at how a formal concatenation can be 'subdivided' into its various parts.
### The Cobar construction
(due to J. F. Adams, see [Felix-Halperin0Thomas 92](#FelixHalperinThomas92))
We define a functor:
\[F :pre \eta CoDGC \to pre CoDGHA\]
so essentially from cocommutative [[differential graded coalgebra|differential graded coalgebras]] to cocommutative [[differential graded Hopf algebra|differential graded Hopf algebras]] (with frills attached in the way of coaugmentations, etc).
Let $(C,\partial,\eta)$ be a cocommutative differential $\mathbb{Z}$-graded coaugmented coalgeba:
$$\partial(C_n) \subseteq C_{n-1}, \quad \overline{C} = C/\eta(k), \quad \overline{\Delta} : \overline{C} \to \overline{C}\otimes \overline{C}.$$
The _Cobar construction_ $F(C,\partial, \eta)$ is the cocommutative pre-dgha defined by
* $F(C,\partial,\eta) = (T(s^{-1}\overline{C}), \delta)$,
where $\delta = \partial_I + \partial_E$.
Here
* $T(s^{-1}\overline{C})$ is the cocommutative Hopf algebra generated by $s^{-1}\overline{C}$, as before(in [[differential graded coalgebra]]) $\overline{C}$ is the cokernel of the coaugmentation, $\eta$)
* $$\partial_I (s^{-1}c_1\otimes \ldots\otimes s^{-1}c_n) = -\sum_{i = 1} ^n\eta(i-1)s^{-1}c_1\otimes \ldots\otimes s^{-1}c_{i-1}\otimes s^{-1}\partial c_i\otimes \ldots s^{-1}c_n,$$
and
* $$\partial_E (s^{-1}c_1\otimes \ldots\otimes s^{-1}c_n) = -\sum_{i = 1} ^n\eta(i-1)\sum_\mu (-1)^{|c'_{i\mu}| +1} (s^{-1}c_1\otimes \ldots\otimes s^{-1}c'_{i\mu}\otimes s^{-1}c^{\prime\prime}_{i\mu}\otimes \ldots\otimes s^{-1}c_n),$$
with $\overline{\Delta}c_i = \sum_\mu c'_{i\mu}\otimes c^{\prime\prime}_{i\mu}$; $\eta(i) = (-1)^{ \sum^i_{k=1}|s^{-1}c_k|}.$
The image of a 1-connected cdgc is a connected cocommutative dgha.
If $C$ is of finite type, $\#F(C,\partial,\eta)$ is isomorphic to $B\#(C,\partial,\eta)$ as a differential $\mathbb{Z}$-graded Hopf algebra.
If $A$ is not (graded) commutative, the differential $d_E$ of $B(A,d,\varepsilon)$ does not respect the shuffle product on $T(s\overline{A})$; $B(A,d,\varepsilon)$ thus becomes merely a differential $\mathbb{Z}$-graded coalgebra. Similarly if $C$ is not (graded) cocommutative $F(C,\partial,\eta)$ is merely a differential $\mathbb{Z}$-graded algebra.
In particular, let
* $\varepsilon-DGA$ be the category of augmented differential graded algebras, ($A = \oplus_{p\geq 0}A_p$).
* $DGC_0$, the category of connected differential graded coalgebras,
then the Bar and Cobar constructions yield functors
$$B: \varepsilon DGA\to DGC_0$$
$$F : DGC_0\to \varepsilon DGA.$$
+-- {: .num_theorem}
###### Proposition
(Husemoller-Moore-[[Jim Stasheff|Stasheff]])
$B$ is right adjoint to $F$.
For any objects $(A,d)$ in $\varepsilon-DGA$, and $(C,\partial)$ of $DGC_0$, the natural adjunction morphisms
$$\hat{\alpha} : FB(A,d) \to (A,d)$$
$$\hat{\beta} : (C,\partial) \to BF(C,\partial)$$
are weak equivalences / [[quasi-isomorphism|quasi-isomorphisms]].
=--
These latter morphisms are defined by
* $\hat{\alpha} : T(s^{-1}\overline{T(s\overline{A})}), \delta)\to (A,d)$ is the zero mapping on $s^{-1}T^{\geq 2}(s\overline{A})$ and the natural isomorphism $s^{-1}s\overline{A} \stackrel{\simeq}{\to} \overline{A}$ on $s^{-1}s\overline{A}$.
* $\hat{\beta} : (C,\partial) \to (T(\overline{sT(s^{-1}\overline{C}}),D)$ is the unique lifting of
$$C\to s^{-1}\overline{C} \to \overline{T(s^{-1}\overline{C})}\to \overline{sT(s^{-1}\overline{C})}.$$
## Related concepts
* [[Koszul duality]]
## References
The source used for the above was
* D. Tanré, _Homotopie rationnelle: Modèles de Chen, Quillen, Sullivan_, Lecture Notes in Maths No. 1025, Springer, 1983.
This was augmented with material from
* H. J. Baues, _Geometry of loop spaces and the cobar construction_, Mem. Amer.
Math. Soc. 25 (230) (1980) ix+171.
See also:
* {#FelixHalperinThomas92} [[Yves Felix]], [[Stephen Halperin]], [[Jean-Claude Thomas]], _Adams' Cobar Equivalence_, Transactions of the American Mathematical Society, Vol. 329, No. 2 (1992), pp. 531-549 ([jstor:2153950](https://www.jstor.org/stable/2153950))
Review:
* [[Dev Sinha]], Section 1 in: _Koszul duality in algebraic topology - an historical perspective_, J. Homotopy Relat. Struct. (2013) 8: 1 ([arXiv:1001.2032](https://arxiv.org/abs/1001.2032))
* [[Mathieu Anel]], [[André Joyal]], _On the bar-cobar duality for algebras and operads_, slides 2012, [pdf](http://mathieu.anel.free.fr/mat/doc/Anel-Joyal-Boston-Operades.pdf)
* [[Mathieu Anel]], [[André Joyal]], _Sweedler theory for (co)algebras and the bar-cobar constructions_, [arXiv:1309.6952](https://arxiv.org/abs/1309.6952)
* [[Kathryn Hess]], _The cobar construction: a modern perspective_, lectures 2007, [pdf](https://sma.epfl.ch/~hessbell/Minicourse_Louvain_Notes.pdf)
Identifying the cobar construction on [[singular homology|singular chains]] on a [[topological space]] with the dg-algebra of chains on its [[loop space]]:
* [[John Frank Adams]], *On the cobar construction*, Proc Natl Acad Sci USA **42** 7 (1956) 409–412 [[doi:10.1073/pnas.42.7.409](https://doi.org/10.1073%2Fpnas.42.7.409)]
* [[Manuel Rivera]], [[Mahmoud Zeinalian]], *Cubical rigidification, the cobar construction, and the based loop space*, Algebr. Geom. Topol. **18** (2018) 3789-3820 [[arXiv:1612.04801](https://arxiv.org/abs/1612.04801), [doi:10.2140/agt.2018.18.3789](https://doi.org/10.2140/agt.2018.18.3789)]
* [[Manuel Rivera]], *Adams' cobar construction revisited*, Documenta Math. **27** (2022) 1213-1223 [[arxiv:1910.08455](https://arxiv.org/abs/1910.08455)]
Generalization of the bar-cobar constructions to [[dg-Hopf algebras]]:
* [[Benoit Fresse]], _The universal Hopf operads of the bar construction_ ([arXiv:math/0701245](https://arxiv.org/abs/math/0701245))
* [[Murray Gerstenhaber]], [[Alexander Voronov]], Section 3.2 of: _Homotopy G-algebras and moduli space operad_, Internat. Math. Research Notices (1995) 141-153 ([arXiv:hep-th/9409063](https://arxiv.org/abs/hep-th/9409063))
* Justin Young, _Brace bar-cobar duality_ ([arXiv:1309.2820](https://arxiv.org/abs/1309.2820))
[[!redirects cobar construction]]
[[!redirects cobar constructions]]
[[!redirects co-bar construction]]
[[!redirects co-bar constructions]]
[[!redirects bar-cobar duality]]
[[!redirects bar-cobar dualities]]
[[!redirects bar-cobar construction]]
[[!redirects bar-cobar constructions]]
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bar construction | https://ncatlab.org/nlab/source/bar+construction | +-- {: .rightHandSide}
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###Context###
#### Higher algebra
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[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The **bar construction** takes a [[monad]] $(T, \mu, \epsilon)$ equipped with an [[algebra over a monad|algebra-over-a-monad]] $(A, \rho)$ to the ([[augmented simplicial set|augmented]]) [[simplicial object]] whose structure maps are given by the structure maps of the monad and its action on its algebra:
$$
\mathrm{B}(T,A)
\coloneqq
\left(
\cdots
\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}
T T A
\stackrel{\stackrel{\mu \cdot Id_A}{\longrightarrow}}{\stackrel{T \cdot \rho}{\longrightarrow}}
T A \stackrel{\rho}{\longrightarrow} A
\right)
\,.
$$
This simplicial object can be viewed as a [[resolution]] of $A$, in a sense explained below.
## Definition
Let $\mathbf{E}$ be a [[category]] and let $(T, m: T T \to T, u: 1_{\mathbf{E}} \to T)$ be a [[monad]] on $\mathbf{E}$. We let $\mathbf{E}^T$ denote the [[Eilenberg-Moore category|category of]] $T$-[[algebra over a monad|algebras]], and $U: \mathbf{E}^T \to \mathbf{E}$ the [[forgetful functor]] which is [[monadic functor|monadic]], with [[left adjoint]] $F$.
Recall that the ([[augmented simplicial set|augmented]]) [[simplex category]] $\Delta_a$, viz. the category consisting of [[finite set|finite]] [[ordinals]][^fine1] and order-preserving maps, is the "[[walking structure|walking]] [[monoid]]", i.e., is [[initial object|initial]] among strict [[monoidal categories]] equipped with a [[monoid object]]. The monoidal product on $\Delta_a$ is ordinal addition $[m]+[n] = [m+n]$. If $[n]$ is the $n$-element ordinal, then the [[terminal object]] $[1]$ carries a unique monoid structure and represents the "generic monoid"[^fine2].
[^fine1]: N.B.: including the empty ordinal.
[^fine2]: If $X: \Delta_a^{op} \to \mathbf{C}$ is a simplicial object, then $X([n])$ is what is usually denoted $X_{n-1}$, the object of cells in dimension $n-1$. Note that $X([0]) = X_{-1}$ is the augmented component. The $n$ can be thought of as the number of vertices of a simplex of dimension $n-1$. We choose the index $n$ over the geometric dimension $n-1$ as it is more convenient for our purposes.
Similarly $\Delta_a^{op}$ is the walking [[comonoid]]. Since the [[comonad]] $F U$ on $\mathbf{E}^T$ can be regarded as a [[comonoid]] in the strict monoidal category of [[endofunctors]] $[\mathbf{E}^T, \mathbf{E}^T]$ (with endofunctor composition as monoidal product), there is a [[strong monoidal functor]] (or in fact a unique [[strict monoidal functor]])
$$\Delta_a^{op} \stackrel{Bar_T}{\to} [\mathbf{E}^T, \mathbf{E}^T]$$
that takes the generic monoid $[1]$ to $F U$ and generally $[n]$ to $(F U)^{\circ n}$.
If furthermore $A$ is a $T$-algebra, there is an [[evaluation]] functor
$$[\mathbf{E}^T, \mathbf{E}^T] \stackrel{eval_A}{\to} \mathbf{E}^T$$
and we have the following definition:
+-- {: .num_defn}
###### Definition
The **bar construction** $Bar_T(A)$ is the simplicial $T$-algebra given by the composite functor
$$\Delta_a^{op} \stackrel{Bar_T}{\to} [\mathbf{E}^T, \mathbf{E}^T] \stackrel{eval_A}{\to} \mathbf{E}^T.$$
The composite
$$\Delta_a^{op} \stackrel{Bar_T(A)}{\to} \mathbf{E}^T \stackrel{U}{\to} \mathbf{E}$$
will here be called the **bar resolution** of $A$.
=--
In the notation of [[two-sided bar constructions]], the bar construction would be written as $Bar_T(A) = B(F, T, A)$, and the bar resolution as $B(T, T, A)$.
## Décalage and resolutions
### Décalage
To explain the sense in which $U Bar_T(A)$ is an *acyclic resolution* of (the constant simplicial object) $A$, we recall the fundamental [[décalage]] construction. Very simply, putting
$$D = [1] + (-): \Delta_a^{op} \to \Delta_a^{op}$$
the décalage functor on simplicial objects $C^{\Delta_a^{op}}$ (valued in a category $\mathbf{C}$) is the functor
$$P: \mathbf{C}^{\Delta_a^{op}} \stackrel{\mathbf{C}^D}{\to} \mathbf{C}^{\Delta_a^{op}}.$$
Note that $D$ has a comonad structure (inherited from the comonoid structure on $[1]$ in $\Delta_a^{op}$, see also at _[décalage -- comonad structure](decalage#DecalageComonad)_), and therefore $P$ also carries a comonad structure. Notice also that there is a comonad map $D \to [1]\circ !$ (where $[1]: 1 \to \Delta_a^{op}$ is left adjoint to $!: \Delta_a^{op} \to 1$ since $[1]$ is initial in $\Delta_a^{op}$), induced by the evident natural inclusion $[1]+i: [1]+[0] \to [1]+[m]$ in $\Delta_a$. This in turn induces a comonad map $P X \to {|X|}$ where
${|-|}$ is the composite ("discretization"):
$$C^{\Delta_a^{op}} \stackrel{ev_{[1]}}{\to} C \stackrel{diag}{\to} C^{\Delta_a^{op}}.$$
The notation $P$ is chosen because décalage is essentially a kind of [[path space]] construction, i.e., in the case $\mathbf{C} = Set$ it is a [[simplicial sets]] analogue of a topological pullback
$$\array{
P X & \to & X^I & \stackrel{ev_1}{\to} X \\
\downarrow & & \downarrow_\mathrlap{ev_0} & \\
{|X|} & \underset{id}{\to} & X
}$$
where $id: {|X|} \to X$ is the identity inclusion of the underlying set with the discrete topology. $P X$ is essentially a sum of spaces of based paths $(\alpha: (I, 0) \to (X, x_0)$ over all possible choices of basepoint $x_0$, fibered over $X$ by taking $\alpha$ to $\alpha(1)$. Each space of based paths is contractible and therefore $P X$ is acyclic.
The following definition names a nonce expression; this author (Todd Trimble) does not know how this is (or might be) referred to in the literature:
+-- {: .num_defn #acyclic}
###### Definition
An **acyclic structure** on a simplicial object $X: \Delta_a^{op} \to C$ is a $P$-coalgebra structure $X \to P X$.
=--
Here a $P$-coalgebra structure on $X$ is the same as a *right* $D$-coalgebra (or $D$-comodule) structure, given by a simplicial map $h: X \to X \circ D$ satisfying evident equations. In more nuts-and-bolts terms, it consists of a series of maps $h_n: X([n]) \to X([n+1])$ satisfying suitable equations.
The map $h: X \to X D$ may be viewed as a homotopy. Again, turning to the topological analogue for intuition, the corresponding $h: X \to P X$ is a homotopy (or rather, the composite $X \to P X \to X^I$ can be turned into a homotopy $I \times X \to X$). The coalgebra structure $h: X \to P X$ has a retraction given by the counit $\varepsilon: P X \to X$, so $X$ becomes a retract of an acyclic space, hence acyclic itself.
+-- {: .num_remark #rem}
###### Remark
Definition \ref{acyclic} gives an *absolute* notion of acyclicity, in the sense that if $X: \Delta_a^{op} \to \mathbf{C}$ carries an acyclic structure $h: X \to X D$ and $G: \mathbf{C} \to \mathbf{C}'$ is any functor, then $G X$ automatically carries an acyclic structure $G h: F X \to F X D$. (For example, $G X$ becomes acyclic in a standard model category sense under any functor $G: \mathbf{C} \to Set$.)
=--
### Resolutions
Returning now to the bar resolution $U Bar_T(A)$: there is a canonical [[natural isomorphism]] $T \circ U Bar_T \cong U Bar_T \circ D$ obtained as the following 2-cell pasting (where $U Bar_T$ abbreviates the top and bottom horizontal composites)
$$\label{strong}\array{
\Delta_a^{op} & \stackrel{Bar_T}{\to} & [\mathbf{E}^T, \mathbf{E}^T] & \stackrel{[id, U]}{\to} & [\mathbf{E}^T, \mathbf{E}] \\
_\mathllap{D = [1] + (-)} \downarrow & \cong & _\mathllap{[id, F U]} \downarrow & \cong & \downarrow_\mathrlap{[id, U F] = [id, T]} \\
\Delta_a^{op} & \underset{Bar_T}{\to} & [\mathbf{E}^T, \mathbf{E}^T] & \underset{[id, U]}{\to} & [\mathbf{E}^T, \mathbf{E}],
}$$
whence there is a homotopy
$$h = (U Bar_T \stackrel{u U Bar_T}{\to} T U Bar_T \cong U Bar_T D).$$
+-- {: .num_prop}
###### Proposition
The map $h$ is an acyclic structure, def. \ref{acyclic}, i.e., a right $D$-coalgebra structure.
=--
+-- {: .proof}
###### Proof
We verify the coassociativity condition for the coaction $h: U Bar_T \to U Bar_T D$; the counit condition is checked along similar lines. The comultiplication of the comonad $F U$ is $\delta \coloneqq F u U$, and putting $\eta = u U: U \to U F U$ for a right $F U$-coaction, the coassociativity of $\eta$ follows from a naturality square
$$\array{
U & \stackrel{\eta}{\to} & U F U \\
_\mathllap{\eta} \downarrow & & \downarrow_\mathrlap{U\delta} \\
U F U & \underset{\eta F U}{\to} & U F U F U.
}$$
Apply $[id_{\mathbf{E}^T}, -]$ to this coassociativity square to get another coassociativity, this time for the comonad $K \coloneqq [id_{\mathbf{E}^T}, F U]$ on $[\mathbf{E}^T, \mathbf{E}^T]$ (with comultiplication denoted $\delta_K$) and coaction $H \coloneqq [id, \eta]: [id, U] \to [id, U] \circ K$. Thus there is an equalizing diagram
$$[id, U] \stackrel{H}{\to} [id, U]K \stackrel{\overset{[id, U]\delta_K}{\to}}{\underset{H K}{\to}} [id, U]K K.$$
Because $Bar_T: \Delta_a^{op} \to [\mathbf{E}^T, \mathbf{E}^T]$ is a strong monoidal functor (see the left isomorphism in (eq:strong)), the squares in
$$\array{
[id, U] Bar_T & \stackrel{H Bar_T}{\to} & [id, U]K Bar_T & \stackrel{\overset{[id, U]\delta_K Bar_T}{\to}}{\underset{H K Bar_T}{\to}} & [id, U]K K Bar_T \\
& _\mathllap{h}{\searrow} & \downarrow_\mathrlap{\cong} & & \downarrow_\mathrlap{\cong} \\
& & [id, U] Bar_T D & \stackrel{\overset{[id, U]Bar_T \delta_D}{\to}}{\underset{h D}{\to}} & [id, U]Bar_T D D
}$$
commute serially, with the triangle commuting by definition of $h$. This completes the verification.
=--
By Remark \ref{rem}, it follows that $U Bar_T(A)$, obtained by applying evaluation at a $T$-algebra $A$, carries an acyclic structure as well. In this sense we may say that $U Bar_T(A)$ (which has $A$ as its augmented component in dimension $-1$) is an acyclic resolution of the constant simplicial $T$-algebra at $A$ that carries a $T$-algebra structure.
## Properties
### Universal property
We now state and prove a [[universal property]] of the bar construction $Bar_T(A)$.
+-- {: .num_defn}
###### Definition
Let $(T, m: T T \to T, u: 1 \to T)$ be a monad on a category $\mathbf{E}$. A $T$-**algebra resolution** is a simplicial object $Y: \Delta_a^{op} \to \mathbf{E}^T$ together with an acyclic structure on $U Y: \Delta_a^{op} \to \mathbf{E}$. A morphism between $T$-algebra resolutions is a natural transformation $\phi: Y \to Y'$ such that $U\phi: U Y \to U Y'$ is a $P$-coalgebra map.
=--
Let $AlgRes_T$ be the category of $T$-algebra resolutions. There is a forgetful functor
$$G: AlgRes_T \to \mathbf{E}^T$$
that takes an algebra resolution $Y$ to its augmentation component $Y[0]$.
+-- {: .num_theorem #universal}
###### Theorem
The functor $\hom_{\mathbf{E}^T}(A, G-): AlgRes_T \to Set$ is represented by $Bar_T(A)$; i.e., $Bar_T(-): \mathbf{E}^T \to AlgRes_T$ is [[left adjoint]] to $G$.
=--
The proof is distributed over two lemmas.
+-- {: .num_lemma #unique}
###### Lemma
Given a $T$-algebra resolution $Y$ and a $T$-algebra map $f: A \to Y[0]$, there is at most one $T$-algebra resolution map $\phi: Bar_T(A) \to Y$ such that
$\phi[0] = f$.
=--
+-- {: .proof}
###### Proof
The $P$-coalgebra structure $h: U Bar_T(A) \to U Bar_T(A) \circ D$ is defined on components $U Bar_T(A)[n] = T^n A$ by $h[n] = u T^n(A): T^n(A) \to T^{n+1}(A)$. Thus in order that $U\phi$ be a $P$-coalgebra map, we must have that the diagram
$$\array{
T^n A & \stackrel{u T^n A}{\to} & U F T^n(A) \\
_\mathllap{U\phi[n]} \downarrow & & \downarrow_\mathrlap{U\phi[n+1]} \\
U Y[n] & \underset{h_Y[n]}{\to} & U Y[n+1]
}$$
commutes. Here $\phi[n]: T^n (A) \to Y[n]$ determines a unique $T$-algebra map $g: F T^n(A) \to Y[n+1]$ such that
$$U(g) \circ u T^n(A) = h_Y[n] \circ U\phi[n]$$
since $F$ is left adjoint to $U$. Thus, starting with $\phi[0] = f$ as given, each algebra map $\phi[n]$ uniquely determines its successor $\phi[n+1] = g$.
=--
+-- {: .num_remark #simp}
###### Remark
The preceding proof does not show that the $\phi[n]$ fit together to form a map $\phi$ of simplicial $T$-algebras (i.e., to respect faces and degeneracies); it merely shows at most one such $T$-algebra resolution map is possible. But once we show that $\phi$ respects faces and degeneracies, the proof of Theorem \ref{universal} will be complete.
=--
+-- {: .num_lemma #exist}
###### Lemma
Given a $T$-algebra resolution $Y$ and an algebra map $f: X \to Y[0]$, there is at least one $T$-algebra resolution map $\phi: Bar_T(X) \to Y$ with $\phi[0] = f$.
=--
+-- {: .proof}
###### Proof
It is enough to produce such a map $\phi: Bar_T(Y[0]) \to Y$ in the case $f = 1_{Y[0]}$, since the case for general $f: X \to Y[0]$ is then given by a composite
$$Bar_T(X) \stackrel{Bar_T(f)}{\to} Bar_T(Y[0]) \stackrel{\phi}{\to} Y.$$
We will do something slightly more general. For any category $\mathbf{C}$, the endofunctor category $[\mathbf{C}^{\Delta_a^{op}}, \mathbf{C}^{\Delta_a^{op}}]$ has a comonoid object $P = - \circ D$, so that there is an induced strong monoidal functor
$$\Delta_a^{op} \stackrel{\tilde{P}}{\to} [\mathbf{C}^{\Delta_a^{op}}, \mathbf{C}^{\Delta_a^{op}}]$$
which, upon evaluating at an object $Y$ of $\mathbf{C}^{\Delta_a^{op}}$, gives a functor
$$B(Y, D, D) \coloneqq eval_Y \circ \tilde{P}: \Delta_a^{op} \to \mathbf{C}^{\Delta_a^{op}}$$
with $B(Y, D, D)[n] = Y D^n$, so that $B(Y, D, D)$ is a double simplicial object. Taking $\mathbf{C} = \mathbf{E}^T$ and taking $Y$ to be a $T$-algebra resolution with acyclic structure $h: Y \to Y D$, we will produce a (double) simplicial map
$$\Phi: B(T, T, Y) \to B(Y, D, D)$$
where $\Phi[n]: T^n Y \to Y D^n$ is defined recursively as in the proof of Lemma \ref{unique}, by setting $\Phi[0] = 1_Y$ and taking $\Phi[n+1]: T^{n+1} Y \to Y D^{n+1}$ the unique simplicial $T$-algebra map such that
$$\array{
T^n Y & \stackrel{u T^n Y}{\to} & T^{n+1} Y \\
_\mathllap{\Phi[n]} \downarrow & & \downarrow_\mathrlap{\Phi[n+1]} \\
Y D^n & \underset{h D^n}{\to} & Y D^{n+1}
}$$
commutes for all $n$. Once we verify the claim that $\Phi$ respects faces and degeneracies, the same will be true for $\phi[n] = \Phi[n][0]: (T^n Y)[0] = T^n(Y[0]) \to (Y D^n)[0] = Y[n]$, whence the proof will be complete by Remark \ref{simp}.
The claim is proved by induction on $n$. Let $\epsilon: D \to 1_{\Delta_a^{op}}$ be the counit and $\delta: D \to D D$ be the comultiplication. We have face maps
$$T^j m T^{n-j-1} Y: T^{n+1} Y \to T^n Y, \qquad Y D^j \epsilon D^{n-j}: Y D^{n+1} \to Y D^n$$
for $j = 0$ to $n$, under the special convention that $m T^{-1}Y: T Y \to Y$ denotes the action $\alpha: T Y \to Y$. We also have degeneracy maps
$$T^j u T^{n-j} Y: T^n Y \to T^{n+1} Y, \qquad Y D^{j-1} \delta D^{n-j}: Y D^n \to Y D^{n+1}$$
for $j = 1$ to $n$. We proceed as follows.
* To check preservation of face maps, we treat separately the cases where $j = 0$ and $j \geq 1$.
* For $j = 0$, we must check commutativity of the square in
$$\array{
T^n Y & \stackrel{u T^n Y}{\to} & T^{n+1} Y & \stackrel{\Phi[n+1]}{\to} & Y D^{n+1} \\
& _\mathllap{id} \searrow & \downarrow_\mathrlap{m T^{n-1} Y} & & \downarrow_\mathrlap{Y\epsilon D^n} \\
& & T^n Y & \underset{\Phi[n]}{\to} & Y D^n.
}$$
Since all the maps are algebra maps and $u T^n Y: T^n Y \to T^{n+1} Y$ exhibits $T^{n+1} Y$ as the free algebra on $T^n Y$, it suffices to check commutativity around the perimeter. (N.B.: the triangle commutes, even in the case where $n=0$ which we need to start the induction.) By definition of $\Phi[n+1]$, commutativity of the perimeter boils down to commutativity of
$$\array{
T^n Y & \stackrel{u T^n Y}{\to} & T^{n+1} Y & \stackrel{\Phi[n+1]}{\to} & Y D^{n+1} \\
& _\mathllap{\Phi[n]} \searrow & & \nearrow_\mathrlap{h D^n} & \downarrow_\mathrlap{Y\epsilon D^n} \\
& & Y D^n & \underset{id}{\to} & Y D^n
}$$
where the triangle commutes by one of the acyclic structure equations.
* For $j \geq 1$, the commutativity of the right square in
$$\array{
T^n Y & \stackrel{u T^n Y}{\to} & T^{n+1} Y & \stackrel{\PhiY[n+1]}{\to} & Y D^{n+1} \\
_\mathllap{T^{j-1}m T^{n-j-1} Y} \downarrow & nat. & \downarrow_\mathrlap{T^j m T^{n-j-1} Y} & & \downarrow_\mathrlap{Y D^j \epsilon D^{n-j}} \\
T^{n-1} Y & \underset{u T^{n-1} Y}{\to} & T^n Y & \underset{\Phi[n]}{\to} & Y D^n \\
& _\mathllap{\Phi[n-1]} \searrow & & \nearrow_\mathrlap{h D^{n-1}} & \\
& & Y D^{n-1} & &
}$$
is again by appeal to a freeness argument where we just need to check commutativity of the perimeter, noting commutativity of the left square by naturality and that of the bottom quadrilateral by the recursive definition of $\Phi[n]$. But the perimeter commutes by examining the diagram
$$\array{
T^n Y & \stackrel{u T^n Y}{\to} & T^{n+1} Y & \stackrel{\Phi[n+1]}{\to} & Y D^{n+1} \\
_\mathllap{T^{j-1}m T^{n-j-1} Y} \downarrow & _\mathllap{\Phi[n]} \searrow & & \nearrow_\mathrlap{h D^n} & \downarrow_\mathrlap{Y D^j \epsilon D^{n-j}} \\
T^{n-1} Y & ind. & Y D^n & nat. & Y D^n \\
& _\mathllap{\Phi[n-1]} \searrow & \downarrow & \nearrow_\mathrlap{h D^{n-1}} & \\
& & Y D^{n-1} & &
}$$
(where the middle vertical arrow is $Y D^{j-1} \epsilon D^{n-j}$) using the inductive hypothesis in the bottom left parallelogram.
* To check preservation of degeneracy maps, we treat separately the cases $j=1$ and $j \geq 2$.
* For $j = 1$, the commutativity of the top right square in
$$\array{
T^{n-1} Y & \stackrel{u T^{n-1} Y}{\to} & T^n Y & \stackrel{\Phi[n]}{\to} & Y D^n \\
_\mathllap{u T^{n-1} Y} \downarrow & nat. & \downarrow _\mathrlap{T u T^{n-1} Y} & & \downarrow_\mathrlap{Y \delta D^{n-1}} \\
T^n Y & \underset{u T^n Y}{\to} & T^{n+1} Y & \underset{\Phi[n+1]}{\to} & Y D^{n+1} \\
& _\mathllap{\Phi[n]} \searrow & & \nearrow_\mathrlap{h D^n} & \\
& & Y D^n & &
}$$
is by appeal to a freeness argument where we just need to check commutativity of the perimeter (the special case $n=1$ being used to start the induction). But this boils down to commutativity of the diagram
$$\array{
T^{n-1} Y & \stackrel{u T^{n-1} Y}{\to} & T^n Y & \stackrel{\Phi[n]}{\to} & Y D^n \\
_\mathllap{u T^{n-1} Y} \downarrow & \searrow_\mathrlap{\Phi[n-1]} & & _\mathllap{h D^{n-1}} \nearrow & \downarrow_\mathrlap{Y \delta D^{n-1}} \\
T^n Y & & Y D^{n-1} & & Y D^{n+1} \\
& _\mathllap{\Phi[n]} \searrow & \downarrow_\mathrlap{h D^{n-1}} & \nearrow_\mathrlap{h D^n} & \\
& & Y D^n & &
}$$
where the bottom right quadrilateral commutes by one of the acyclic structure equations.
* For $j \geq 2$, the commutativity of the top right square in
$$\array{
T^{n-1} Y & \stackrel{u T^{n-1} Y}{\to} & T^n Y & \stackrel{\Phi[n]}{\to} & Y D^n \\
_\mathllap{T^{j-1} u T^{n-j} Y} \downarrow & nat. & \downarrow _\mathrlap{T^j u T^{n-j} Y} & & \downarrow_\mathrlap{Y D^{j-1} \delta D^{n-j}} \\
T^n Y & \underset{u T^n Y}{\to} & T^{n+1} Y & \underset{\Phi[n+1]}{\to} & Y D^{n+1} \\
& _\mathllap{\Phi[n]} \searrow & & \nearrow_\mathrlap{h D^n} & \\
& & Y D^n & &
}$$
is once again by appeal to a freeness argument where we just need to check commutativity of the perimeter. Here it boils down to commutativity of
$$\array{
T^{n-1} Y & \stackrel{u T^{n-1} Y}{\to} & T^n Y & \stackrel{\Phi[n]}{\to} & Y D^n \\
_\mathllap{T^{j-1} u T^{n-j} Y} \downarrow & \searrow_\mathrlap{\Phi[n-1]} & & _\mathllap{h D^{n-1}} \nearrow & \downarrow_\mathrlap{Y D^{j-1} \delta D^{n-j}} \\
T^n Y & ind. & Y D^{n-1} & nat. & Y D^{n+1} \\
& _\mathllap{\Phi[n]} \searrow & \downarrow & \nearrow_\mathrlap{h D^n} & \\
& & Y D^n & &
}$$
where the middle vertical arrow is $Y D^{j-2} \delta D^{n-j}$.
This completes the proof.
=--
## Special cases
### For modules over an algebra
Let $A$ be a commutative [[associative algebra]]s over some [[ring]] $k$. Write $A Mod$ for the category of [[connective chain complex|connective]] [[chain complex]]es of [[module]]s over $A$.
For $N$ a right module, also $N \otimes_k A$ is canonically a module. This construction extends to a functor
$$
A \otimes_k (-) :
A Mod
\to
A Mod
\,.
$$
The [[monoid]]-structure on $A$ makes this a [[monad]] in [[Cat]]: the monad product and unit are given by the product and unit in $A$.
For $N$ a module its right action $\rho :N \otimes A \to N$ makes the module an [[algebra over a monad|algebra over this monad]].
The bar construction $\mathrm{B}(A,N)$ is then the simplicial module
$$
\cdots
\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}
N \otimes_k A \otimes_k A \stackrel{\overset{Id \otimes \mu}{\longrightarrow}}{\underset{\rho \otimes Id}{\longrightarrow}} N \otimes_k A
\,.
$$
Under the [[Moore complex]] functor of the [[Dold-Kan correspondence]] this is identified with a [[chain complex]] whose [[differential]] is given by the alternating sums of the face maps indicated above.
This chain complex is what originally was called the **bar complex** in [[homological algebra]]. Because the first authors denoted its elements using a notation involving vertical bars ([Ginzburg](#Ginzburg))!!
This chain complex provides a resolution that computes the [[Tor]]
$$
Tor(N, A \times A)
\,.
$$
This gives the [[Hochschild homology]] of $A$. See there for more details.
### For differential graded (Hopf) algebras
See [[bar and cobar construction]].
### For $E_\infty$-algebras
See ([Fresse](#Fresse)).
##Related entries
* [[simplicial resolution]]; this is essentially the same concept but from a slightly different perspective.
* [[homotopy coherent nerve]]
* [[canonical resolution]]
* [[monadic cohomology]]
## References and Literature
A general discussion of bar construction for monads is at
* {#Trimble} [[Todd Trimble]], _On the Bar Construction_ ([blog](http://golem.ph.utexas.edu/category/2007/05/on_the_bar_construction.html))
Textbook accounts can be found at:
* [[Saunders Mac Lane]], section IV.5 of _Homology_
* [[Charles Weibel]], section 8.6 of _[[An Introduction to Homological Algebra]]_ (1994)
The bar complex of a bimodule is reviewed for instance in
* {#Ginzburg} [[Victor Ginzburg]], _Lectures on noncommutative geometry_ ([arXiv:math/0506603](http://arxiv.org/abs/math.AG/0506603))
around page 16.
The bar complex for [[E-infinity algebra]]s is discussed in
* [[Benoit Fresse]], _The bar complex of an E-infinity algebra_, Advances in Mathematics Volume 223, Issue 6, 1 April 2010, Pages 2049-2096
{#Fresse}
The compositional structure of the bar construction of several monads, as well as its interpretation in terms of [[partial evaluation|partial evaluations]] is studied in
* [[Carmen Constantin]], [[Tobias Fritz]], [[Paolo Perrone]] and [[Brandon Shapiro]], _Partial evaluations and the compositional structure of the bar construction_. ([arXiv](https://arxiv.org/abs/2009.07302))
[[!redirects bar constructions]]
[[!redirects bar complex]]
[[!redirects bar resolution]]
[[!redirects bar resolutions]] |
Barak Kol | https://ncatlab.org/nlab/source/Barak+Kol |
* [webpage](http://www.phys.huji.ac.il/~barak_kol/)
## Selected writings
On [[(p,q)5-brane webs]]:
* [[Ofer Aharony]], [[Amihay Hanany]], [[Barak Kol]], _Webs of $(p,q)$ 5-branes, Five Dimensional Field Theories and Grid Diagrams_, JHEP 9801:002,1998 ([arXiv:hep-th/9710116](http://arxiv.org/abs/hep-th/9710116))
* [[Ofer Aharony]], [[Amihay Hanany]], _Branes, Superpotentials and Superconformal Fixed Points_, Nucl. Phys. B504:239-271, 1997 ([arXiv:hep-th/9704170](https://arxiv.org/abs/hep-th/9704170))
## Related $n$Lab entries
* [[(p,q)5-brane]]
* [[brane intersection]]
* [[orientifold]], [[orientifold plane]]
category: people |
Barbara Fantechi | https://ncatlab.org/nlab/source/Barbara+Fantechi | __Barbara Fantechi__ is an algebraic geometer with professorship in Trieste.
* [Website](http://people.sissa.it/~fantechi/)
## Selected writing
On basics of [[algebraic geometry]], following [[Grothendieck]]'s [[FGA]]:
* [[Barbara Fantechi]], [[Lothar Göttsche]], [[Luc Illusie]], [[Steven L. Kleiman]], [[Nitin Nitsure]], [[Angelo Vistoli]]:
**[[FGA explained]]** (ICTP, Trieste 2003--2005)
Mathematical Surveys and Monographs __123__
Amer. Math. Soc. 2005. x+339 pp.
* [ISBN:978-0-8218-4245-4](https://bookstore.ams.org/surv-123-s)
[MR2007f:14001](http://www.ams.org/mathscinet-getitem?mr=2007f:14001)
[lecture notes](http://indico.ictp.it/event/a0255/other-view?view=ictptimetable)
## Related $n$Lab entries
* [[perfect obstruction theory]]
* [[Gromov-Witten invariants]]
category: people
|
Barcan formula | https://ncatlab.org/nlab/source/Barcan+formula | Under construction
>If it is possible that something is $F$, then something is such that it is possible that it is $F$.
## Related concepts
* [[de dicto and de re]]
[[!redirects Barcan's formula]] |
Bargmann-Segal transform | https://ncatlab.org/nlab/source/Bargmann-Segal+transform | See [[coherent state]].
Bargmann-Segal transform is the integral transform whose kernel is the overlap between the projective measure corresponding to the coherent states and
a measure corresponding to an orthonormal basis comeing from some polarization for $L^2$-sections. The kernel is a special case of a
[[Bergman kernel]] in complex analysis.
Classical case of Heisenberg group:
* V. Bargmann, _On a Hilbert space of analytic functions and an associated integral transform_, Communications on Pure and Applied Mathematics __14__ (1961) 187-214 [MR0157250](http://www.ams.org/mathscinet-getitem?mr=157250) [doi](ttp://dx.doi.org/10.1002/cpa.3160140303)
Vectors in the Hilbert space can be represented in the coherent state representation: $|f\rangle = \int |z\rangle\langle z|f\rangle d\mu$; if $f$ is in $L^2$ then $\langle z|f\rangle$ is a holomorphic function and this passage is called the Bargmann-Segal transform (referring to [Irving Segal](http://en.wikipedia.org/wiki/Irving_Segal)); this way certain Hilbert space of holomorphic function appears, the Bargmann-Fock space.
Further generalization is to Perelomov coherent states.
More recent generalized Segal-Bargmann transform of Hall:
* Brian Charles Hall. _The Segal-Bargmann coherent state transform for Lie groups_. J. Funct. Anal. __122__:103–151, 1994, [doi](http://dx.doi.org/10.1006/jfan.1994.1064); _Geometric quantization and the generalized Segal–Bargmann transform for Lie groups of compact type_, Comm. Math. Phys., __226__:233–268, 2002. [doi](http://dx.doi.org/10.1007/s002200200607)
[[!redirects Segal-Bargmann transform]]
[[!redirects Bargmann transform]] |
Barnes G-function | https://ncatlab.org/nlab/source/Barnes+G-function |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Arithmetic
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Much like the [[Gamma function]] generalizes the functional equation
$$
\Gamma(z + 1) \;=\; z \, \Gamma(z)
$$
to non-integer values of $z$, so the *Barnes $G$-function* $G(-)$ corresponds to the functional equation
\[
\label{TranslationFormula}
G(z + 1) \;=\; \Gamma(z) \, G(z)
\,.
\]
Just as for $z =n \in \mathbb{N}$, $\Gamma(n+1) = n!$, so $G(n+2) = n!(n-1)! \cdots 1!$.
## Definition
(...)
## Properties
### Special values
\[
\label{GOf1}
G(1) \;=\; 1
\]
$$
G(1/2)
\;=\;
2^{1/24}
\cdot
e^{ \tfrac{3}{2} \zeta^'(-1) }
\cdot
\pi^{ - 1/4 }
\,.
$$
([WP here](https://en.wikipedia.org/wiki/Barnes_G-function#Value_at_1/2))
### Stirling-like asymptotic expansion
{#AsymptoticExpansion}
$$
ln G(1 + z)
\;=\;
z^2
\left(
\tfrac{1}{2} ln(z)
-
\tfrac{3}{4}
\right)
+
\tfrac{1}{2}
ln(2 \pi)
z
-
\tfrac{1}{12} ln(z)
+
\zeta^'(-1)
+
\mathcal{O}(z^{-1})
\,,
$$
where $\zeta$ denotes the [[Riemann zeta function]].
(e.g. [WP here](https://en.wikipedia.org/wiki/Barnes_G-function#Asymptotic_expansion), [WMW (14)](#WMW))
### Relations to the Gamma-function
A version of the [[Gauss multiplication formula]] for the [[Gamma function]]:
\begin{prop}\label{MultiplicationFormulaForGammaFunctionInTermsOfGFunction}
$$
\underoverset
{j = 1}
{N}
{\prod}
\Gamma(j/2)
\;=\;
\frac
{
G(N/2+ 1)
\cdot
G(N/2 + 1/2)
}
{ G(1/2) }
\,.
$$
\end{prop}
([Kotěšovec 13, p. 2](#Kotesovec))
\begin{proof}
By repeated use of the translation formula (eq:TranslationFormula)
and using the initial value $G(1) = 1$ (eq:GOf1):
In the case that $N = 2M$ in an [[even number]]:
$$
\underoverset
{j = 1}
{N}
{\prod}
\Gamma(j/2)
\;=\;
\underoverset
{j = 1}
{M}
{\prod}
\Gamma(j)
\cdot
\underoverset
{j = 1}
{M}
{\prod}
\Gamma(j - \frac{1}{2})
\;=\;
G(M+1) \cdot G(M +1/2) /G(1/2)
\;=\;
G(N/2 + 1) \cdot G(N/2 + 1/2)/G(1/2).
$$
In the case that $N = 2M+1$ is an [[odd number]]:
$$
\underoverset
{j = 1}
{N}
{\prod}
\Gamma(j/2)
\;=\;
\underoverset
{j = 1}
{M}
{\prod}
\Gamma(j)
\cdot
\underoverset
{j = 1}
{M+1}
{\prod}
\Gamma(j - \frac{1}{2})
\;=\;
G(M+1) \cdot G(M +3/2) /G(1/2)
\;=\;
G(N/2 + 1/2) \cdot G(N/2 + 1)/G(1/2).
$$
\end{proof}
## References
See also:
* Wikipedia, *[Barnes G-function](https://en.wikipedia.org/wiki/Barnes_G-function)*
* {#WMW} WolframMathWorld, *[Barnes G-Function](https://mathworld.wolfram.com/BarnesG-Function.html)*
In the context of [counting](semistandard+Young+tableau#NumberOfSYTWithBoundedNumberOfRows) of [[standard Young tableaux]] of bounded height:
* {#Kotesovec13} Václav Kotěšovec, *Asymptotic of Young tableaux of bounded height*, 2013 ([pdf](http://www.kotesovec.cz/math_articles/kotesovec_young_tableaux_conjecture.pdf), [[KotesovecAsymptoticsOfBoundedYoungTableaux.pdf:file]])
[[!redirects Barnes G-functions]]
|
Barr embedding theorem | https://ncatlab.org/nlab/source/Barr+embedding+theorem |
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+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
#Contents#
* toc
{: toc}
[[Barr]] proved a theorem about [[embedding of categories|embedding]] [[regular categories]] into [[presheaf category|categories of]] [[small presheaves]], and also a strengthening for [[Barr exact categories]].
## Idea
Barr's embedding theorem has the classical form of many embedding[^note] theorems in mathematics: if a structure $\mathcal{C}$ has certain good properties, then it admits an embedding with certain other good properties into another structure $\mathcal{D}$ which is somehow more explicit than $\mathcal{C}$.
For example,
* by Tychonoff's embedding theorem, if a $T_0$ space $X$ is [[completely regular]], then there exists an embedding of $X$ into a product of [[metric space]]s
* by the [[Whitney embedding theorem]], if an abstract $d$-dimensional real manifold $M$ is smooth, then there exists an embedding, which is an [[embedding of smooth manifolds]], of $M$ into the explicit real manifold that is $\mathbb{R}^{2d}$
* by the [[Freyd-Mitchell embedding theorem]], if a category is [[small]] and [[abelian category|abelian]], then there exist an embedding, which is [[exact functor|exact]], into the category of modules of a (not necessarily commutative) [[ring]].
and,
* by Barr's embedding theorem, if a category is [[small]] and [[regular category|regular]], then there exists an [[embedding]], which is a full, faithful and [[regular functor|regular]], into a category of [[presheaves]] over some [[small category]].
## A proof of Makkai
The proof of (a version of) Barr's theorem given by Makkai in [Makkai1980](#Makkai80) is a nice example of a non-trivial application of [[ultraproduct]]s in category theory.
## References
It has been proved in
* [[Michael Barr|M. Barr]], _Exact categories_, Lecture Notes in Math. __236__, (Springer, Berlin, 1971), 1-120.
and, in a different way, in
* M. Barr, _Representation of categories_, J. Pure Appl. Alg. __41__ (1986) 113-137 (this article has supposedly some fixable errors).
* F. Borceux, _A propos d'un théorème de Barr_, Séminaire de mathématique (nouvelle série) Rapport 28 - Mai 1983, Institute de Mathematique Pure et Appliqee, Univ. Cath. de Louvain.
* M. Makkai, _A theorem on Barr-exact categories, with an infinitary generalization_, Ann. Pure Appl. Logic __47__ (1990), no. 3, 225-268.
[[Michael Barr]]'s full exact embedding theorem for [[Barr exact categories]], proved in (?)
* Michael Barr, _Embedding of categories_, Proc. Amer. Math. Soc. __37__, No. 1 (Jan., 1973), pp. 42-46 ([jstor](http://www.jstor.org/stable/2038702), [pdf](http://www.math.mcgill.ca/barr/papers/embed.pdf))
generalizes the Lubkin-Freyd-Mitchell embedding theorems for abelian categories. The Giraud's theorem for topoi is not much more than a special case of that theorem.
* {#Makkai80} M. Makkai, _On full embeddings I_, Journal of Pure and Applied Algebra __16__, (1980), pp. 183-195
* M. Makkai, _Full continuous embeddings of toposes_, Trans. Amer. Math. Soc. __269__, No. 1 (Jan., 1982), pp. 167-196 [jstor](http://www.jstor.org/stable/1998599)
One can also consider categories enriched over a locally finitely presentable closed symmetric monoidal category $V$. Such a $V$-category $C$ is regular if it is finitely complete, admits the coequalizers of kernel pairs all [[regular epimorphism]]s are universal (i.e. stable under pullbacks) and stable under cotensors with the finite objects. A $V$-functor is regular if it preserves finite limits and regular epimorphisms. The following generalizes the Barr's embedding theorem for regular categories to the regular enriched categories:
* Dimitri Chikhladze, _Barr's embedding theorem for enriched categories_, J. Pure Appl. Alg. __215__, n. 9 (2011) 2148-2153, [arxiv/0903.1173](http://arxiv.org/abs/0903.1173), [doi](http://dx.doi.org/10.1016/j.jpaa.2010.12.004)
Its main result is
> Theorem 10. For a small regular $V$-category $C$ there exists a small category $T$ and a regular fully faithful functor $E : C \longrightarrow [T, V]$.
[[!redirects Barr's embedding theorem]] |
Barr's theorem | https://ncatlab.org/nlab/source/Barr%27s+theorem |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### Foundations
+--{: .hide}
[[!include foundations - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
**Barr's theorem** was originally conjectured by [[William Lawvere]] as an infinitary generalization of the [[Deligne completeness theorem]] for [[coherent toposes]] which can be expressed as the existence of a surjection $\mathcal{S}/K\to\mathcal{E}$ for a coherent topos $\mathcal{E}$ with set of points $K$. General toposes $\mathcal{E}$ may fail to have [[enough points]] but [[Michael Barr]] showed that a surjection from a suitable [[Boolean topos]] still exists.
## Statement
+-- {: .num_theorem}
###### Theorem
If $\mathcal{E}$ is a [[Grothendieck topos]], then there is a [[surjective geometric morphism]]
$$
\mathcal{F} \to \mathcal{E}
$$
where $\mathcal{F}$ satisfies the [[axiom of choice]].
=--
## Remark
[[Deligne completeness theorem|Deligne's completeness theorem]] says that a coherent Grothendieck topos has enough points in $Set$ and this corresponds to the Gödel-Henkin completeness theorem for first-order theories. Similarly, Barr's theorem can interpreted as saying that a Grothendieck topos has sufficient _Boolean-valued_ points and is in turn closely related to Mansfield's **Boolean-valued completeness theorem** for infinitary first-order theories.
## Constructive proof and classical detour
Let $\mathbb{T}$ be a [[geometric theory]] and $U_\mathbb{T}$ its [[classifying topos|universal model]]. Recall that $U_\mathbb{T}$ represents deducibility in geometric logic in the sense that a geometric sequent $ \sigma$ is deducible from $\mathbb{T}$ precisely iff $U_\mathbb{T}\models \sigma $.
Suppose that $M=f^*(U_\mathbb{T})$ is a $\mathbb{T}$-model in a topos $\mathcal{E}$ where $f:\mathcal{E}\to Set[U_\mathbb{T}]$ is a [[surjective geometric morphism]] to the [[classifying topos]] of $\mathbb{T}$ and let $ (\varphi \vdash_{\vec{x}} \psi) $ be a geometric sequent such that $M\models (\varphi \vdash_{\vec{x}} \psi) $. Now by the definition of the satisfaction relation, this is the same as to say that the monomorphism $\{\vec{x}.\varphi\wedge\psi\}_M\hookrightarrow\{\vec{x}.\varphi\}_M$ is an isomorphism but, $f$ being surjective, $f^*$ is [[conservative functor|conservative]] whence $U_\mathbb{T}\models (\varphi \vdash_{\vec{x}} \psi) $ and, accordingly, $(\varphi \vdash_{\vec{x}} \psi)$ is deducible from $\mathbb{T}$ in geometric logic.
This together with the existence of a Boolean cover assured by Barr's theorem implies the following important
+-- {: .num_theorem}
###### Corollary
Let $\mathbb{T}$ be a [[geometric theory]] and $\sigma$ a geometric sequent that holds in all $\mathbb{T}$-models in Boolean toposes. Then $\sigma$ is deducible from $\mathbb{T}$ in geometric logic.
=--
In other words, if a statement in [[geometric logic]] is deducible from a [[geometric theory]] using classical [[logic]] and the [[axiom of choice]], then it is also deducible from it in [[constructive mathematics]]. Unfortunately, the proof of Barr's theorem itself is highly non-constructive whence is of no direct help in finding such constructive replacements for classical proofs of geometric statements.
## Related entries
* [[Deligne completeness theorem]]
* [[Boolean topos]]
## Link
* MO discussion on topos without points: _[Topos Without point, from the point of view of logic.](http://mathoverflow.net/questions/98729/topos-without-point-from-the-point-of-view-of-logic)_
## References
* [[Michael Barr|M. Barr]], _Toposes without points_ , JPAA **5** (1974) pp.265-280. ([preprint](http://www.math.mcgill.ca/barr/papers/top.no.pt.pdf)) doi:[10.1016/0022-4049(74)90037-1](http://dx.doi.org/10.1016/0022-4049(74%2990037-1)
Extensive discussion of the context of Barr's theorem is in chapter 7 of:
* [[P. T. Johnstone]], _Topos Theory_ , Academic Press New York 1977 (Dover reprint 2014).
A proof sketch and a survey of its model-theoretic context is in
* [[Gonzalo Reyes|Gonzalo E. Reyes]], _Sheaves and concepts: A model-theoretic interpretation of Grothendieck topoi_ , Cah. Top. Diff. Géo. **Cat. XVIII** no.2 (1977) pp.405-437. ([numdam](http://www.numdam.org/item?id=CTGDC_1977__18_2_105_0))
For a discussion of the importance of the corollary in constructive algebra see also
* [[Gavin Wraith]], _Intuitionistic algebra: some recent developments in topos theory_ In: Proceedings of the International Congress of Mathematicians (Helsinki, 1978), pages 331–337, Helsinki, 1980. Acad. Sci. Fennica. ([pdf](http://www.mathunion.org/ICM/ICM1978.1/Main/icm1978.1.0331.0338.ocr.pdf))
{#Wraith}
For proof-theoretic approaches to Barr's theorem see
* Sara Negri, _Contraction-free sequent calculi for geometric theories with an
application to Barr's theorem_ , Archive for Mathematical Logic **42** (2003) pp.389–401.
For the connection with the Boolean-valued completeness theorem see also
* R. Goldblatt, _Topoi - The Categorical Analysis of Logic_ , North-Holland 1982². (Dover reprint New York 2006; [project euclid](http://projecteuclid.org/euclid.bia/1403013939))
* R. Mansfield, _The Completeness Theorem for Infinitary Logic_ , JSL **37** no.1 (1972) pp.31-34.
* [[Michael Makkai]], [[Gonzalo E. Reyes]], _First-order Categorical Logic_ , LNM **611** Springer Heidelberg 1977.
|
Barr-exact category > history | https://ncatlab.org/nlab/source/Barr-exact+category+%3E+history | < [[Barr-exact category]]
[[!redirects Barr-exact category -- history]] |
Barratt-Eccles operad | https://ncatlab.org/nlab/source/Barratt-Eccles+operad |
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###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _Barratt-Eccles operad_ $\mathcal{E}$ is a specific realization of an [[E-infinity operad]], a cofibrant [[resolution]] of the [[commutative operad]] (in the [[model structure on operads]]).
As a [[topological operad]] it is given by $\mathcal{E}_n := E \Sigma_n$, the [[universal principal bundle]] for the [[symmetric group]] $\Sigma_n$. As an [[sSet]]-operad it has $\mathcal{E}_n = N(\Sigma_n // \Sigma_n)$, the [[nerve]] of the [[action groupoid]] of $\Sigma_n$ acting on itself.
## Definition
### As a simplicial operad
We give the definition of the Barratt-Eccles operad as an object of the category of multi-colored [[symmetric operad|symmetric]] [[simplicial operads]] ([[sSet]]-[[enriched category|enriched]] [[symmetric multicategories]]). (See [Berger-Fress, section 1.1.5 ](#BerFre01).)
The **Barratt-Eccles operad** $\mathcal{E}$ is the operad defined as follows.
It has a single color.
For $n \in \mathbb{N}$, its [[simplicial set]] $\mathcal{E}(n)$ of $n$-ary operations is the [[nerve]] $N(\Sigma_n//\Sigma_n)$ of the [[action groupoid]] $\Sigma_n//\Sigma_n$ of the [[symmetric group]] $\Sigma_n$ permuting $n$ elements acting by right multiplication on itself.
$$
\mathcal{E}(n) := N(\Sigma_n // \Sigma_n)
\,.
$$
Explicitly, this is the [[simplicial set]] whose $k$-cells are $(k+1)$-tuples of group elements
$$
\mathcal{E}(n)_k =_{iso} (\Sigma_n)^{\times k+1}
\,.
$$
Regarded as the nerve of the action groupoid, the face maps on $\mathcal{E}(n)$ are given by multiplication in $\Sigma_n$
$$
d_i (g_0, g_1, \cdots, g_k)
=
g_0, \cdots, g_{i-1}, g_i \cdot g_{i+1}, g_{i+2}, \cdots, g_k
\;\;\;\;\;
0 \leq i \leq k
\,.
$$
But, alternatively, we can parameterize $\mathcal{E}(n)_k$ by the tuples
$$
(w_0, w_1, \cdots, w_k) :=
(g_0, g_0 \cdot g_1, g_0 \cdot g_1 \cdot g_2,
\cdots, \prod_{i=0}^{k} g_i)
\,.
$$
In terms of this the $i$th face map is given simply by omitting the $i$th entry
$$
d_i(w_0, \cdots, w_k) = (w_0, \cdots, \hat w_i, \cdots, w_k)
\,.
$$
The $i$th degeneracy map is given by repeating the $i$th entry
$$
s_i (w_0, \cdots, w_n)
:=
(w_0, \cdots, w_{i-1}, w_i, w_i, w_{i+1}, \cdots, w_n)
\;\;\;
0 \leq i \leq n
\,.
$$
In terms of this, the $\Sigma_n$-[[action]] on $\mathcal{E}(n)$ (giving the structure of a [[symmetric operad]]) is then the [[diagonal]] action
$$
\sigma \cdot (w_0, w_1, \cdots, w_k)
:=
(\sigma \cdot w_0, \sigma \cdot w_1, \cdots, \sigma \cdot w_k)
\,.
$$
The composition operations in the operad
$$
\mathcal{E}(r) \times (\mathcal{E}(n_1) \times \cdots \times \mathcal{E}(n_r))
\to
\mathcal{E}(n_1 + \cdots + n_r)
$$
are the morphisms of simplicial sets which in degree $k$ are maps on tuples, which in each degree $i$ are given by the natural function
$$
\Sigma_r \times (\Sigma_{n_1} \times \cdots \times \Sigma_{n_r})
\to
\Sigma_{n_1 + \cdots + n_r}
$$
that composes $r$ permutations with a permutation of $r$ elements to a permutation of $\sum_{i = 0}^r n_r$ elements.
(This function is in fact that which gives the composition in [[Assoc]] when regarded as a [[symmetric operad]], $Assoc := Symm(*)$.)
### As a dg-operad
(...)
See ([Berger-Fresse](#BerFre01)).
## Properties
### As a simplicial operad
Each of the simplicial sets $\mathcal{E}(n)$ for $n \in \mathbb{N}$ is [[contractible]]. One way to see this is to observe that $\Sigma_n // \Sigma_n$ is (the [[nerve]] of) the [[pullback]]
$$
\array{
\Sigma_n // \Sigma_n
&\to&
*
\\
\downarrow && \downarrow
\\
(\mathbf{B} \Sigma_n)^I &\to& \mathbf{B}\Sigma_n
}
\,,
$$
where $\mathbf{B}\Sigma_n$ is one-object groupoid with $\Sigma_n$ as its morphisms, $(\mathbf{B}\Sigma_n)^I$ is its [[arrow category]] and the bottom vertical map is evaluation at the source. Since this is an acyclic fibration, so is the top vertical morphism.
It follows that the canonical morphism of simplicial operads
$$
\mathcal{E} \to Comm
$$
to the [[commutative operad]] (which has $Comm(n) = *$ for all $n \in \mathbb{N}$) is a weak equivalence (in the [[model structure on operads]]). In fact, it is a cofibrant resolution.
### As a dg-operad
(...)
## Related concepts
* [[E-infinity algebra]]
* [[(2,1)-algebraic theory of E-infinity algebras]]
## References
As a [[simplicial operad]], the Barratt-Eccles operad was introduced in
* [[M. Barratt]], [[P. Eccles]], _On $\Gamma^+$-structures I. A free group functor for stable homotopy theory_, Topology 13 (1974), 25-45.
Its realization as a [[dg-operad]] is discussed in detail in
* C. Berger, _Combinatorial models for real configuration spaces and $E_n$-operads_, [pdf](http://math.unice.fr/~cberger/config.pdf)
* [[Clemens Berger]], [[Benoit Fresse]] _Combinatorial operad actions on cochains_, Math. Proc. Cambridge Philos. Soc. 137 (2004), 135-174, [arXiv:math.AT/0109158](http://arxiv.org/abs/math/0109158), [doi](http://10.1017/S0305004103007138)
{#BerFre01}
* [[Clemens Berger]], [[Benoit Fresse]], _A prismatic decomposition of the Barratt-Eccles operad_, [math.AT/0204326](http://arxiv.org/abs/math/0204326)
Elmendorf and Mandell show that the Barratt-Eccles operad $E\Sigma_*$ is obtained by applying functor $E$ to certain operad $\Sigma_*$ in
* A.D. Elmendorf, M.A. Mandell, _Rings, modules, and algebras in infinite loop space theory_, Advances in Mathematics __205__, n.1, 2006, pp 163–228, [arxiv](http://front.math.ucdavis.edu/0403.5403), [doi](http://dx.doi.org/10.1016/j.aim.2005.07.007)
[[!redirects Barratt-Eccles operads]]
|
barreled topological vector space | https://ncatlab.org/nlab/source/barreled+topological+vector+space | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Functional analysis
+--{: .hide}
[[!include functional analysis - contents]]
=--
=--
=--
# Contents
* the following line creates the automatic table of contents
{:toc}
## Idea
Barreled spaces are [[topological vector spaces]] for which the theorem of [[Banach-Steinhaus theorem|Banach–Steinhaus]] is valid. This theorem says, roughly, that for a set of continuous linear maps $L(E, F)$ from a barreled space $E$ to a [[locally convex space|locally convex]] [[TVS]] boundedness in the topology of pointwise convergence implies boundedness in the topology of bounded convergence.
## Definition
A subset $T \subset E$ of a [[TVS]] E is a **barrel** if it is
* absorbing
* balanced
* closed
* convex
A [[TVS]] $E$ is **barreled** (or __barrelled__) if every barrel is a neighborhood of zero.
Sometimes [[locally convex]] is included in the definition, this is not implied by barreled as defined above, i.e. there are barreled spaces that are not locally convex.
In the definition of **quasibarreled** or **infrabarreled** the barrels are replaced by sets that are barrels and which absorb all bounded sets (sets with the latter property are also called **bornivorous**).
## Properties
+-- {: .un_prop}
###### Proposition
A [[locally convex space|locally convex]] [[TVS]] which is a [[Baire space]] is barreled.
=--
+-- {: .un_prop}
###### Proposition
A [[locally convex space|locally convex]] [[TVS]] is barreled iff its topology is the [[strong topology]].
=--
## Examples
Since all [[locally convex space|locally convex]] [[TVS]]es that are [[Baire space|Baire spaces]] are barreled, the examples naturally include [[Fréchet spaces]], [[Banach spaces]] and [[Hilbert spaces]].
## References
See the [[functional analysis bibliography]].
The definition of quasibarreled is from
* S.M. Khaleelulla: _Counterexamples in Topological Vector Spaces_.
It is called _infrabarreled_ in
* H.H. Schaefer: _Topological vector spaces_.
[[!redirects barreled]]
[[!redirects barrelled]]
[[!redirects barreled space]]
[[!redirects Barreled space]]
[[!redirects barreled spaces]]
[[!redirects Barreled spaces]]
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|
Barry Fawcett | https://ncatlab.org/nlab/source/Barry+Fawcett |
* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=37865)
## Selected writings
On [[completely distributive lattices]]:
* {#WoodFawcett} [[Barry Fawcett]], [[Richard J. Wood]], *Constructive complete distributivity I* , Math. Proc. Camb. Phil. Soc. **107** (1990) 81-89 [[doi:10.1017/S0305004100068377](https://doi.org/10.1017/S0305004100068377)]
category: people
[[!redirects B. Fawcett]] |
Barry Kurtz | https://ncatlab.org/nlab/source/Barry+Kurtz |
* [personal page](https://cs.appstate.edu/~blk/)
## Selected writings
On [[syntax]] and [[semantics]] of [[programming languages]]:
* [[Kenneth Slonneger]], [[Barry Kurtz]], *Formal Syntax and Semantics of Programming Languages*, Addison-Wesley (1995) [[webpage](https://homepage.divms.uiowa.edu/~slonnegr/plf/Book/), [pdf](https://doc.lagout.org/science/Artificial%20Intelligence/Natural%20Language%20Processing/Formal%20Syntax%20and%20Semantics%20of%20Programming%20Languages%20-%20Kenneth%20Slonneger.pdf)]
## Related entries
* [[denotational semantics]]
category: people
|
Barry Mazur | https://ncatlab.org/nlab/source/Barry+Mazur | Barry Charles Mazur is a mathematician at Harvard University.
* [website](http://www.math.harvard.edu/~mazur/)
## Selected writings
Early occurence of the argument that later came to be known as the *[[Eilenberg swindle]]*:
* {#Mazur61} [[Barry C. Mazur]], _On embeddings of spheres_, Acta Mathematica **105** 1–2 (1961) 1–17 [[doi:10.1007/BF02559532](https://doi.org/10.1007%2FBF02559532)]
Introducing [[Artin-Mazur formal groups]]:
* {#ArtinMazur77} [[Michael Artin]], [[Barry Mazur]], _Formal Groups Arising from Algebraic Varieties_, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 10 no. 1 (1977), p. 87-131 ([numdam:ASENS_1977_4_10_1_87_0](http://www.numdam.org/item?id=ASENS_1977_4_10_1_87_0), [MR56:15663](http://www.ams.org/mathscinet-getitem?mr=56:15663))
On [[elliptic curves]] over general [[commutative rings]] (in [[arithmetic geometry]]):
* [[Nicholas Katz]], [[Barry Mazur]], _Arithmetic moduli of elliptic curves, Annals of Mathematics Studies_, vol. 108, Princeton University Press, 1985 ([jstor:j.ctt1b9s05p](https://www.jstor.org/stable/j.ctt1b9s05p))
[[!redirects B. Mazur]]
[[!redirects Barry C. Mazur]]
[[!redirects Barry Charles Mazur]]
category: people |
Barry Mitchell | https://ncatlab.org/nlab/source/Barry+Mitchell | Barry Mitchell (1933-2021) was an American mathematician who wrote one of the earliest books on category theory (1965).
He proved what is now known as the *[[Freyd-Mitchell embedding theorem]]*, which shows that any [[abelian category]] has an exact embedding into a [[category of modules]].
* [Local newspaper obituary](https://www.legacy.com/obituaries/dailyfreeman/obituary.aspx?n=barry-m-mitchell&pid=197672237&fhid=3910)
## Selected writings
Introducing the [[Mitchell embedding theorem]]:
* [[Barry Mitchell]], *The Full Imbedding Theorem*, American Journal of Mathematics **86** 3 (1964) 619-637 [[doi:10.2307/2373027](https://doi.org/10.2307/2373027)]
On [[category theory]]:
* [[Barry Mitchell]], *Theory of categories*, Pure and Applied Mathematics **17**, Academic Press (1965) [[ISBN:978-0-12-499250-4](https://www.elsevier.com/books/theory-of-categories/mitchell/978-0-12-499250-4)]
See also:
* [[Barry Mitchell]], *The dominion of Isbell*, Transactions of the American Mathematical Society **167** (1972) 319-331 [[doi:10.2307/1996142](https://doi.org/10.2307/1996142)]
## Related entries
* [[Mitchell embedding theorem]]
* [[semicategory]]
|
Barry Simon | https://ncatlab.org/nlab/source/Barry+Simon |
* [webpage](http://www.math.caltech.edu/people/simon.html)
* [Wikipedia entry](http://en.wikipedia.org/wiki/Barry_Simon)
## Selected writings
On [[mathematical physics]]:
* [[Michael Reed]] and [[Barry Simon]]: *[[Methods of Modern Mathematical Physics]]*, Academic Press (1978)
On the [[Berry phases]] as the [[holonomy]] of [[connections]] on [[fiber bundles]]:
* {#Simon83} [[Barry Simon]], *Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase*, Phys. Rev. Lett. **51** (1983) 2167 ([doi:10.1103/PhysRevLett.51.2167](https://doi.org/10.1103/PhysRevLett.51.2167))
category: people
|
Bart Jacobs | https://ncatlab.org/nlab/source/Bart+Jacobs | * [papers](http://www.cs.ru.nl/B.Jacobs/PAPERS/index.html)
## Selected writings
On [[dependent type theory]] and its [[categorical semantics]] ([[categorical semantics of dependent types]]):
* [[Bart Jacobs]], *Comprehension categories and the semantics of type dependency*, Theoret. Comput. Sci. **107** 2 (1993) 169-207 [[MR1201808](http://www.ams.org/mathscinet-getitem?mr=1201808), <a href="https://doi.org/10.1016/0304-3975(93)90169-T">doi:10.1016/0304-3975(93)90169-T</a>]
* [[Bart Jacobs]], *Categorical Logic and Type Theory*, Studies in Logic and the Foundations of Mathematics **141**, Elsevier (1998) [[ISBN:978-0-444-50170-7](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/141), [pdf](https://people.mpi-sws.org/~dreyer/courses/catlogic/jacobs.pdf), [webpage](http://www.cs.ru.nl/B.Jacobs/CLT/bookinfo.html)]
On [[category theory|category theoretic]] [[probability theory]] (cf. *[[convex powerset of distributions monad]]*):
* [[Bart Jacobs]], *Coalgebraic trace semantics for combined possibilitistic and probabilistic systems*, Proceedings of CMCS 2008, ENTCS **203** 5 (2008) 131-152 [[doi:10.1016/j.entcs.2008.05.023](https://doi.org/10.1016/j.entcs.2008.05.023)]
On [[CD-categories]] ([[gs-monoidal categories]]):
* Kenta Cho, [[Bart Jacobs]], _Disintegration and Bayesian inversion via string diagrams_, Mathematical Structures in Computer Science **29** 7 (2019) 938-971 [[arXiv:1709.00322](https://arxiv.org/abs/1709.00322), [doi:10.1017/S0960129518000488](https://doi.org/10.1017/S0960129518000488)]
## Related pages
* [[definability (fibred category theory)]]
category: people |
Bartek | https://ncatlab.org/nlab/source/Bartek | I am a PhD student at the University of Auckland, working on [[Cartan geometry]], and more specifically [[Parabolic geometry]]. |
Bartel Leendert van der Waerden | https://ncatlab.org/nlab/source/Bartel+Leendert+van+der+Waerden | Bartel Leendert van der Waerden was a mathematician at the universities of Groningen, Leipzig, and Zürich.
He got his PhD degree in 1926 at the University of Amsterdam, advised by Hendrick de Vries.
## Selected works
* [[Moderne Algebra]]. Teil I, Die Grundlehren der mathematischen Wissenschaften, vol. 33, Berlin, New York: Springer-Verlag, 1930, ISBN 978-3-540-56799-8
* [[Moderne Algebra]]. Teil II, Die Grundlehren der mathematischen Wissenschaften, vol. 34, Springer-Verlag, 1931, ISBN 978-3-540-56801-8
Based on lectures by [[Emil Artin]] and [[Emmy Noether]],
it was the first textbook using the modern approach to [[algebra]].
category: reference |
Bartlomiej Czech | https://ncatlab.org/nlab/source/Bartlomiej+Czech |
* [spire page](https://inspirehep.net/authors/1024822)
* [institute page](https://www.ias.tsinghua.edu.cn/en/info/1059/1176.htm)
## Selected writings
On [[quantum information theory]] in view of [[AdS-CFT duality]] ([[holographic entanglement entropy]]):
* [[Bowen Chen]], [[Bartlomiej Czech]], [[Zi-zhi Wang]], *Quantum Information in Holographic Duality*, Rept. Prog. Phys. **85** (2022) 4, 046001 [[arXiv:2108.09188](https://arxiv.org/abs/2108.09188), [doi:10.1088/1361-6633/ac51b5](https://doi.org/10.1088/1361-6633/ac51b5)]
category: people
[[!redirects Bart Czech]] |
Bartomeu Monserrat | https://ncatlab.org/nlab/source/Bartomeu+Monserrat |
* [institute page](https://www.msm.cam.ac.uk/people/monserrat)
* [Monserrat Lab](https://www.tcm.phy.cam.ac.uk/~bm418/)
## Selected writings
On [[braid group statistics|anyonic braiding]] of nodal points in the [[Brillouin zone]] of [[semi-metals]] ("braiding in momentum space"):
* Siyu Chen, [[Adrien Bouhon]], [[Robert-Jan Slager]], [[Bartomeu Monserrat]], *Non-Abelian braiding of Weyl nodes via symmetry-constrained phase transitions* (formerly: *Manipulation and braiding of Weyl nodes using symmetry-constrained phase transitions*),
Phys. Rev. B **105** (2022) L081117 $[$[arXiv:2108.10330](https://arxiv.org/abs/2108.10330), [doi:10.1103/PhysRevB.105.L081117](https://doi.org/10.1103/PhysRevB.105.L081117)$]$
* [[Bo Peng]], [[Adrien Bouhon]], [[Robert-Jan Slager]], [[Bartomeu Monserrat]], *Multi-gap topology and non-Abelian braiding of phonons from first principles*, Phys. Rev. B **105** (2022) 085115 ([arXiv:2111.05872](https://arxiv.org/abs/2111.05872), [doi;10.1103/PhysRevB.105.085115](https://doi.org/10.1103/PhysRevB.105.085115))
* [[Bo Peng]], [[Adrien Bouhon]], [[Bartomeu Monserrat]], [[Robert-Jan Slager]], *Phonons as a platform for non-Abelian braiding and its manifestation in layered silicates*, Nature Communications volume 13, Article number: 423 (2022) ([doi:10.1038/s41467-022-28046-9](https://doi.org/10.1038/s41467-022-28046-9))
category: people
|
Barton Zwiebach | https://ncatlab.org/nlab/source/Barton+Zwiebach |
* [website](http://web.mit.edu/physics/people/faculty/zwiebach_barton.html)
[Wikipedia entry](http://en.wikipedia.org/wiki/Barton_Zwiebach)
## Selected writings
On [[bosonic string|bosonic]] [[closed string|closed]] [[string field theory]] and introducing its relation to [[L-infinity algebras|$L_\infty$-algebras]]:
* [[Barton Zwiebach]], *Issues In Covariant Closed String Theory*, pp 192-200 in: *Proceedings of [10th and Final Workshop on Grand Unification](http://inspirehep.net/record/966930)*, 20-22 Apr 1989. Chapel Hill, North Carolina [[spire:282685](http://inspirehep.net/record/282685)]
* [[Barton Zwiebach]], *Closed string field theory: Quantum action and the Batalin-Vilkovisky master equation*, Nucl. Phys. B **390** 1 (1993) 33-152 [[arXiv:hep-th/9206084](http://arxiv.org/abs/hep-th/9206084), <a href="https://doi.org/10.1016/0550-3213(93)90388-6">doi:10.1016/0550-3213(93)90388-6</a>]
On [[string junctions]] in relation to [[Lie algebras]] and [[Lie algebra representations]]:
* Oliver DeWolfe, [[Barton Zwiebach]], _String Junctions for Arbitrary Lie Algebra Representations_, Nucl. Phys. B541 (1999) 509-565 ([arXiv:hep-th/9804210](https://arxiv.org/abs/hep-th/9804210))
* Oliver DeWolfe, Tamas Hauer, [[Amer Iqbal]], [[Barton Zwiebach]], _Uncovering the Symmetries on $[p,q]$ 7-branes: Beyond the Kodaira Classification_, Adv. Theor. Math. Phys. 3 (1999) 1785-1833 ([arXiv:hep-th/9812028](https://arxiv.org/abs/hep-th/9812028))
## Related entries
* [[string field theory]]
category: people |
Bartosz Milewski | https://ncatlab.org/nlab/source/Bartosz+Milewski |
### Contact ###
bartosz@relisoft.com
### Publications ###
* [Bartosz Milewski's Programming Cafe](http://bartoszmilewski.com).
* [Profunctor optics: a categorical update](https://arxiv.org/abs/2001.07488)
* [Category Theory for Programmers](https://github.com/hmemcpy/milewski-ctfp-pdf)
### Selected writings
On [[categorical semantics]] of [[functional programming languages]] (such as for [[monads in computer science]]):
* [[Bartosz Milewski]] (compiled by Igal Tabachnik), *Category Theory for Programmers*, Blurb (2019) [[pdf](https://github.com/hmemcpy/milewski-ctfp-pdf/releases/download/v1.3.0/category-theory-for-programmers.pdf), [github](https://github.com/hmemcpy/milewski-ctfp-pdf), [webpage](https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/), [ISBN:9780464243878](https://www.blurb.com/b/9621951-category-theory-for-programmers-new-edition-hardco)]
* [[Bartosz Milewski]], *The Dao of Functional Programming* (2023) [[pdf](https://github.com/BartoszMilewski/Publications/blob/master/TheDaoOfFP/DaoFP.pdf), [github](https://github.com/BartoszMilewski/Publications/tree/master/TheDaoOfFP)]
On [[optics (in computer science)]]:
* [[Bryce Clarke]], Derek Elkins, [[Jeremy Gibbons]], [[Fosco Loregian]], [[Bartosz Milewski]], Emily Pillmore, [[Mario Román]], _Profunctor optics, a categorical update_, 2020. ([arXiv:2001.07488](https://arxiv.org/abs/2001.07488))
category: people |
Baruch Spinoza | https://ncatlab.org/nlab/source/Baruch+Spinoza |
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### Context
#### Philosophy
+-- {: .hide}
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=--
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=--
* [Wikipedia entry](http://en.wikipedia.org/wiki/Baruch_Spinoza)
* [Stanford Encycl. of Phil. entry](http://plato.stanford.edu/entries/spinoza/)
## Hegel on Spinoza
From [[Georg Hegel]], _[Lectures on the history of Philosophy -- Spinoza](https://www.marxists.org/reference/archive/hegel/works/hp/hpspinoz.htm)_:
> The philosophy of Descartes underwent a great variety of unspeculative developments, but in Benedict Spinoza a direct successor to this philosopher may be found, and one who carried on the Cartesian principle to its furthest logical conclusions. For him soul and body, thought and Being, cease to have separate independent existence. The dualism of the Cartesian system Spinoza, as a Jew, altogether set aside. For the profound unity of his philosophy as it found expression in Europe, his manifestation of Spirit as the identity of the finite and the infinite in God, instead of God's appearing related to these as a Third — all this is an echo from Eastern lands. The Oriental theory of absolute identity was brought by Spinoza much more directly into line, firstly with the current of European thought, and then with the European and Cartesian philosophy, in which it soon found a place.
> First of all we must, however, glance at the circumstances of Spinoza's life. He was by descent a Portuguese Jew, and was born at Amsterdam in the year 1632; the name he received was Baruch, but he altered it to Benedict. In his youth he was instructed by the Rabbis of the synagogue to which he belonged, but he soon fell out with them, their wrath having been kindled by the criticisms which he passed on the fantastic doctrines of the Talmud. He was not, therefore, long in absenting himself from the synagogue, and as the Rabbis were in dread lest his example should have evil consequences, they offered him a yearly allowance of a thousand gulden if he would keep away from the place and hold his tongue. This offer he declined; and the Rabbis thereafter carried their persecution of him to such a pitch that they were even minded to rid themselves of him by assassination. After having made a narrow escape from the dagger, he formally withdrew from the Jewish communion, without, however, going over to the Christian Church. He now applied himself particularly to the Latin language, and made a special study of the Cartesian philosophy. Later on he went to Rhynsburg, near Leyden, and from the year 1664 he lived in retirement, first at Voorburg, a village near the Hague, and then at the Hague itself, highly respected by numerous friends: he gained a livelihood for himself by grinding optical glasses. It was no arbitrary choice that led him to occupy himself with light, for it represents in the material sphere the absolute identity which forms the foundation of the Oriental view of things. Although he had rich friends and mighty protectors, among whom even generals were numbered, he lived in humble poverty, declining the handsome gifts offered to him time after time. Nor would he permit Simon von Vries to make him his heir; he only accepted from him an annual pension of three hundred florins; in the same way he gave up to his sisters his share of their father's estate. From the Elector Palatine, Carl Ludwig, a man of most noble character and raised above the prejudices of his time, he received the offer of a professor's chair at Heidelberg, with the assurance that he would have liberty to teach and to write, because "the Prince believed he would not put that liberty to a bad use by interfering with the religion publicly established." Spinoza (in his published letters) very wisely declined this offer, however, because "he did not know within what limits that philosophic liberty would have to be confined, in order that he might not appear to be interfering with the publicly established religion." He remained in Holland, a country highly interesting in the history of general culture, as it was the first in Europe to show the, example of universal toleration, and afforded to many a place of refuge where they might enjoy liberty of thought; for fierce as was the rage of the theologians there against Bekker, for example (Bruck. Hist. crit. phil. T. IV. P. 2, pp. 719, 720), and furious as were the attacks of Voetius on the Cartesian philosophy, these had not the consequences which they would have had in another land. Spinoza died on the 21st of February, 1677, in the forty-fourth year of his age. The cause of his death was consumption, from which he had long been a sufferer; this was in harmony with his system of philosophy, according to which all particularity and individuality pass away in the one substance. A Protestant divine, Colerus by name, who published a biography of Spinoza, inveighs strongly against him, it is true, but gives nevertheless a most minute and kindly description of his circumstances and surroundings — telling how he left only about two hundred thalers, what debts he had, and so on. A bill included in the inventory, in which the barber requests payment due him by M. Spinoza of blessed memory, scandalizes the parson very much, and regarding it he makes the observation: "Had the barber but known what sort of a creature Spinoza was, he certainly would not have spoken of his blessed memory." The German translator of this biography writes under the portrait of Spinoza: characterem reprobationis in vultu gerens, applying this description to a countenance which doubtless expresses the melancholy of a profound thinker, but is otherwise wild and benevolent. The reprobatio is certainly correct; but it is not a reprobation in the passive sense; it is an active disapprobation on Spinoza's part of the opinions, errors and thoughtless passions of mankind.(1)
> Spinoza used the terminology of Descartes, and also published an account of his system. For we find the first of Spinoza's works entitled "An Exposition according to the geometrical method of the principles of the Cartesian philosophy." Some time after this he wrote his Tractatus theologico-politicus, and by it gained considerable reputation. Great as was the hatred which Spinoza roused amongst his Rabbis, it was more than equalled by the odium which he brought upon himself amongst Christian, and especially amongst Protestant theologians — chiefly through the medium of this essay. It contains his views on inspiration, a critical treatment of the books of Moses and the like chiefly from the point of view that the laws therein contained are limited in their application to the Jews. Later Christian theologians have written critically on this subject, usually making it their object to show that these books were compiled at a later time, and that they date in part from a period subsequent to the Babylonian captivity; this has become a crucial point with Protestant theologians, and one by which the modern school distinguishes itself from the older, greatly pluming itself thereon. All this, however, is already to be found in the above-mentioned work of Spinoza. But Spinoza drew the greatest odium upon himself by his philosophy proper, which we must now consider as it is given to us in his Ethics. While Descartes published no writings on this subject, the Ethics of Spinoza is undoubtedly his greatest work; it was published after his death by Ludwig Mayer, a physician, who had been Spinoza's most intimate friend. It consists of five parts; the first deals with God (De Deo). General metaphysical ideas are contained in it, which include the knowledge of God and nature. The second part deals with the nature and origin of mind (De natura et origine mentis). We see thus that Spinoza does not treat of the subject of natural philosophy, extension and motion at all, for he passes immediately from God to the philosophy of mind, to the ethical point of view; and what refers to knowledge, intelligent mind, is brought forward in the first part, under the head of the principles of human knowledge. The third book of the Ethics deals with the origin and nature of the passions (De oriqine et natura affectuum); the fourth with the powers of the same, or human slavery (De servitute humana seu de affectuum viribus); the fifth, lastly, with the power of the understanding, with thought, or with human liberty (De potentia intellectus seu de libertate humana). (2) Kirchenrath Professor Paulus published Spinoza's works in Jena; I had a share in the bringing out of this edition, having been entrusted with the collation of French translations.
## related entries
* [[Spinoza's system]]
category: people
[[!redirects Spinoza]] |
barycenter | https://ncatlab.org/nlab/source/barycenter | ##Barycenter of a simplex
+--{: .un_defn}
######Definition
If $\sigma = \{ v_0, \ldots, v_q\} \in K_q$, the set of $q$-simplices of a [[simplicial complex]], $K$, then its **barycentre**, $b(\sigma)$, is the point
$$b(\sigma) = \sum_{0\leq i \leq q}\frac{1}{q + 1} v_i \in |K|.$$
=--
For the use of barycenters in the **barycentric subdivision**, see [[classical triangulation]] or
* wikipedia: [barycentric subdivision](http://en.wikipedia.org/wiki/Barycentric_subdivision), [barycentric coordinates](http://en.wikipedia.org/wiki/Barycentric_coordinates_%28mathematics%29)
* [barycentric coordinates homepage](http://www.inf.usi.ch/hormann/barycentric)
* Wolframworld: [[barycentric coordinates](http://mathworld.wolfram.com/BarycentricCoordinates.html) |
baryogenesis | https://ncatlab.org/nlab/source/baryogenesis |
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### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
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=--
#Contents#
* table of contents
{:toc}
## Idea
In [[cosmology]] and [[particle physics]], by _baryogenesis_ one refers to the process by which an abundance of [[baryons]] is produced in the early [[observable universe]], leading to the baryonic [[matter]] seen all around. Since in the [[standard model of particle physics]] [[matter]] and [[antimatter]] are _essentially_ always produced symmetrically ("conservation of baryon number"), the topic of baryogenesis is to a large extent the study of how the matter/antimatter symmetry may be [[broken symmetry|broken]].
([Sakharov 67](#Sakharov67)) formulated three conditions jointly necessary for matter/antimatter asymmetry leading to baryogenesis in the early universe, now known as the _Sakharov conditions_
1. violation of baryon number conservation;
1. [[CP problem|C and CP violation]]
1. departure from thermal equilibrium.
Indeed, by a [[quantum anomaly]] in the [[standard model of particle physics]], the _[[axial anomaly]]_ (a [[chiral anomaly]]) (see e.g. [Jackiw 08](#Jackiw08)), the [[divergence]] of the [[baryon current]] (the chiral version of the [[Dirac current]]) does not quite vanish but is ([t'Hooft 76](#Hooft76)) in traditional physics notation
$$
\partial_\mu j_b^\mu
=
\partial_\mu j_L^\mu
=
n_f
\left(
\frac{g^2}{32 \pi^2}
W_{\mu \nu}^\alpha \tilde W^{\alpha \mu \nu}
-
\frac{{g'}^2}{32\pi^2} \epsilon_{\mu \nu \rho \sigma} F^{\mu \nu} F^{\rho \sigma}
\right)
\,,
$$
where $F_\nabla$ is the [[field strength]] of the [[electroweak field]] (the [[curvature]] of a $SU(2)$-[[principal connection]] $\nabla$).
(see e.g. [Dine 90, (4)](#Dine90) [Trodden-Carroll, 4.9.3](#TroddenCarroll), [Shu 08, (7.6)](#Shu08)).
In more intrinsic notation and ignoring numerical constants, the statement is that
$$
\mathbf{d} \star j_b = \langle F_\nabla \wedge F_\nabla \rangle
\,.
$$
Properly interpreted (see at _[Yang-Mills instanton -- from the correct maths to the traditional physics story](Yang-Mills+instanton#FromTheMathsToThePhysicsStory)_) this is the _local_ expression for the [[connection on a 3-bundle|3-connection]] on the [[Chern-Simons line 3-bundle]] associated with the gauge field. If the latter is globally nontrivial in that it is in an _[[instanton]]_-sector (has nontrivial [[second Chern class]]), then the [[integral]] of $\langle F\wedge F\rangle$ over closed manifolds is an integer -- the "[[instanton]] number" -- and in conclusion there is baryon generation proportional to this number (physics lingo also: _sphaleron transitions_).
Notice that this would mean that baryons "are" a kind of topological twist, different from, but not entirely unlike to the old idea of _[[On Vortex Atoms]]_. See there at _[Similarlity to concepts of modern particle physics](On+Vortex+Atoms#SimilarityToParticlePhysics)_
That this process was at least one source of baryogenesis in the early universe is plausible but not certain. The process is heavily suppressed by inverse energy/temperature (e.g. [Dine 90, around (9)](#Dine90)) and out of reach of conceivable experiments in the present age of the universe, but plausibly may have occurred in the very early universe -- just as it should be for realistic baryogenesis.
## Exposition
{#Exposition}
One needs to be careful that the concept of "particle" itself is only well defined on Minkowski spacetime, hence not on cosmological scales. But nevertheless there is a fermion current 3-form $J$, such that its integral
$$
Q_\Sigma \coloneqq \int_\Sigma J \in \mathbb{R}
$$
has the interpretation of the total net _charge_ carried by the stuff measured by $J$, hence counts the net "fermion number" in all of space" as seen by $\Sigma$.
This is familiar from, say, Maxwell's equations, where such a $J$ appears on the right hand side as the electromagnetic source form
$$
d \star F = J
\,.
$$
Now given any such current 3-form, then its de Rham differential $d J$ measures how the charges $Q$ changes as time progresses. Because let $\Sigma_1$ and $\Sigma_2$ be two spatial slices of spacetimes, and $X_4$ a piece of spacetime connecting them, i.e. such that
$$
\partial X_4 = \Sigma_2 - \Sigma_1
\,.
$$
Then [[Stokes's theorem]] gives that $d J$ is the local change of charge, because:
$$
\begin{aligned}
Q_{\Sigma_2} - Q_{\Sigma_1}
&=
\int_{\Sigma_2} J - \int_{\Sigma_1} J
\\
&=
\int_{\partial X_4} J
\\
& =
\int_{X_4} d J
\end{aligned}
$$
So if $J$ is closed, then total charge is conserved, while if $J$ is not closed, then $d J $ is the local measure for how charge is being "created" or how it disappears.
Now the remarkable fact is that in the standard model of particle physics, $J$ comes out non-closed. Its differetial instead comes out proportial to the 4-form which measures instanton number
$$
d J \propto \langle F_\nabla \wedge F_\nabla \rangle
\,.
$$
This is the famous _[[chiral anomaly]]_.
So far this is highlighted in every textbook. But the following further crucial subtlety tends not to be recognized for what it is.
Namely on cosmological spacetimes that carry [[instantons]], then (as discussed at _[Yang-Mills instanton -- from the correct maths to the traditional physics story](Yang-Mills+instanton#FromTheMathsToThePhysicsStory)_) the 4-form $\langle F_\nabla \wedge F_\nabla\rangle$ is not in fact globally exact. The above formulas hold only locally, on a chart of spacetime. But on intersections two such pieces of data need to be glued by a gauge transformation. If we do make the usual simple assumptions (for simplicity of the discussion), then this gauge transformation is that famous "Chern-Simons winding number" $S^3 \to SU(2)$, which they keep handing the physics students without properly explaining it (such as the one who I am reacting to [here](http://physicsoverflow.org/38243/topological-number-integral-theory-boundary-volume-forms)).
As a result, in the presence of instantons, the integral of $\langle F_{\nabla} \wedge F_{\nabla}\rangle$ may be non-zero on a _closed_ cosmological spacetime.
The usual picture (which you see displayed in many popular accounts) is: imagine a 4d cup $D^4$ (a unit disk thought of as a "cup" cobordism from nothing to $S^3$) where the big bang expands spacetime from nothing. Then glue on something like $S^3 \times LongInterval$ and think of the result as being a simple model for the universe. Then assume some boundary condition saying that far in the future from the big bang the gauge fields $\nabla$ that carry those instantons decay away, hence are [[vanishing at infinity]]. So then for computing the total fermion charge in this universe, we are effectively dealing with its one-point compactification. In the present simple example this is the 4-sphere $S^4$.
So since the four form vanishes "at infinity", we may just as well assume that it is supported on that cup $D^4$ in our model, the neighbourhood of the big bang.
We learn that that:
1. The total net fermion charge in the universe is $\int_{S^4} \langle F_\nabla \wedge F_\nabla\rangle \in \mathbb{Z}$;
1. this fermion number is picked up incrementally increasing from zero (at the origin of our $D^4$-cup, the "big bang singularity) and reaching at "comoving time $t \lt 1$" the value
$$
Q_t
=
\int_{D^4_{t}} \langle F_\nabla \wedge F_\nabla\rangle
=
\int_{S^3} J_t
$$
(where $D^4_t$ is the disk of radius $t \lt 1$).
Then at $t = 1$ (in this paramneterization) the four form goes to zero (and in this fashion eventually extends to a global 4-form on our one-point compactified cosmological spacetime).
So after the comoving time $t = 1$ there is no net particle creation anymore. Moreover, the net particle number picked up until then is equal to the second Chern class of the cosmological gauge field, hence equal to the number that masure the "knottedness" of the cosmological gauge field (which we here think of as all being concentrated around the big big).
One particle per knottedness of the cosmological gauge field.
## Related concepts
* [[leptogenesis]]
* [[nucleosynthesis]]
* [[On Vortex Atoms]]
* [[comoving time]]
## References
The original article stating the _Sakharov conditions_ is
* {#Sakharov67} [[Andrei Sakharov]], _Violation of CP invariance, C asymmetry, and baryon asymmetry of the universe_. Journal of Experimental and Theoretical Physics 5: 24–27. 1967, republished as A. D. Sakharov (1991). _Violation of CP invariance, C asymmetry, and baryon asymmetry of the universe_. Soviet Physics Uspekhi 34 (5): 392–393. Bibcode:1991SvPhU..34..392S. doi:[10.1070/PU1991v034n05ABEH002497](http://dx.doi.org/10.1070/PU1991v034n05ABEH002497). 1967
reviewed in
* [[Dennis Perepelitsa]], _Sakharov Conditions for Baryogenesis_, 2008 ([pdf](http://phys.columbia.edu/~dvp/dvp-sakharov.pdf))
The derivation that [[instantons]] lead to baryon number violation is due to
* {#Hooft76} [[Gerard 't Hooft]], _Symmetry Breaking through Bell-Jackiw Anomalies_ Phys. Rev. Lett. 37 (1976) ([pdf](http://www.staff.science.uu.nl/~hooft101/gthpub/symm_br_bell_jackiw.pdf))
* [[Gerard 't Hooft]] _Computation of the quantum effects due to a four-dimensional pseudoparticle_, Phys. Rev. D14:3432-3450 (1976)
A good review of the axial anomaly is in
* {#Jackiw08} [[Roman Jackiw]], _Axial anomaly_, (2008), Scholarpedia, 3(10):7302. ([web](http://www.scholarpedia.org/article/Axial_anomaly))
and of the resulting [[phenomenology]] of baryon number non-conservation is in
* {#Dine90} [[Michael Dine]], _Baryon number violation at high-energy in the standard model: Fact or fiction?_ Sep 1990, ([spire](https://inspirehep.net/record/299206?ln=de), [pdf](http://www.slac.stanford.edu/cgi-wrap/getdoc/ssi90-020.pdf))
Review:
* Heidi Kuismanen, section 2 of _Leptogenesis as the origin of matter-antimatter asymmetry in extra dimensionaL and supersymmetric modeLs_ ([[KuismanenBaryogenesis.pdf:file]])
* V. A. Rubakov, [[Mikhail Shaposhnikov]], *Electroweak Baryon Number Non-Conservation in the Early Universe and in High Energy Collisions*, Usp. Fiz. Nauk **166** (1996) 493-537 Phys. Usp. 39 (1996) 461-502 7lbrack;[arXiv:hep-ph/9603208](https://arxiv.org/abs/hep-ph/9603208)]
* Antonio Riotto, Mark Trodden, _Recent Progress in Baryogenesis_, Ann.Rev.Nucl.Part.Sci.49:35-75,1999 ([arXiv:hep-ph/9901362](http://arxiv.org/abs/hep-ph/9901362))
* Laurent Canetti, Marco Drewes, [[Mikhail Shaposhnikov]], _Matter and Antimatter in the Universe_, New J. Phys. 14 (2012) 095012 ([arXiv:1204.4186](https://arxiv.org/abs/1204.4186))
* Yanagida, _The origin of matter_ ([[YanagidaBaryogenesis.pdf:file]])
* {#TroddenCarroll} Mark Trodden, Sean Carroll, TASI Lectures: Introduction to Cosmology _[4.9 Baryon Number violation](http://ned.ipac.caltech.edu/level5/Sept03/Trodden/Trodden4_9.html)_
* James M. Cline, _Baryogenesis_, lectures at Les Houches Summer School, Session 86: Particle Physics and Cosmology: the Fabric of Spacetime, 7-11 Aug. 2006 ([arXiv:hep-ph/0609145](https://arxiv.org/abs/hep-ph/0609145))
* {#Shu08} Jing Shu, section 7 of _Connecting LHC Signals with Deep Physics at the TeV Scale and Baryogenesis_ 2008
Discussion with focus on [[QCD]]-effects:
* Dominik J. Schwarz, _The first second of the Universe_, Annalen Phys.12:220-270, 2003 ([arXiv:astro-ph/0303574](https://arxiv.org/abs/astro-ph/0303574))
|
baryon | https://ncatlab.org/nlab/source/baryon | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In the [[standard model of particle physics]] (specifically in [[QCD]]), a _baryon_ is [[bound state]] of three [[quarks]] via the [[strong nuclear force]]. Baryons are the "heavy" types of [[hadrons]], the other being the [[mesons]].
Examples of baryons are the [[nucleons]]: [[protons]] and [[neutrons]]. Other examples are [[Lambda baryons]].
Also [[pentaquarks]] are counted as baryons.
## Properties
### Conceptualization and computation in AdS/QCD
{#ConceptualizationAndComputationInAdSQCD}
In the [[Witten-Sakai-Sugimoto model]] for [[non-perturbative effect|strongly coupled]] [[QCD]] via an [[intersecting D-brane model]], the [[hadrons]] in [[QCD]] correspond to [[string theory|string-theoretic]]-phenomena in an ambient [[bulk field theory]] on an approximately [[anti de Sitter spacetime]]:
1. the [[mesons]] ([[bound states]] of 2 [[quarks]]) correspond to [[open strings]] in the bulk, whose two endpoints on the [[asymptotic boundary]] correspond to the two [[quarks]];
1. [[baryons]] ([[bound states]] of $N_c$ [[quarks]]) appear in two different but equivalent ([Sugimoto 16, 15.4.1](#Sugimoto16)) guises:
1. as [[wrapped brane|wrapped]] [[D4-branes]] with $N_c$ [[open strings]] connecting them to the [[D8-brane]]
([Witten 98b](#Witten98b), [Gross-Ooguri 98](#GrossOoguri98))
1. as [[skyrmions]]
([Sakai-Sugimoto 04, section 5.2](#SakaiSugimoto04), [Sakai-Sugimoto 05, section 3.3](#SakaiSugimoto05), see [Bartolini 17](#Bartolini17)).
For review see [Sugimoto 16](#Sugimoto16), also [Rebhan 14, around (18)](#Rebhan14).
<center>
<img src="https://ncatlab.org/nlab/files/BaryonsAsD4Branes.jpg" width="700">
</center>
> graphics grabbed from [Sugimoto 16](#Sugimoto16)
This produces [[baryon]] [[mass]] spectra with moderate quantitative agreement with [[experiment]] ([HSSY 07](#HSSY07)):
<center>
<img src="https://ncatlab.org/nlab/files/BaryonSpectrumInSakaiSugimoto.jpg" width="700">
</center>
> graphics grabbed from [Sugimoto 16](#Sugimoto16)
## Related concepts
* [[constituent quark]]
* [[diquark]]
* [[quantum hadrodynamics]]
* [[hadron supersymmetry]], [[hadron Kaluza-Klein theory]]
* [[flavor brane]]
* [[Walecka model]]
* [[baryogenesis]]
* [[skyrmion]]
* [[baryon-lepton symmetry]]
## References
### General
Introduction and review:
* [[Franz Gross]], [[Eberhard Klempt]] et al., Chapter 9 of: *50 Years of Quantum Chromodynamics*, EJPC [[arXiv:2212.11107](https://arxiv.org/abs/2212.11107)]
Baryons as 3-[[constituent quark]] [[bound states]]:
* Gernot Eichmann, Helios Sanchis-Alepuz, Richard Williams, Reinhard Alkofer, Christian S. Fischer, _Baryons as relativistic three-quark bound states_, Progress in Particle and Nuclear Physics Volume 91, November 2016, Pages 1-100 ([arXiv:1606.09602](https://arxiv.org/abs/1606.09602), [doi:10.1016/j.ppnp.2016.07.001](https://doi.org/10.1016/j.ppnp.2016.07.001))
Baryons as [[quark]]/[[diquark]] [[bound states]]:
* Martin Oettel, _Baryons as Relativistic Bound States of Quark and Diquark_ ([arXiv:nucl-th/0012067](https://arxiv.org/abs/nucl-th/0012067), [spire:538966](https://inspirehep.net/literature/538966))
Experiment:
* Eberhard Klempt, Jean-Marc Richard, _Baryon spectroscopy_, Rev. Mod. Phys. 82:1095-1153, 2010 ([arXiv:0901.2055](https://arxiv.org/abs/0901.2055)
See also
* Wikipedia, _[Baryon](http://en.wikipedia.org/wiki/Baryon)_
* Aarts, _Baryons at finite temperature_ ([[AartsBaryonsAtFiniteTemperature.pdf:file]])
[[!include baryon chiral perturbation theory -- references]]
### Baryons as Skyrmions
The [[Skyrmion]]-model for baryons (see there for more references):
* {#Weigel96} [[Herbert Weigel]], _Baryons as Three Flavor Solitons_, Int. J. Mod. Phys. A11:2419-2544, 1996 ([arXiv:hep-ph/9509398](https://arxiv.org/abs/hep-ph/9509398), [cds:288541](http://cds.cern.ch/record/288541), [doi:10.1142/S0217751X96001218](https://doi.org/10.1142/S0217751X96001218))
* {#Weigel08} [[Herbert Weigel]], _Chiral Soliton Models for Baryons_, Lecture Notes in Physics book series, volume 743, Springer 2008 ([doi:10.1007/978-3-540-75436-7](https://doi.org/10.1007/978-3-540-75436-7))
* {#RhoZahed16} [[Mannque Rho]], [[Ismail Zahed]] (eds.) _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))
[[!include hadrons as KK-modes of 5d Yang-Mills theory -- references]]
### In the large $N$ limit
In the [[large N limit]]:
* {#Witten79} [[Edward Witten]], _Baryons in the $1/n$ Expansion_, Nucl. Phys. B160 (1979) 57-115 ([spire:140391](http://inspirehep.net/record/140391), <a href="https://doi.org/10.1016/0550-3213(79)90232-3">doi:10.1016/0550-3213(79)90232-3</a>)
[[!redirects baryons]]
|
baryon chiral perturbation theory -- references | https://ncatlab.org/nlab/source/baryon+chiral+perturbation+theory+--+references |
### Baryon chiral perturbation theory
{#BaryonChiralPerturbationTheoryReferences}
Discussion of [[baryon chiral perturbation theory]], i.e of [[chiral perturbation theory]] with explicit [[effective field theory|effective]] (as opposed to or in addition to implicit [[Skyrmion|skyrmionic]]) [[baryon]] [[field (physics)|fields]] included (see also _[[Walecka model]]_ and _[[quantum hadrodynamics]]_):
Review:
* [[Ulf-G. Meissner]], _Chiral QCD: Baryon dynamics_, in: At The Frontier of Particle Physics, pp. 417-505 (2001) ([arxiv:hep-ph/0007092](https://arxiv.org/abs/hep-ph/0007092))
* [[Véronique Bernard]], _Chiral Perturbation Theory and Baryon Properties_, Prog. Part. Nucl. Phys. 60:82-160, 2008 ([arXiv:0706.0312](https://arxiv.org/abs/0706.0312))
* [[Stefan Scherer]], _Baryon chiral perturbation theory_, PoS CD09:075, 2009 ([arXiv:0910.0331](https://arxiv.org/abs/0910.0331))
Original articles:
* Elizabeth Jenkins, Aneesh V. Manohar, _Baryon chiral perturbation theory using a heavy fermion lagrangian_, Physics Letters B Volume 255, Issue 4, 21 February 1991, Pages 558-562 (<a href="https://doi.org/10.1016/0370-2693(91)90266-S">doi:10.1016/0370-2693(91)90266-S</a>)
* Robert Baur, Joachim Kambor, _Generalized Heavy Baryon Chiral Perturbation Theory_, Eur. Phys. J. C7:507-524, 1999 ([arXiv:hep-ph/9803311](https://arxiv.org/abs/hep-ph/9803311))
Higher order terms:
* José Antonio Oller, Michela Verbeni, Joaquim Prades, _Meson-baryon effective chiral Lagrangians to $\mathcal{O}(q^3)$_, Journal of High Energy Physics, Volume 2006, JHEP09(2006) ([arXiv:hep-ph/0608204](https://arxiv.org/abs/hep-ph/0608204), [doi:10.1088/1126-6708/2006/09/079](https://iopscience.iop.org/article/10.1088/1126-6708/2006/09/079))
* Matthias Frink, [[Ulf-G. Meissner]], _On the chiral effective meson-baryon Lagrangian at third order_, Eur. Phys. J. A29:255-260, 2006 ([arXiv:hep-ph/0609256](https://arxiv.org/abs/hep-ph/0609256))
* Jose Antonio Oller, Joaquim Prades, Michela Verbeni, _Meson-Baryon Effective Chiral Lagrangians at $\mathcal{O}(q^3)$ Revisited_ ([arXiv:hep-ph/0701096](https://arxiv.org/abs/hep-ph/0701096), [spire:742291](https://inspirehep.net/literature/742291))
See also:
* Lisheng Geng, _Recent developments in $SU(3)$ covariant baryon chiral perturbation theory_, Front. Phys., 2013, 8(3): 328-348 ([arXiv:1301.6815](https://arxiv.org/abs/1301.6815))
|