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Yoneda lemma for tricategories | https://ncatlab.org/nlab/source/Yoneda+lemma+for+tricategories | +--{: .rightHandSide}
+--{: .toc .clickDown tabindex="0"}
### Context
#### Yoneda lemma
+--{: .hide}
[[!include Yoneda lemma - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* automatic table of contents goes here
{:toc}
## Idea
The **Yoneda lemma for tricategories** is a version of the [[Yoneda lemma]] that applies to [[tricategories]], the most common algebraic sort of weak [[3-category]] ([Buhné, Thrm 2.12](#Buhné)).
## References
* {#Buhné} Lukas Buhné, _Topics in three-dimensional descent theory_, ([pdf](https://d-nb.info/1072553694/34))
[[!redirects Yoneda lemma for 3-categories]] |
Yoneda reduction | https://ncatlab.org/nlab/source/Yoneda+reduction |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Yoneda lemma
+--{: .hide}
[[!include Yoneda lemma - contents]]
=--
=--
=--
#Contents#
* automatic table of contents goes here
{:toc}
## Idea
The term _Yoneda reduction_ was coined by [[Todd Trimble]] in his (unpublished) thesis. It refers to a technique based on the [[Yoneda lemma]] for performing a number of [[end|end and coend]] calculations which arise in [[coherence]] theory and [[enriched category theory]].
## The module perspective on the Yoneda lemma
There are various formulations of the Yoneda lemma. One says that given a [[presheaf]] $F: C^{op} \to Set$, there is a canonical isomorphism
$$F(c) \cong Nat(\hom_C(-, c), F)$$
where "Nat" refers to the set of natural transformations between presheaves $C^{op} \to Set$; in other words, the hom
$$Set^{C^{op}}(\hom_C(-, c), F)$$
appropriate to the presheaf category.
There is an $V$-[[enriched category]] version, whenever $C$ is a category enriched in a [[complete category|complete]], [[cocomplete category|cocomplete]], [[symmetric monoidal category|symmetric]] [[closed monoidal category|monoidal closed category]] $V$. Here "Nat" is constructed as an enriched [[end]] (an example of a [[weighted limit]]):
$$V^{C^{op}}(C(-, c), F) = \int_d F(d)^{C(d, c)}$$
and therefore the enriched Yoneda lemma gives an [[isomorphism]]
$$F(c) \cong \int_d F(d)^{C(d, c)} \qquad (1)$$
which is ($V$-)natural in $c$; we may therefore write
$$F(-) \cong \int_d F(d)^{C(d, -)} \qquad (2)$$
and this isomorphism is $V$-natural in $F$.
We pause to give an instance of the Yoneda lemma which is both familiar and which serves to inform much of the module-theoretic terminology in the discussion below. Let $V = Ab$; let $R$ be a ring (conceived as an $Ab$-enriched category with exactly one object $\bullet$). Then $Ab^{R^{op}}$ is the ($Ab$-enriched) category of right $R$-modules, or equivalently, left $R^{op}$-modules). The presheaf $\hom_R(-, \bullet)$ is just the underlying abelian group of $R$ seen as a right [[module]] over the ring $R$, also known as the _[[regular representation]]_.
The first formulation (1) of the Yoneda lemma would simply say that at the level of _abelian groups_, we have for any right $R$-module $M$
$$M(\bullet) \cong RightMod_R(R, M)$$
Further taking into account the "naturality" in the argument bullet, the formulation (2) says that actually we have an isomorphism at the level of _right $R$-modules_
$$M \cong RightMod_R(R, M)$$
where the module structure on the right side arises by considering the argument $R$ now as a _[[bimodule]]_ over the (ring) $R$.
The (enriched) Yoneda lemma is nothing but a far-reaching extrapolation of this basic isomorphism: it says
$$F \cong RightMod_C(\hom_C, F)$$
where the $C$-presheaf or right $C$-module hom on the right is appropriately constructed as an enriched end, and $\hom_C: C^{op} \otimes C \to V$ is a treated as a $V$-enriched "bimodule" over $C$, and plays the role of the "[[regular representation]]" of $C$.
## Calculus of bimodules
The analogy between [[presheaves]] and [[module]]s can be pursued considerably further. Again, we start with the perhaps more familiar context of rings and modules.
In the first place, given a ring $R$, there is a familiar monoidal category of $R$-bimodules (and bimodule morphisms). If $M, N$ are bimodules over $R$, with left $R$-actions denoted by $\lambda$'s and the right actions by $\rho$'s, their tensor product $M \otimes_R N$, defined by the coequalizer
$$M \otimes R \otimes N \stackrel{\to}{\to} M \otimes N \to M \otimes_R N$$
(where the two parallel arrows are $M \otimes \lambda$, $\rho \times N$) carries an evident $R$-bimodule structure. Each of the functors $M \otimes_R -$ and $- \otimes_R N$ admits a [[adjunction|right adjoint]] expressed by natural isomorphisms of abelian groups
$$Bimod(N, Left_R(M, Q)) \cong Bimod(M \otimes_R N, Q) \cong Bimod(M, Right_R(N, Q))$$
where $Left_R(M, Q)$ denotes the abelian group of left $R$-module maps $M \to Q$, equipped with its natural $R$-bimodule structure; $Right(N, Q)$ is similar. Thus the monoidal category of $R$-bimodules is biclosed.
More generally, there is a bicategory whose objects or 0-cells are rings $R, S, \ldots$, and whose morphisms or 1-cells $R \to S$ are left $R$-, right $S$-bimodules. 2-cells are homomorphisms of bimodules. If $M: R \to S$ and $N: S \to T$ are bimodules, then their bimodule composite is $M \otimes_S N: R \to T$. This too is a biclosed bicategory, meaning that
This generalized module theory can be pursued much further.
(Lost a bunch of work, due to vagaries of computers. Sigh. Will return later.)
## Examples
+-- {: .un_theorem}
###### Relative coherence theorem for symmetric monoidal categories
If $V$ is symmetric monoidal, then the monoid of endomorphisms on the $n$-fold tensor functor
$$\bigotimes^n: V^{\otimes n} \to V$$
is in bijection with the monoid of endomorphisms on the unit object $I$.
=--
+-- {: .un_proof}
###### Proof
By fully and faithfully embedding $V$ (as a symmetric monoidal category) into $Set^{V^{op}}$, we may without loss of generality suppose $V$ is complete, cocomplete, symmetric monoidal closed.
The result is by induction on $n$: observe that a map
$$x_1 \otimes \x_2 \otimes \ldots \otimes x_n \to x_1 \otimes x_2 \otimes \ldots \otimes x_n$$
natural in all the arguments $x_i$, in particular in $x_n$, corresponds to a map dinatural in $x_n$:
$$x_1 \otimes \ldots \otimes x_{n-1} \to x_1 \otimes \ldots \otimes x_{n-1} \otimes x_n)^{x_n}$$
and hence to a map to the end
$$x_1 \otimes \ldots \otimes x_{n-1} \to \int_{x_n} (x_1 \otimes \ldots \otimes x_{n-1} \otimes x_n)^{x_{n}^{I}}$$
where the end exists and is isomorphic to
$$x_1 \otimes \ldots \otimes x_{n-1} \otimes I \cong x_1 \otimes \ldots \otimes x_{n-1}$$
by Yoneda reduction. This completes the induction.
=--
(It's been ages since I've thought about this. I need to think through the argument carefully again.)
## Blog resources
[[Todd Trimble]] talks about Yoneda reduction on the $n$Café [here](http://golem.ph.utexas.edu/category/2008/01/the_concept_of_a_space_of_stat.html#c014365). |
Yoneda structure | https://ncatlab.org/nlab/source/Yoneda+structure |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### 2-Category theory
+-- {: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
# Contents
* automatic table of contents goes here
{: toc}
## Idea
The concept of a **Yoneda structure** provides in a general 2-categorical setting the axiomatic description of the formal properties of the usual [[category of presheaves|presheaf construction]] and [[Yoneda embedding]] of ([[locally small category|locally]]) [[small category|small categories]]. The size issues arising in that context are absorbed directly into the structure via a class of (locally) "small" maps.
## Preliminaries
The axioms of a Yoneda structure are out to capture the properties of the presheaf construction with [[CAT]] replaced by general 2-category $\mathcal{K}$. In order to handle size issues a class of "legitimate" or "admissible" 0-cells is singled out in $|\mathcal{K}|$ as well as a class of 1-cells that behave well with respect to this class and the presheaf construction. In fact, it suffices to describe the admissible 1-cells since one can then identify the admissible 0-cells with the admissible identity 1-cells.
In $CAT$ relative to the usual presheaf construction one should think of the [[locally small category|locally small categories]] as the admissible 0-cells i.e. those categories $\mathcal{C}$ with all Hom-sets $\mathcal{C}(x,y)$ contained in a category $Set$ of "small" sets itself contained as object in a larger [[Grothendieck universe]] $U_0$. In this setting admissible functors $f:\mathcal{A}\to\mathcal{B}$ are those with all relative Hom-sets $\mathcal{B}(f(a),b)\in Set$. Furthermore, one can show ([Freyd-Street 1995](FS95)) that _a category_ $\mathcal{C}\in CAT$ _is small_ i.e. $|\mathcal{C}|\in Set$ precisely if $\mathcal{C}$ and $Set^{\mathcal{C}^{op}}$ _are locally small_.
Admissible functors $f$ in this sense in $CAT$ are closed under precomposition not only among themselves but with respect to arbitrary (composable) $g$ since the relative Hom-sets $\mathcal{B}(f{}g(x),b)$ are simply a subclass of $\mathcal{B}(f(a),b)$ namely those for which $a\in im(g)$. Whence these "relatively small" functors form a [[right ideal]]. Given the close connection between [[KZ doctrines]] and Yoneda structures it will nevertheless be useful to consider the more general case of closure in itself under composition as well, a situation which we acknowledge terminologically with the prefix "proto".
+-- {: .num_defn #admissible_1cell}
###### Definition
Let $\mathcal{K}$ be a [[2-category]] and $\mathbb{A}$ be a class of 1-cells. The 1-cells $f\in \mathbb{A}$ (and by abuse, the class $\mathbb{A}$ as well) are called _admissible_ if for all $f\in \mathbb{A}$ and composable 1-cells $g\in \mathcal{K}$, $f\circ g\in \mathbb{A}$. $f\in \mathbb{A}$ and $\mathbb{A}$ are called _proto-admissible_ if this closure property holds for $g\in \mathbb{A}$.
=--
+-- {: .num_defn #admissible_0cell}
###### Definition
Let $\mathcal{K}$ be a [[2-category]] and $\mathbb{A}$ be an admissible (resp. proto-admissible) class of 1-cells. A 0-cell $C\in |\mathcal{K}|$ is called _admissible_ (resp. _proto-admissible_) if $id_C$ is admissible (resp. proto-admissible). We denote the corresponding class of 0-cells by $|\mathbb{A}|$.
=--
For admissible $\mathbb{A}$, $C\in|\mathbb{A}|$ iff all 1-cells with codomain $C$ are admissible. This formulation has the advantage that it makes sense for [[semi-category|semi-categories]] as well.
Having now "taken care" of the size issues we recall/introduce some terminology concerning [[Kan extensions]] and [[relative adjoint functor|relative adjoint functors]] that will prove effective in yielding a surprisingly concise axiomatic description of the presheaf construction.
+-- {: .num_defn #left_extension}
###### Definition
Let $\mathcal{K}$ be a [[2-category]] and $\eta:f\Rightarrow e\circ g$ be a 2-cell:
$$
\array{
A& & \overset{f}{\longrightarrow} & &B
\\
&{}_g\searrow & \Downarrow _\eta& \nearrow _e&
\\
& & C & &
}
$$
We say that $\eta$ (or, by abuse, the diagram) exhibits $e:C\to B$ as a _left extension_ of $f:A\to B$ _along_ $g:A\to C$ if for all parallel maps $k:C\to B$ pasting with $\eta$ induces a bijection between 2-cells $\sigma:e\Rightarrow k$ and 2-cells $f\Rightarrow k\circ g$.
We say that a 1-cell $h:B\to D$ _preserves_ this left extension if the following diagram exhibits $h\circ e$ as a left extension of $h\circ f$ along $g$ :
$$
\array{ & &D & &
\\
&{}^{h{}f}{\nearrow}&\seArrow^{id} &{\nwarrow}^h&
\\
A& & \overset{f}{\longrightarrow} & &B
\\
&{}_g\searrow & \Downarrow _\eta& \nearrow _e&
\\
& & C & &
}
$$
The left extension is called _absolute_ if it is preserved by all 1-cells with domain $B$.
=--
+-- {: .num_defn #left_lifting}
###### Definition
Let $\phi:f\Rightarrow g\circ l$ be a 2-cell:
$$
\array{
A& & \overset{l}{\longrightarrow} & &C
\\
&{}_f\searrow & \neArrow _\phi& \swarrow _g&
\\
& & B & &
}
$$
We say that $\phi$ (or, by abuse, the diagram) exhibits $l:A\to C$ as a _left lifting_ of $f:A\to B$ _through_ $g:C\to B$ iff for all parallel maps $k:A\to C$ pasting with $\phi$ induces a bijection between 2-cells $\sigma:l\Rightarrow k$ and 2-cells $f\Rightarrow g\circ k$.
We say that a 1-cell $j:D\to A$ preserves this left lifting if the following diagram exhibits $l\circ j$ as a left lifting of $f\circ j$ through $g$ :
$$
\array{
D &\overset{j}{\longrightarrow} & A& \overset{l}{\longrightarrow} &C
\\
&{}_{f{}j}\searrow{}^{id}{\neArrow} &{}_f\downarrow & \overset{\phi}{\Rightarrow} \swarrow_g &
\\
& & B & &
}
$$
We say that the left lifting is _absolute_ if it is preserved by all 1-cells with codomain $A$.
=--
+-- {: .num_example #fully_faithfull}
###### Example
The following diagram
$$
\array{
A& & \overset{id_A}{\longrightarrow} & &A
\\
&{}_f\searrow & \neArrow _{id}& \swarrow _f&
\\
& & B & &
}
$$
exhibits $id_A$ as an absolute left lifting of $f:A\to B$ through itself iff $f:A\to B$ is _representably fully-faithful_ i.e. the functor "postcomposition with $f$" $\mathcal{K}(X,f):\mathcal{K}(X,A)\to\mathcal{K}(X,B)$ is fully-faithful for all $X\in|\mathcal{K}|$. This holds since the Hom-set $Hom_{\mathcal{K}(X,A)}(k,g)$ is precisely the set of 2-cells $k\Rightarrow g$ and $\mathcal{K}(X,f)$ acts on them by pasting with $id_f$.
=--
+-- {: .num_example #adjunction}
###### Example
The following diagram
$$
\array{
B& & \overset{g}{\longrightarrow} & &A
\\
&{}_{id_B}\searrow & \neArrow _\eta& \swarrow _f&
\\
& & B & &
}
$$
exhibits $g$ as an absolute left lifting of $id_B$ through $f$ iff there exists a 2-cell
$$
\array{
A& & \overset{id_A}{\longrightarrow} & &A
\\
&{}_{f}\searrow & \Uparrow _\epsilon& \nearrow _g&
\\
& & B & &
}
$$
such that the pasting of $\epsilon$ on $\eta$ at $g$
$$
\array{
& &A & &
\\
&{}_{f}\swarrow & \neArrow _\epsilon& \searrow _{id_A}&
\\
B& & \overset{g}{\longrightarrow} & &A
\\
&{}_{id_B}\searrow & \neArrow _\eta& \swarrow _f&
\\
& & B & &
}
$$
and the pasting of $\epsilon$ on $\eta$ at $f$
$$
\array{
B &\overset{g}{\longrightarrow} & A& \overset{id_A}{\longrightarrow} &A
\\
&{}_{id_B}\searrow{}^{\eta}{\neArrow} &{}_f\downarrow & \overset{\epsilon}{\neArrow} \nearrow_g &
\\
& & B & &
}
$$
yield identity 2-cells.
Of course, this situation expresses an **adjunction** $g\dashv f$ with _unit_ $\eta$ and _counit_ $\epsilon$. Here the absolute left lifting property of $\eta$ is furthermore equivalent to the left lifting property of $\eta$ plus preservation by $f \colon A\to B$ (cf. [Street-Walters 1978](#SW78), prop.2).
=--
## Definition
We are now ready to give the definition of a Yoneda structure:
+-- {: .num_defn #yoneda_structure}
###### Definition
Let $\mathcal{K}$ be a [[2-category]] and $\mathbb{A}$ be an admissible class of 1-cells.
A _presheaf construction_ $\mathcal{P}$ for $\mathbb{A}$ assigns to every admissible object $A\in |\mathbb{A}|$ an object $\mathcal{P}A\in |\mathcal{K}|$ called its _object of presheaves_ and an admissible 1-cell $y_A:A\to\mathcal{P}A$ called its _Yoneda morphism_ subject to the following conditions:
* (**YS1**) For each admissible 1-cell $f:A\to B$ with admissible domain $A\in |\mathbb{A}|$ there is given a 2-cell $\chi_f:y_A\Rightarrow e_f\circ f$ such that the following diagram
$$
\array{
A& & \overset{f}{\longrightarrow} & &B
\\
&{}_{y_A}\searrow & {}^{\chi_f}\neArrow& \swarrow _{e_f}&
\\
& & \mathcal{P}A & &
}
$$
exhibits $f$ as an absolute left lifting of $y_A$ through $e_f$ and $e_f$ as a left extension of $y_A$ along $f$.
* (**YS2**) For all $A\in|\mathbb{A}|$, $id_{\mathcal{P}A}$ is the left extension of $y_A$ along itself as exhibited in
$$
\array{
A& & \overset{y_A}{\longrightarrow} & &\mathcal{P}A
\\
&{}_{y_A}\searrow & \Downarrow _{id}& \nearrow _{id_{\mathcal{P}A}}&
\\
& & \mathcal{P}A & &
}
$$
* (**YS3**) For all $i:A\to B$, $j:B\to C$ such that $A,B\in |\mathbb{A}|$ and $i,j\in\mathbb{A}$ the following diagram
$$
\array{
A &\overset{i}{\rightarrow}& B& \overset{j}{\rightarrow} & C
\\
{}_{y_A}\downarrow &\overset{\chi_{{y_B}\circ i}}{\Rightarrow}&\downarrow_{y_B}&\overset{\chi_j}{\Rightarrow}\swarrow &{}_{e_j}
\\
\mathcal{P}A&\underset{e_{y_B\circ i}}{\leftarrow} &\mathcal{P}B& &
}
$$
exhibits $e_{{y_B}i}\circ e_j$ as the left extension of $y_A$ along $j\circ i$.
The pair $(\mathbb{A},\mathcal{P})$ is called a **Yoneda structure** on the 2-category $\mathcal{K}$.
=--
We desisted from tracking the prefix 'proto' through the foregoing but it should clear that a _proto-Yoneda structure_ results from replacing 'admissible' by 'proto-admissible' throughout the definition. Indeed, in (YS3) the assumption that $i\in\mathbb{A}$ was made in proviso for the case of proto-admissible 1-cells (cf. [Walker 2017](#Walker17)) since in presence of the right ideal property this already follows from the admissibility of $id_B$.
In cases where we need to keep track of from which (proto-)Yoneda structure the various structural 1- and 2-cells come from we will use the presheaf construction as a superscript for disambiguation: for (proto-)Yoneda structure $(\mathbb{A},\mathcal{P})$ we write $y_A^\mathcal{P}$ and $\chi_f^{\mathcal{P}}$ etc.
In Street-Walters ([1978](#SW78)) a stronger axiom (YS2') is also considered:
* (**YS2'**) If the following diagram
$$
\array{
A& & \overset{f}{\longrightarrow} & B&
\\
&{}_{y_A}\searrow & \overset{\chi_f}{\Rightarrow}& \swarrow \overset{\sigma}{\Rightarrow}\swarrow_g
\\
& & \mathcal{P}A & &
}
$$
exhibits $f$ as an absolute left lifting of $y_A$ through $g$ then $\sigma:e_f\Rightarrow g$ is an isomorphism.
It is shown there (prop.11) that (YS1) and (YS2') imply (YS2) and (YS3).
Together with a finite-completeness assumption on $\mathcal{K}$ and pointwiseness of the left extensions the resulting Yoneda structures are called **good** in Weber ([2007](#Weber07)) where it is also shown that Yoneda structures arising from presheaf constructions of the form $\mathcal{P}({}_-) = [({}_-)^{op},\Omega]$ have this property, $\Omega$ being an abstract "object of small sets" in a cartesian closed 2-category $\mathcal{K}$ with an involution $({}_-)^{op}$.
## Properties
....
## Examples
* The primordial example is $CAT$ with $\mathcal{P}:C\mapsto Set^{C^{op}}$ with $y_C:a\mapsto Hom_C({}_-,a)$ , the Yoneda embedding. A functor $f:C\to D$ is admissible if the relative $Hom_D(f{}_-,{}_-):D\to SET^{C^{op}}$ factors through $Set^{C^{op}}$. Admissible categories are precisely the locally small categories.
## The relation to KZ doctrines{#extension_KZ-doctrine}
Whereas it was shown already in the 1970s that ordinary monads on a 1-category can be brought into an [[extension system|extension form]] that avoids the iteration of the endofunctor similar presentations for 2-dimensional monad theory evolved more recently. For the comparison of [[lax-idempotent 2-monad|lax-idempotent 2-monads]] aka _KZ doctrines_ to Yoneda structures such presentations provided in the work of Marmolejo and Wood ([2012](#MW12)) come in handy. The link to Yoneda structures has been made in Walker ([2017](#Walker17)).
+-- {: .num_defn #KZ_doctrine}
###### Definition
Let $\mathcal{K}$ be a [[2-category]]. A _KZ-doctrine (in extension form)_ on $\mathcal{K}$ is a pair $(P,y)$ where $P$ assigns to every 0-cell $A\in\mathcal{K}$ a 0-cell $P(A)\in\mathcal{K}$ and $y$ is a family of 1-cells $y_A:A\to P(A)$ indexed by the 0-cells $A\in\mathcal{K}$ such that
* For every pair of 0-cells $A$, $B$ and 1-cells $f:A\to P(B)$ there exists an invertible 2-cell $\epsilon_f$
$$
\array{
A& & \overset{f}{\longrightarrow} & &P(B)
\\
&{}_{y_A}\searrow & \Downarrow _{\epsilon_f}& \nearrow _\overline{f}&
\\
& & P(A) & &
}
$$
that exhibits $\overline{f}$ as left extension of $f$ along $y_A$. Furthermore, in case $B=A$ and $f=y_A$ then $\epsilon_{y_A}$ is given by the identity 2-cell $id_{y_A}:y_A\Rightarrow id_{P(A)}\circ y_A$.
* For any 1-cell $g:B\to P(C)$, the 1-cell $\overline{g}:P(B)\to P(C)$ given itself by the left extension along $y_B$ preserves the left extension of $f:A\to B$ along $y_A$ exhibited by $\epsilon_f$.
=--
Clearly, with $(P,y)$ the presheaf construction of a Yoneda structure comes into sight though we still need to define a suitable class of admissible maps from $(P,y)$.
But before we do this we will introduce the concept that corresponds to the familiar notion of a pseudoalgebra for a lax-idempotent 2-monad thereby hopefully making it plausible that $(P,y)$ indeed is equivalent to the usual algebraic concept.
The main idea of the following definition is that the "pseudoalgebras" $X\in|\mathcal{K}|$ mimic the extension properties of the $P(A)$, in particular, all $P(A)$ satisfy the condition trivially and should be thought of as free algebras.
+-- {: .num_defn #P-complete}
###### Definition
Given a KZ-doctrine $(P,y)$ on $\mathcal{K}$. A 0-cell $X\in|\mathcal{K}|$ is called _P-cocomplete_ if for every $g:B\to X$ there exists an invertible 2-cell $\epsilon_g$
$$
\array{
B& & \overset{g}{\longrightarrow} & &X
\\
&{}_{y_B}\searrow & \Downarrow _{\epsilon_g}& \nearrow _\overline{g}&
\\
& & P(B) & &
}
$$
that exhibits $\overline{g}$ as left extension of $g$ along $y_B$. Moreover, this left extension $\overline{g}:P(B)\to X$ preserves all the left extensions $\overline{f}:P(A)\to P(B)$ along $y_A$ of arbitrary $f:A\to P(B)$.
=--
+-- {: .num_defn #P-cell}
###### Definition
A 1-cell $h:X\to Y$ between two P-cocomplete objects $X$, $Y$ is called a _P-homomorphism_ (, or a _P-cell_) if $h$ preserves the left extension $\overline{f}:P(A)\to X$ along $y_A:A\to P(A)$ for every $f:A\to X$.
=--
+-- {: .num_prop #P-algebra}
###### Proposition
Given a KZ-doctrine $(P,y)$ on $\mathcal{K}$. The following are equivalent:
* $A\in|\mathcal{K}|$ is P-cocomplete;
* $y_A:A\to P(A)$ has a left adjoint with invertible counit;
* $A$ is the underlying object of a pseudoalgebra.
=--
**Proof**. A combination of results from Bunge-Funk ([1999](#BF99)) and Marmolejo-Wood ([2012](#MW12)). $\qed$
We now attend the problem of defining a class $\mathbb{A}$ of admissible maps for $(P,y)$.
+-- {: .num_defn #P-admissible}
###### Definition
Given a KZ-doctrine $(P,y)$ on $\mathcal{K}$. A 1-cell $a:B\to C$ is called _P-admissible_ if the left extension of $y_B:B\to P(B)$ along $a$ exists and moreover is preserved by all left extensions $\overline{h}:P(B)\to X$ along $y_B$ of 1-cells $h:B\to X$ into a P-cocomplete 0-cell $X$.
=--
A crucial property of the Yoneda embedding is of course that is in fact an _embedding_ whence we must demand the same for the units of a KZ-doctrine:
+-- {: .num_defn #locally_fully_faithful}
###### Definition
A KZ-doctrine $(P,y)$ on a 2-category $\mathcal{K}$ is called _locally fully-faithful_ if all $y_A: A\to P(A)$ are representably fully-faithful (cf. [ex.](#fully_faithful)).
=--
Now we are ready to state the main result of this section.
+-- {: .num_prop #Yoneda_monad}
###### Theorem (Walker)
Let $(P,y)$ be a locally fully-faithful KZ-doctrine on 2-category $\mathcal{K}$ with $\mathbb{A}_P$ the class of P-admissible 1-cells. The pair $(\mathbb{A}_P,P)$ defines a proto-Yoneda structure on $\mathcal{K}$.
=--
**Proof**. cf. Walker ([2017](#Walker17), p.9). $\qed$
Jokingly, we may say that _a general KZ-doctrine is nothing but unfaithful Yoneda-structure without size problems!_ Conversely, the main difference between a locally fully-faithful KZ-doctrine and a Yoneda structure concerns size: For the KZ-doctrine every identity morphism is admissible and, accordingly, its presheaf construction is total whereas this need not be the case for general Yoneda structures.
## In retrospective
The following is a quote from [[Ross Street|Ross Street's]] "Australian conspectus of higher categories" ([Street 2010](#Street10), p. 241):
>In 1971 Bob Walters and I began work on Yoneda structures on 2-categories $[$[KS1](#KS73)$]$, $[$[StW](#SW78)$]$. The idea was to axiomatize the deeper aspects of categories beyond their merely being algebraic structures. This work centred on the Yoneda embedding $A\to \mathcal{P}A$ of a category $A$ into its presheaf category $\mathcal{P}A =[ A^{op} ,Set]$ . We covered the more general example of categories enriched in a base $\mathcal{V}$ where $\mathcal{P}A = [A^{op} ,\mathcal {V}]$ . Clearly size considerations needed to be taken seriously although a motivating size-free example was preordered sets with $\mathcal{P}A$ the inclusion-ordered set of right order ideals in $A$. Size was just an extra part of the structure. With the advent of elementary topos theory and the stimulation of the work of Anders Kock and Christian Mikkelsen, we showed that the preordered objects in a topos provided a good example. We were happy to realize $[$[KS1](#KS73)$]$ that an elementary topos was precisely a finitely complete category with a _power object_ (that is, a relations classifier). This meant that my work with Walters could be viewed as a higher-dimensional version of topos theory. As usual when raising dimension, what we might mean by a 2-dimensional topos could be many things, several of which could be useful. I looked $[$[St6](#Street74)$]$, $[$[St8](#Street80)$]$ at those special Yoneda structures where $\mathcal{P}A$ classified two-sided discrete fibrations.
## Related entries
* [[Yoneda embedding]]
* [[Yoneda lemma]]
* [[pro-arrow equipment]]
* [[2-topos]]
* [[cosmos]]
* [[relative adjoint functor]]
* [[Kan extension]]
* [[lax-idempotent 2-monad]]
* [[exact square]]
## Link
* n-café seminar blog on the Street-Walters paper: ([link](http://golem.ph.utexas.edu/category/2014/03/an_exegesis_of_yoneda_structur.html))
## References
The original sources are
* {#W71}[[Bob Walters]], _Yoneda 2Categories_ , talk at the University of New South Wales December 1971. ([manuscript](https://web.archive.org/web/20170211062709/https://dl.dropboxusercontent.com/u/92056191/Archive/temporary_new_material/yoneda.pdf))
* {#KS73}[[Max Kelly]], [[Ross Street]] (eds.), _Abstracts of the Sydney Category Theory Seminar 1972/73_ , Macquarie University.
* {#SW78} [[Ross Street]], [[Bob Walters]], _Yoneda structures on 2-categories_, JPAA **50** (1978) 350-379 [<a href="https://doi.org/10.1016/0021-8693(78)90160-6">doi:10.1016/0021-8693(78)90160-6</a>]
Early variations on the theme are in
* {#Street74}[[Ross Street]], _Elementary cosmoi I_ , pp.104-133 in Springer LNM **420** 1974.
* {#Street80}[[Ross Street]], _Cosmoi of internal categories_ , Transactions AMS **258** (1980) pp.271-318.
The Street quote stems from
* {#Street10}[[Ross Street]], _An Australian conspectus of higher categories_ , pp.237-264 in Baez, May (eds.), _Towards Higher Categories_ , Springer Heidelberg 2010. ([draft](http://maths.mq.edu.au/~street/Minneapolis.pdf))
The result on locally small categories suggesting the definition of a small object is reported in
* {#FS95}[[Peter Freyd]], [[Ross Street]], _On the size of categories_ , TAC **1** (1995) pp.174-185. ([abstract](http://www.tac.mta.ca/tac/volumes/1995/n9/1-09abs.html))
Exact squares in Yoneda structures are studied in
* [[René Guitart]], _Relations et carrés exacts_ , Ann. Sc. Math. Québec **IV** no.2 (1980) pp.103-125. ([draft](http://rene.guitart.pagesperso-orange.fr/textespublications/rg42.pdf))
* L. Van den Bril, _Exactitude dans les Yoneda-structures_ , Cah. Top. Géo. Diff. Cat. **XXIII** no.2 (1982) pp.215-224.
The following explores Yoneda structures arising from 2-categories with a discrete-opfibration classifier (such as [[2-toposes]]):
* {#Weber07} [[Mark Weber]], *Yoneda structures from 2-toposes*, Appl Categor Struct **15** (2007) 259–323 [[doi:10.1007/s10485-007-9079-2](https://doi.org/10.1007/s10485-007-9079-2), [pdf](https://sites.google.com/site/markwebersmaths/home/yoneda-structures-from-2-toposes)]
The following two investigate the connections with KZ doctrines:
* {#Walker17} C. Walker, _Yoneda Structures and KZ Doctrines_ , arXiv:1703.08693 (2017). ([abstract](http://arxiv.org/pdf/1703.08693v3))
* {#Walker18} C. Walker, _Distributive Laws via Admissibility_ , arXiv:1706.09575 (2018). ([abstract](https://arxiv.org/abs/1706.09575))
The interplay of Yoneda structures and KZ doctrines is employed to some effect in
* {#DL18} Ivan Di Liberti, [[Fosco Loregian]], _Accessibility and Presentability in 2-Categories_ , arXiv:1804.08710 (2018). ([abstract](https://arxiv.org/abs/1804.08710))
The pertaining technical ingredients on KZ doctrines are due to the following two papers
* {#BF99} [[Marta Bunge]], [[Jonathon Funk]], _On a bicomma object condition for KZ-doctrines_ , JPAA **143** (1999) pp.69-105.
* {#MW12} [[Francisco Marmolejo]], [[Richard J. Wood]], *Kan extensions and lax idempotent pseudomonads*, [[TAC]] **26** 1 (2012) 1-29 [[26-01](http://www.tac.mta.ca/tac/volumes/26/1/26-01abs.html)]
The relation to pro-arrow equipments, the presheaf construction and Isbell duality is discused in
* {#DL19} [[Ivan Di Liberti]], [[Fosco Loregian]], _On the Unicity of Formal Category Theories_ , arXiv:1901.01594 (2019). ([abstract](https://arxiv.org/abs/1901.01594))
[[!redirects yoneda structure]]
[[!redirects Yoneda Structure]]
[[!redirects Yoneda-structure]]
[[!redirects Yoneda structures]]
[[!redirects admissible 1-cell]]
|
Yong-Baek Kim | https://ncatlab.org/nlab/source/Yong-Baek+Kim |
* [research group page](https://sites.google.com/view/ybkimgroup/yong-baek-kim)
* [institute page](https://www.physics.utoronto.ca/members/kim-yong-baek/)
## Selected writings
On ([[non-perturbative effect|strongly]]) [[interaction|interacting]] [[topological phases of matter]] (such as [[topological order]]):
* [[Jason Alicea]], [[Matthew Fisher]], [[Marcel Franz]], [[Yong-Baek Kim]], *Strongly Interacting Topological Phases*, report on Banff workshop [15w5051](http://www.birs.ca/events/2015/5-day-workshops/15w5051) (2015) [[pdf](https://www.birs.ca/workshops/2015/15w5051/report15w5051.pdf), [[AliceaEtAl-InteractingTopPhases.pdf:file]]]
category: people |
Yong-Geun Oh | https://ncatlab.org/nlab/source/Yong-Geun+Oh |
* [website](http://www.math.wisc.edu/~oh/)
category: people |
Yong-Shi Wu | https://ncatlab.org/nlab/source/Yong-Shi+Wu |
* [institute page](https://faculty.utah.edu/u0028335-YONG-SHI_WU/research/index.hml)
* [institute page](https://web.physics.utah.edu/~wu/)
* [GoogleScholar page](https://scholar.google.com/citations?user=9NM7cUkAAAAJ&hl=en)
## Selected writings
On [[anyon]]-[[wavefunctions]] as [[multi-valued functions]] on a [[configuration space of points]]:
* {#Wu84} [[Yong-Shi Wu]], *Multiparticle Quantum Mechanics Obeying Fractional Statistics*, Phys. Rev. Lett. **53** (1984) 111 $[$[doi:10.1103/PhysRevLett.53.111](https://doi.org/10.1103/PhysRevLett.53.111), [pdf](https://core.ac.uk/download/pdf/276286925.pdf)$]$
category: people |
Yongbin Ruan | https://ncatlab.org/nlab/source/Yongbin+Ruan |
* [webpage](http://www-personal.umich.edu/~ruan/)
## Selected writings
Introducing [[quantum cohomology]]:
* [[Yongbin Ruan]], [[Gang Tian]], _A mathematical theory of quantum cohomology_, Mathematical Research Letters **1** 2 (1994) 269-278 [[doi:10.4310/MRL.1994.v1.n2.a15](https://doi.org/10.4310/MRL.1994.v1.n2.a15), [pdf](https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/1994/0001/0002/MRL-1994-0001-0002-a015.pdf)]
* [[Yongbin Ruan]], [[Gang Tian]], _A mathematical theory of quantum cohomology_, J. Diff. Geometry __42__ 2 (1995) 259-367 [[doi:10.4310/jdg/1214457234](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-42/issue-2/A-mathematical-theory-of-quantum-cohomology/10.4310/jdg/1214457234.full)]
Introducing [[Chen-Ruan cohomology]] for [[orbifolds]]:
* {#ChenRuan00} [[Weimin Chen]], [[Yongbin Ruan]], _A New Cohomology Theory for Orbifold_, Commun. Math. Phys. 248 (2004) 1-31 ([arXiv:math/0004129](https://arxiv.org/abs/math/0004129))
On [[orbifolds]], [[orbifold cohomology]] and specifically on [[Chen-Ruan cohomology]] and [[equivariant K-theory]]:
* {#ALR07} [[Alejandro Adem]], [[Johann Leida]], [[Yongbin Ruan]], _Orbifolds and Stringy Topology_, Cambridge Tracts in Mathematics **171** (2007) ([doi:10.1017/CBO9780511543081](https://doi.org/10.1017/CBO9780511543081), [pdf](http://www.math.colostate.edu/~renzo/teaching/Orbifolds/Ruan.pdf))
On [[discrete torsion]] in relation to twisted [[Chen-Ruan cohomology|Chen-ruan]] [[orbifold cohomology]]:
* [[Yongbin Ruan]], *Discrete torsion and twisted orbifold cohomology*, J. Sympl. Geom. 2 (2003), 1–24 ([arXiv:math/0005299](https://arxiv.org/abs/math/0005299))
On [[twisted equivariant K-theory|twisted]] [[orbifold K-theory]]:
* {#AdemRuan01} [[Alejandro Adem]], [[Yongbin Ruan]], _Twisted Orbifold K-Theory_, Commun. Math. Phys. 237 (2003) 533-556 ([arXiv:math/0107168](https://arxiv.org/abs/math/0107168))
* [[Alejandro Adem]], [[Yongbin Ruan]], [[Bin Zhang]], _A Stringy Product on Twisted Orbifold K-theory_, Morfismos (10th Anniversary Issue), Vol. 11, No 2 (2007), 33-64. ([arXiv:math/0605534](https://arxiv.org/abs/math/0605534), [Morfismos pdf](www.morfismos.cinvestav.mx/Portals/morfismos/SiteDocs/Articulos/Volumen11/No2/Zhang/arz.pdf))
On [[orbifolds]] in [[mathematical physics]] and in particular in [[string theory]]:
* [[Alejandro Adem]], [[Jack Morava]], [[Yongbin Ruan]], _[[Orbifolds in Mathematics and Physics]]_, Contemporary Mathematics 310, American Mathematical Society, 2002
On [[gauged linear sigma-models]]:
* Huijun Fan, Tyler Jarvis, [[Yongbin Ruan]], _A Mathematical Theory of the Gauged Linear Sigma Model_ ([arXiv:1506.02109](https://arxiv.org/abs/1506.02109))
category: people
|
Yoram Moses | https://ncatlab.org/nlab/source/Yoram+Moses |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Yoram_Moses)
* [GoogleScholar page](https://scholar.google.com/citations?user=X90c-SYAAAAJ&hl=en)
## Selected writings
On [[S5 modal logic]] as [[epistemic logic]]:
* [[Joseph Y. Halpern]], [[Yoram Moses]], *A guide to completeness and complexity for modal logics of knowledge and belief*, Artificial Intelligence **54** 3 (1992) 319-379
* {#FaginHalpernMosesVardi95} Ronald Fagin, [[Joseph Y. Halpern]], [[Yoram Moses]], Moshe Y. Vardi, *Reasoning About Knowledge*, The MIT Press (1995) $[$[ISBN:9780262562003](https://mitpress.mit.edu/9780262562003/reasoning-about-knowledge/)$]$
category: people |
Yoseph Ayoub > history | https://ncatlab.org/nlab/source/Yoseph+Ayoub+%3E+history |
* [[Joseph Ayoub]]
|
Yoshinori Matsuo | https://ncatlab.org/nlab/source/Yoshinori+Matsuo |
* [Institute page](http://www.dma.jim.osaka-u.ac.jp/view?l=en&u=10010757&a2=0000005&a3=0000134&o=affiliation&sm=affiliation&sl=en&sp=1)
## Selected writings
On the [[nuclear matrix model]] for [[baryons]]/[[nucleons]] the [[Witten-Sakai-Sugimoto model]]:
* {#HashimotoMatsuoMorita19} [[Koji Hashimoto]], [[Yoshinori Matsuo]], [[Takeshi Morita]], _Nuclear states and spectra in holographic QCD_, JHEP12 (2019) 001 ([arXiv:1902.07444](https://arxiv.org/abs/1902.07444))
Discussion of [[nucleon]] binding energies with the [[nuclear matrix model]]:
* [[Koji Hashimoto]], [[Yoshinori Matsuo]], _Nuclear binding energy in holographic QCD_ ([arXiv:2103.03563](https://arxiv.org/abs/2103.03563))
category: people |
Yosida duality | https://ncatlab.org/nlab/source/Yosida+duality | ## Idea
An analogue of the [[Gelfand duality]] for [[Riesz spaces]].
## Statement
The category of [[compact Hausdorff topological spaces]]
and [[continuous maps]] is contravariantly [[equivalent]]
to the category of uniformly complete Archimedean unital [[Riesz spaces]].
## Related concepts
* [[Gelfand duality]]
* [[Riesz space]]
## References
* [[Bas Westerbaan]], _Yosida Duality_, [arXiv:1612.03327](https://arxiv.org/abs/1612.03327).
* [[Kôsaku Yosida]], _On the representation of the vector lattice_, Proceedings of the Imperial Academy, Tokyo 18 (1942), 339–342. [ProjectEuclid](http://projecteuclid.org/euclid.pja/1195573861). |
You Qi | https://ncatlab.org/nlab/source/You+Qi |
* [personal page](https://you-qi2121.github.io/mypage/index.html)
## Selected writings
Introduction to [[categorification]] with focus on [[Khovanov homology]]:
* [[Mikhail Khovanov]] (notes by [[You Qi]]), *Introduction to categorification*, lecture notes, Columbia University (2010, 2020) [[web](https://www.math.columbia.edu/~khovanov/cat2020/), [web](https://you-qi2121.github.io/mypage/categorificationnotes.html), full:[pdf](https://www.dropbox.com/scl/fi/wdesax1c8il6tgwbi20t4/KhovanovYouQi-Categorification.pdf?rlkey=l5cm3khnzu604ijdnl06o89od&dl=0)]
[[!redirects Qi You]]
category: people
|
Younesse Kaddar | https://ncatlab.org/nlab/source/Younesse+Kaddar | A theoretical computer science student at the École Normale Supérieure Paris-Saclay
* [Web page](http://younesse.net)
## Related pages
* [[homotopy type theory]]
category: people |
Young diagram | https://ncatlab.org/nlab/source/Young+diagram |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## The idea
Young diagrams are used to describe many objects in algebra and combinatorics, including:
* integer [[partitions]]. For example, the integer partition
$$ 17 = 5 + 4 + 4 + 2 + 1 + 1 $$
is drawn as the Young diagram
+-- {: #Young style="text-align:center"}
<svg width="120" height="140" xmlns="http://www.w3.org/2000/svg" se:nonce="39384" xmlns:se="http://svg-edit.googlecode.com" xmlns:xlink="http://www.w3.org/1999/xlink">
<!-- Created with SVG-edit - http://svg-edit.googlecode.com/ -->
<desc>Young diagram (5,4,4,2,1,1)</desc>
<g>
<title>Layer 1</title>
<rect x="10" y="10" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_1"/>
<rect x="30" y="10" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_2"/>
<rect x="50" y="10" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_3"/>
<rect x="70" y="10" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_4"/>
<rect x="90" y="10" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_5"/>
<rect x="10" y="30" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_6"/>
<rect x="30" y="30" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_7"/>
<rect x="50" y="30" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_8"/>
<rect x="70" y="30" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_9"/>
<rect x="10" y="50" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_10"/>
<rect x="30" y="50" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_11"/>
<rect x="50" y="50" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_12"/>
<rect x="70" y="50" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_13"/>
<rect x="10" y="70" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_14"/>
<rect x="30" y="70" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_15"/>
<rect x="10" y="90" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_16"/>
<rect x="10" y="110" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_17"/>
</g>
</svg>
=--
* [[conjugacy classes]] in $S_n$.
* [[irreducible representations]]:
* [[irreducible representations]] of the [[symmetric groups]] $S_n$ over any field of characteristic zero
* [[irreducible representations|irreducible]] (algebraic) representation of the [[special linear groups]] $SL(N,\mathbb{C})$
* [[irreducible representation|irreducible]] [[unitary representations]] of the [[special unitary group|special unitary groups]] $SU(N)$
* [[elementary symmetric functions]]
* [[Schur functors]]
* [[linear basis|basis]] vectors for the free [[lambda-ring]] on one generator, $\Lambda$
* [[flag varieties]] for the [[special linear groups]] $SL(N,k)$, where $k$ is any field
* [[characteristic classes]] for complex [[vector bundles]]: that is, [[cohomology]] classes on the [[classifying spaces]] of the [[general linear groups]] $GL(N,\mathbb{C})$
* [[characteristic classes]] for hemitian vector bundles: that is, [[cohomology]] classes on the [[classifying spaces]] of the [[unitary groups]] $U(N)$
* finite-dimensional [[C*-algebras]]: any such algebra is of the form $M_{n_1}(\mathbb{C}) \oplus \cdots \oplus M_{n_k}(\mathbb{C})$ for some unique list of natural numbers $n_1 \ge n_2 \ge \cdots \ge n_k$.
* finite [[abelian group|abelian]] [[p-groups]]: any such group is of the form $\mathbb{Z}/p^{n_1} \oplus \cdots \oplus \mathbb{Z}/p^{n_k}$ for some unique list of natural numbers $n_1 \ge n_2 \ge \cdots \ge n_k$.
* finite [[commutative algebra|commutative]] [[semisimple algebras]] over a [[prime field]] $\mathbb{F}_p$: any such algebra is of the form $\mathbb{F}_{p^{n_1}} \oplus \cdots \oplus \mathbb{F}_{p^{n_k}}$ for some unique list of natural numbers $n_1 \ge n_2 \ge \cdots \ge n_k$.
* the [[trace of a category|trace of the category of finite sets]] has isomorphism classes of objects corresponding to Young diagrams.
## Young diagram
A **Young diagram** $F^\lambda$, also called **Ferrers diagram**, is a graphical representation of an unordered integer partition $\lambda = (\lambda_1\ge\lambda_2\ge\ldots\ge\lambda_l$). If $\lambda\vdash n$ is a partition of $n$ then the Young diagram has $n$ boxes. A partition can be addressed as a multiset over $\mathbb{N}$.
There are two widely used such representations. The _English_ one uses matrix-like indices, and the _French_ one uses Cartesian coordinate-like indices for the boxes $x_{i,j}$ in the diagram $F^\lambda$.
In the English representation the boxes are adjusted to the north-west in the 4th quadrant of a 2-dimensional Cartesian coordinate system, with the 'y'-axis being downward oriented. For instance the diagram $F^{(5,4,4,2,1,1)}$ representing the partition $(5,4,4,2,1,1)$ of $17$ is given in the English representation as:
+-- {: #Young style="text-align:center"}
<svg width="120" height="140" xmlns="http://www.w3.org/2000/svg" se:nonce="39384" xmlns:se="http://svg-edit.googlecode.com" xmlns:xlink="http://www.w3.org/1999/xlink">
<!-- Created with SVG-edit - http://svg-edit.googlecode.com/ -->
<desc>Young diagram (5,4,4,2,1,1)</desc>
<g>
<title>Layer 1</title>
<rect x="10" y="10" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_1"/>
<rect x="30" y="10" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_2"/>
<rect x="50" y="10" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_3"/>
<rect x="70" y="10" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_4"/>
<rect x="90" y="10" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_5"/>
<rect x="10" y="30" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_6"/>
<rect x="30" y="30" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_7"/>
<rect x="50" y="30" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_8"/>
<rect x="70" y="30" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_9"/>
<rect x="10" y="50" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_10"/>
<rect x="30" y="50" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_11"/>
<rect x="50" y="50" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_12"/>
<rect x="70" y="50" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_13"/>
<rect x="10" y="70" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_14"/>
<rect x="30" y="70" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_15"/>
<rect x="10" y="90" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_16"/>
<rect x="10" y="110" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_17"/>
</g>
</svg>
=--
Let $\mathbb{Y}$ be the set of Young diagrams. Important functions on Young
diagrams include:
* {#Conjugation} **conjugation**: denoted by a prime $\prime : \mathbb{Y} \rightarrow \mathbb{Y}$
reflects the Young diagram along its main diagonal (north-west to south-east).
In the above example the conjugated partition would be $\lambda^\prime=(6,4,3,3,1)$.
* **weight**: $wt \colon \mathbb{Y} \rightarrow \mathbb{N}$ provides the number
of boxes.
* **length**: $\ell \colon \mathbb{Y} \rightarrow \mathbb{N}$ provides the number
of rows or equivalently the number of positive parts of the partition $\lambda$.
The length of the conjugated diagram gives the number of columns.
* **plus**: $+ \colon \mathbb{Y}\times \mathbb{Y} \rightarrow \mathbb{Y} ::
(\mu,\nu) \mapsto \mu + \nu = (\mu_1+\nu_1,\ldots,\mu_l+\nu_l)$
* **times**: $\times \colon \mathbb{Y}\times \mathbb{Y} \rightarrow \mathbb{Y} ::
(\mu,\nu) \mapsto (\mu \cup \nu)_{\ge}$ the unordered union of the multisets. It
follows that $\mu\times \nu =(\mu^\prime + \nu^\prime)^\prime$.
A **filling** of a Young diagram with elements from a set $S$ is called a [[Young tableau]].
## Skew Young diagram
A generalization of a Young diagram is a skew Young diagram. Let $\mu,\nu$ be two partitions, and let $\nu \le \mu$ be defined as $\forall i : \nu_i\le \mu_i$ (possibly adding trailing zeros). The **skew Young diagram** $F^{\mu/\nu}$ is given by the Young diagram $F^\mu$ with all boxes belonging to $F^\nu$ when superimposed removed. If $\mu=(5,4,4,2,1,1)$ and $\nu=(3,3,2,1)$ then $F^{\mu/\nu}$ looks like:
+-- {: #SkewYoung style="text-align:center"}
<svg width="120" height="140" xmlns="http://www.w3.org/2000/svg" se:nonce="39424" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:se="http://svg-edit.googlecode.com">
<!-- Created with SVG-edit - http://svg-edit.googlecode.com/ -->
<desc>Skew Young diagram (5,4,4,2,1,1)/(3,3,2,1)</desc>
<g>
<title>Layer 1</title>
<rect id="svg_39424_1" stroke-width="2" stroke="#cccccc" fill="none" height="20" width="20" y="10" x="0"/>
<rect id="svg_39424_2" stroke-width="2" stroke="#cccccc" fill="none" height="20" width="20" y="10" x="20"/>
<rect id="svg_39424_3" stroke-width="2" stroke="#cccccc" fill="none" height="20" width="20" y="10" x="40"/>
<rect id="svg_39424_4" stroke-width="2" stroke="#000000" fill="#ffdddd" height="20" width="20" y="10" x="60"/>
<rect id="svg_39424_5" stroke-width="2" stroke="#000000" fill="#ffdddd" height="20" width="20" y="10" x="80"/>
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<rect id="svg_39424_17" stroke-width="2" stroke="#000000" fill="#ffdddd" height="20" width="20" y="110" x="0"/>
</g>
</svg>
=--
* A skew diagram is called **connected** if all boxes share an edge.
* A skew diagram is called a **horizontal strip** if every column contains at most one box.
* A skew diagram is called a **vertical strip** if every row contains at most one box.
* conjugation, weight, length extend to skew diagrams accordingly.
## Young tableau
...
## Related concepts
* [[semistandard Young tableau]],
* [[hook length formula]], [[hook-content formula]]
* [[Schur polynomial]]
* [[representation theory of the symmetric group]]
* [[representation theory of the general linear group]]
* [[Schur-Weyl duality]]
* [[Schur-Weyl measure]]
## References ##
Quick introduction
* Alexander Yong, What is ... a Young Tableau, _Notices of the American Mathematical Society_ *54* (February 2007), 240–241. ([pdf](http://www.ams.org/notices/200702/whatis-yong.pdf))
Textbook accounts are in any book on [[representation theory]] in general and on the [[representation theory of the symmetric group]] in particular; such as:
* {#FultonHarris91} [[William Fulton]], [[Joe Harris]], _Representation Theory: a First Course_, Springer, Berlin, 1991 ([pdf](http://isites.harvard.edu/fs/docs/icb.topic1381051.files/fulton-harris-representation-theory.pdf))
* {#Sagan2001} [[Bruce E. Sagan]], _The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions_, Springer, 2001
More details:
* Kazuhiko Koike, Itaru Terada, _Young-diagrammatic methods for the representation theory of the classical groups of type $B_n$, $C_n$, $D_n$_, Journal of Algebra, Volume 107, Issue 2, May 1987, Pages 466-511
* [[William Fulton]], _Young Tableaux, with Applications to Representation Theory and Geometry_, Cambridge U. Press, 1997 ([doi:10.1017/CBO9780511626241](https://doi.org/10.1017/CBO9780511626241))
* [[Ron M. Adin]], [[Yuval Roichman]], *Enumeration of Standard Young Tableaux*, Chapter 14 in: Miklós Bóna, *Handbook of Enumerative Combinatorics*, CRC Press 2015 ([arXiv:1408.4497](https://arxiv.org/abs/1408.4497), [ISBN:9781482220858](https://www.routledge.com/Handbook-of-Enumerative-Combinatorics/Bona/p/book/9781482220858))
Connection to [[algebraic geometry]]:
* C. de Concini, [[D. Eisenbud]], C. Procesi, _Young diagrams and determinantal varieties_, Invent. Math. __56__ (1980), 129-165.
With an eye towards application to (the [[standard model of particle physics|standard model]] of) [[particle physics]]:
* [[Howard Georgi]], §12 in: *Lie Algebras In Particle Physics*, Westview Press (1999), CRC Press (2019) [[doi:10.1201/9780429499210](https://doi.org/10.1201/9780429499210)]
category: combinatorics
[[!redirects Young diagram]]
[[!redirects Young diagrams]]
[[!redirects Ferrers diagram]]
[[!redirects Ferrers diagrams]]
[[!redirects Young tableau]]
[[!redirects Young tableaus]]
[[!redirects Young tableaux]]
|
Young measure | https://ncatlab.org/nlab/source/Young+measure | Related entries: [[calculus of variations]], [[microlocal defect functional]], [[measure theory]]
* L. C. Young, _Generalized curves and the existence of an attained absolute minimum in the calculus of variations_, Comptes Rendus de la Soc. des Sciences et des Lettres de Varsovie, classe III, 30, 212-234 (1937); _Generalized surfaces in the calculus of variations_, Ann. of Math. (2) 43 (1942), 84–103 [MR0006023]
(http://www.ams.org/mathscinet-getitem?mr=0006023); _Generalized surfaces in the calculus of variations. II_, Ann. of Math. (2) 43 (1942), 530–544 [MR0006832](http://www.ams.org/mathscinet-getitem?mr=0006832); _Lectures on the calculus of variations and optimal control theory_, Foreword by Wendell H. Fleming, W. B. Saunders Co., Philadelphia-London-Toronto 1969 [MR0259704](http://www.ams.org/mathscinet-getitem?mr=0259704)
* Pablo Pedregal, _Optimization, relaxation and Young measures_, Bull. Amer. Math. Soc. 36 (1999), 27-58 [link](http://www.ams.org/journals/bull/1999-36-01/S0273-0979-99-00774-0)
* Luc Tartar, _On mathematical tools for studying partial differential equations of continuum physics: $H$-measures and Young measures_, Developments in partial differential equations and applications to mathematical physics (Ferrara, 1991) Plenum, New York, 1992, pp. 201–217 [MR1213932 (94c:35031)](http://www.ams.org/mathscinet-getitem?mr=1213932)
category: analysis |
Young subgroup | https://ncatlab.org/nlab/source/Young+subgroup | ## Idea
Every composition of an integer $n$ induces a [[subgroup]] of $\mathfrak{S}_{n}$.
## Definition
Let $n \ge 1$. Given a [[composition]] $(\lambda_{1},...,\lambda_{k})$ of the integer $n$ (where $\lambda_{i} \ge 1$), we have the Young subgroup:
$\mathfrak{S}_{\lambda_{1}} \times ... \times \mathfrak{S}_{\lambda_{k}} \hookrightarrow \mathfrak{S}_{n}$
associated to this composition.
## Example
If $(k,1,l)$ is a composition of $n$ as above, then:
$\mathfrak{S}_{k} \times \mathfrak{S}_{1} \times \mathfrak{S}_{l} \hookrightarrow \mathfrak{S}_{n}$
is a subgroup of permutations $\sigma \in \mathfrak{S}_{n}$ which verify that $k+1$ is a [[fixed point]] of $\sigma$. However some permutations fix $k+1$ without being an element of this subgroup.
## Related notions
[[symmetric group]] |
Young's inequality | https://ncatlab.org/nlab/source/Young%27s+inequality |
\tableofcontents
## Young's inequality for products
\begin{proposition}
Given
* $a, \,b \,\in\, \mathbb{R}_{\gt 0}$, non-[[negative number|negative]] [[real numbers]];
* $p, \, q \,\in\, \mathbb{R}_{\gt 1}$ such that
$$
\frac{1}{p}
+
\frac{1}{q}
=
1
$$
then the following [[inequality]] holds:
$$
a b
\;\leq\;
\frac{a^p}{p}
+
\frac{b^q}{q}
\,,
$$
which is an [[equality]] if and only if $a^p = b^q$.
\end{proposition}
One proof is by [[convex function|convexity]] of the [[exponential function]]: choosing $x, y, t$ such that $\exp(x) = a^p$, $\exp = b^q$ and $t = \frac1{p}$, Young's inequality is identical to the convexity constraint
$$\exp(tx + (1-t)y) \leq t\exp(x) + (1-t)\exp(y).$$
## Related concepts
* from Young's inequality follows [[Hölder's inequality]]
## References
See also:
* Wikipedia, *[Young's inequality for products](https://en.m.wikipedia.org/wiki/Young%27s_inequality_for_products)*
[[!redirects Young's inequality for products]]
|
Young–Fibonacci lattice | https://ncatlab.org/nlab/source/Young%E2%80%93Fibonacci+lattice |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A kind of [[modular lattice]].
## References
* Wikipedia, _[Young–Fibonacci lattice](http://en.wikipedia.org/wiki/Young%E2%80%93Fibonacci_lattice)_
|
Yu Qiu | https://ncatlab.org/nlab/source/Yu+Qiu |
* [webpage](https://www.math.cuhk.edu.hk/people/academic-staff/yqiu)
## related $n$Lab entries
* [[Bridgeland stability condition]]
* [[Dynkin quiver]]
category: people |
Yu-tin Huang | https://ncatlab.org/nlab/source/Yu-tin+Huang |
* [Institute page](https://www.phys.ntu.edu.tw/enphysics/yutin.html)
* [InSpire page](https://inspirehep.net/authors/1038150)
## Selected writings
On [[scattering amplitudes]] in [[perturbative quantum field theory]]:
* [[Henriette Elvang]], [[Yu-tin Huang]], *Scattering Amplitudes*, Cambridge University Press (2015) [[arXiv:1308.1697](http://arxiv.org/abs/1308.1697), [doi:10.1017/CBO9781107706620]( https://doi.org/10.1017/CBO9781107706620)]
category: people |
Yuan Feng | https://ncatlab.org/nlab/source/Yuan+Feng |
* [Institute page](https://profiles.uts.edu.au/Yuan.Feng)
* [Research group page](https://quantum-lab.org/yuan/)
## Selected writings
Introducing the [[QPMC]] quantum protocol model checker:
* [[Yuan Feng]], Ernst Moritz Hahn, Andrea Turrini, Lijun Zhang, *`QPMC`: A Model Checker for Quantum Programs and Protocols*, in *Formal Methods. FM 2015*, Lecture Notes in Computer Science **9109**, Springer (2015) [[doi:10.1007/978-3-319-19249-9_17](https://doi.org/10.1007/978-3-319-19249-9_17)]
On [[software verification]] in [[quantum computing]], hence with/of [[quantum programming languages]]:
* {#YingFeng18} [[Mingsheng Ying]], [[Yuan Feng]], *Model Checking Quantum Systems --- A Survey* [[arXiv:1807.09466](https://arxiv.org/abs/1807.09466)]
* Ji Guan, [[Yuan Feng]], Andrea Turrini, [[Mingsheng Ying]], *Model Checking Applied to Quantum Physics* [[arXiv:1902.03218](https://arxiv.org/abs/1902.03218)]
* {#YingFeng21} [[Mingsheng Ying]], [[Yuan Feng]], *Model Checking Quantum Systems -- Principles and Algorithms*, Cambridge University Press (2021) [[ISBN:9781108484305](https://www.cambridge.org/ae/academic/subjects/computer-science/programming-languages-and-applied-logic/model-checking-quantum-systems-principles-and-algorithms?format=HB)]
On [[quantum computing|quantum]] [[software verification]] with the [[Coq]] [[proof assistant]]:
* Wenjun Shi, Qinxiang Cao, Yuxin Deng, Hanru Jiang, [[Yuan Feng]], *Symbolic Reasoning about Quantum Circuits in Coq*, Journal of Computer Science and Technology (JCST), **36** 6 (2021) 1291-1306 $[$[arXiv:2005.11023](https://arxiv.org/abs/2005.11023), [doi:10.1007/s11390-021-1637-9](https://doi.org/10.1007/s11390-021-1637-9)$]$
category: people |
Yufei Zhao | https://ncatlab.org/nlab/source/Yufei+Zhao |
* [personal page](https://yufeizhao.com/)
## Selected writings
On [[Young tableaux]] and the [[representation theory of the symmetric group]]:
* [[Yufei Zhao]], _Young Tableaux and the Representations of the Symmetric Group_ ([pdf](https://yufeizhao.com/research/youngtab-hcmr.pdf), [[ZhaoYoungTableaux.pdf:file]])
category: people |
Yuhma Asano | https://ncatlab.org/nlab/source/Yuhma+Asano |
* [SPIRE page](https://inspirehep.net/authors/1274238)
## Selected writings
On [[M2-M5-brane bound states]] in the [[BMN matrix model]]:
* {#AIST17b} [[Yuhma Asano]], [[Goro Ishiki]], [[Shinji Shimasaki]], [[Seiji Terashima]], _Spherical transverse M5-branes from the plane wave matrix model_, JHEP 02 (2018) 076 $[$[arXiv:1711.07681](https://arxiv.org/abs/1711.07681), <a href="https://doi.org/10.1007/JHEP02(2018)076">doi:10.1007/JHEP02(2018)076</a>$]$
* {#AIST17a} [[Yuhma Asano]], [[Goro Ishiki]], [[Shinji Shimasaki]], [[Seiji Terashima]], _On the transverse M5-branes in matrix theory_, Phys. Rev. D **96** 126003 (2017) $[$[arXiv:1701.07140](https://arxiv.org/abs/1701.07140), [doi:10.1103/PhysRevD.96.126003](https://doi.org/10.1103/PhysRevD.96.126003)$]$
On the emergence of [[higher spin gauge theory|higher spin]] [[gravity]] from the [[IKKT matrix model]]:
* [[Yuhma Asano]], [[Harold Steinacker]], *Spherically symmetric solutions of higher-spin gravity in the IKKT matrix model* ([arXiv:2112.08204](https://arxiv.org/abs/2112.08204)) |
Yuho Sakatani | https://ncatlab.org/nlab/source/Yuho+Sakatani |
* [inspire page](https://inspirehep.net/authors/1077874)
## Selected writings
On [[M-brane]] [[sigma-models]] on [[exceptional geometry|exceptional]] [[exceptional generalized geometry|generalized geometric]] [[target spacetimes]]:
* [[Yuho Sakatani]], [[Shozo Uehara]], _Branes in Extended Spacetime: Brane Worldvolume Theory Based on Duality Symmetry_, Phys. Rev. Lett. **117** 191601 (2016) [[arXiv:1607.04265](https://arxiv.org/abs/1607.04265), [talk slides](http://www2.yukawa.kyoto-u.ac.jp/~qft.web/2016/slides/sakatani.pdf)]
* [[Yuho Sakatani]], [[Shozo Uehara]], _Exceptional M-brane sigma models and $\eta$-symbols_ [[arXiv:1712.10316](https://arxiv.org/abs/1712.10316)]
On [[exotic branes]] in [[string theory]]/[[M-theory]]:
* [[Jose J. Fernandez-Melgarejo]], [[Tetsuji Kimura]], [[Yuho Sakatani]], *Weaving the Exotic Web*, JHEP **2018** 72 (2018) [<a href="https://doi.org/10.1007/JHEP09(2018)072">doi:10.1007/JHEP09(2018)072</a>]
category: people |
Yuji Tachikawa | https://ncatlab.org/nlab/source/Yuji+Tachikawa |
* [webpage](http://member.ipmu.jp/yuji.tachikawa/)
## Selected writings
On the [[AGT correspondence]]:
* {#AGT09} [[Luis Alday]], [[Davide Gaiotto]], [[Yuji Tachikawa]], _Liouville Correlation Functions from Four-dimensional Gauge Theories_, Lett.Math.Phys.91:167-197, 2010 ([arXiv:0906.3219](http://arxiv.org/abs/0906.3219))
On [[5-brane webs]] and [[D=4 N=2 super Yang-Mills theory]]:
* [[Francesco Benini]], [[Sergio Benvenuti]], [[Yuji Tachikawa]], _Webs of five-branes and $\mathcal{N}=2$ superconformal field theories_, JHEP 0909:052 (2009) [[arXiv:0906.0359](https://arxiv.org/abs/0906.0359)]
On [[D=4 N=2 super Yang-Mills theory]]:
* [[Yuji Tachikawa]], *$\mathcal{N}=2$ supersymmetric dynamics for pedestrians*, Lecture Notes in Physics, vol. 890, 2014 ([arXiv:1312.2684](https://arxiv.org/abs/1312.2684), [doi:10.1007/978-3-319-08822-8](https://link.springer.com/book/10.1007%2F978-3-319-08822-8), [web version](http://www-hep.phys.s.u-tokyo.ac.jp/~yujitach/tmp/dummies/dummies.html#dummiesse7.html))
On [[anomaly cancellation]] in [[D=6 N=(2,0) SCFT]] and [[D=6 N=(1,0) SCFT]]:
and their [[anomaly cancellation]] via the [[Green-Schwarz mechanism]]:
* [[Kantaro Ohmori]], [[Hiroyuki Shimizu]], [[Yuji Tachikawa]], _Anomaly polynomial of E-string theories_, J. High Energ. Phys. 2014, 2 (2014) ([arXiv:1404.3887](https://arxiv.org/abs/1404.3887))
* [[Kantaro Ohmori]], [[Hiroyuki Shimizu]], [[Yuji Tachikawa]], [[Kazuya Yonekura]], _Anomaly polynomial of general 6d SCFTs_, Progress of Theoretical and Experimental Physics, Volume 2014, Issue 10, October 2014, 103B07 ([arXiv:1408.5572](https://arxiv.org/abs/1408.5572))
On [[generalized global symmetries]] and their [[gauge symmetry|gauging]]:
* {#BT17} [[Lakshya Bhardwaj]], [[Yuji Tachikawa]]. *On finite symmetries and their gauging in two dimensions*, J. High Energ. Phys. **2018** 189 (2018) [[arXiv:1704.02330](https://arxiv.org/abs/1704.02330), <a href="https://doi.org/10.1007/JHEP03(2018)189">doi:10.1007/JHEP03(2018)189</a>]
Relation of the [[GSO projection]] to the [[K-theory classification of topological phases of matter]]:
* Justin Kaidi, Julio Parra-Martinez, [[Yuji Tachikawa]], _Topological Superconductors on Superstring Worldsheets_ ([arXiv:1911.11780](https://arxiv.org/abs/1911.11780))
On the [[WZW term]] in [[chiral perturbation theory]]:
* Yasunori Lee, [[Kantaro Ohmori]], [[Yuji Tachikawa]], _Revisiting Wess-Zumino-Witten terms_ ([arXiv:2009.00033](https://arxiv.org/abs/2009.00033))
On lifting the [[Witten genus]] of the [[heterotic string]] to [[topological modular forms]]:
* [[Yuji Tachikawa]], *Topological modular forms and the absence of a heterotic global anomaly*, Progress of Theoretical and Experimental Physics, **2022** 4 (2022) 04A107 $[$[arXiv:2103.12211](https://arxiv.org/abs/2103.12211), [doi:10.1093/ptep/ptab060](https://doi.org/10.1093/ptep/ptab060)$]$
* [[Yuji Tachikawa]], [[Mayuko Yamashita]], *Topological modular forms and the absence of all heterotic global anomalies*, Comm. Math. Phys. **402** (2023) 1585-1620 [[arXiv:2108.13542](https://arxiv.org/abs/2108.13542), [doi:10.1007/s00220-023-04761-2](https://doi.org/10.1007/s00220-023-04761-2)]
and in relation to [[Anderson duality]] of [[tmf]]:
* [[Yuji Tachikawa]], [[Mayuko Yamashita]], *Anderson self-duality of topological modular forms, its differential-geometric manifestations, and vertex operator algebras* [[arXiv:2305.06196](https://arxiv.org/abs/2305.06196)]
On [[anomaly cancellation]] via the [[Green-Schwarz mechanism]] from the point of view of [[higher gauge theory]]:
* [[Yasunori Lee]], [[Kantaro Ohmori]], [[Yuji Tachikawa]], *Matching higher symmetries across Intriligator-Seiberg duality* ([arXiv:2108.05369](https://arxiv.org/abs/2108.05369))
On the mod-2 [[elliptic genus]], both as a [[partition function]] of a [[superstring]] as well as via the [[string-orientation of tmf]]:
* [[Yuji Tachikawa]], [[Mayuko Yamashita]], [[Kazuya Yonekura]], *Remarks on mod-2 elliptic genus* $[$[arXiv:2302.07548](https://arxiv.org/abs/2302.07548)$]$
## Related entries
* [[AGT correspondence]]
* [[FQFT]]
category: people
|
Yuji Terashima | https://ncatlab.org/nlab/source/Yuji+Terashima |
* [Institute page](https://tohoku.pure.elsevier.com/en/persons/yuji-terashima)
## Selected writings
On [[fiber integration in differential cohomology]] (fiberwise [[Stokes theorem]] lifted to [[Deligne cohomology]]):
* {#GomiTerashima00} [[Kiyonori Gomi]], [[Yuji Terashima]], _A Fiber Integration Formula for the Smooth Deligne Cohomology_, International Mathematics Research Notices 2000, No. 13 ([pdf](http://imrn.oxfordjournals.org/content/2000/13/699.full.pdf), [pdf](http://numr.wdfiles.com/local--files/differential-cohomology/gomi-terashima.pdf), [doi:10.1155/S1073792800000386](https://doi.org/10.1155/S1073792800000386))
On [[higher parallel transport]] for [[circle n-bundles with connection]]:
* [[Kiyonori Gomi]], [[Yuji Terashima]], _Higher dimensional parallel transport_ Mathematical Research Letters 8, 25–33 (2001) ([pdf](http://mrlonline.org/mrl/2001-008-001/2001-008-001-004.pdf))
On [[twisted K-theory]] via [[vectorial bundles]]:
* {#Gomi} [[Kiyonori Gomi]], _Twisted K-theory and finite-dimensional approximation_ ([arXiv:0803.2327](http://arxiv.org/abs/0803.2327))
* {#GomiTerashima} [[Kiyonori Gomi]], [[Yuji Terashima]], _Chern-Weil Construction for Twisted K-Theory_, Communication ins Mathematical Physics, Volume 299, Number 1, 225-254 ([doi:10.1007/s00220-010-1080-1](https://doi.org/10.1007/s00220-010-1080-1))
On [[discrete torsion]]:
* [[Kiyonori Gomi]], [[Yuji Terashima]], _Discrete Torsion Phases as Topological Actions_, Commun. Math. Phys. (2009) 287: 889 ([doi:10.1007/s00220-009-0736-1](https://doi.org/10.1007/s00220-009-0736-1))
Introducing the [[3d-3d correspondence]]:
* [[Yuji Terashima]], [[Masahito Yamazaki]], _$SL(2,\mathbb{R})$ Chern-Simons, Liouville, and Gauge Theory on Duality Walls_, JHEP 1108:135, 2011 ([arXiv:1103.5748](https://arxiv.org/abs/1103.5748))
* [[Yuji Terashima]], [[Masahito Yamazaki]], _Semiclassical Analysis of the 3d/3d Relation_, Phys.Rev.D88:026011, 2013 ([arXiv:1106.3066](https://arxiv.org/abs/1106.3066))
category: people |
Yukawa coupling | https://ncatlab.org/nlab/source/Yukawa+coupling |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In the [[standard model of particle physics]], the _Yukawa couplings_ encode the [[interaction]] between the fundamental [[fermion fields]] and the [[Higgs field]], and thus, via the [[Higgs mechanism]], the [[masses]] of the [[fermion field]] after [[electroweak symmetry breaking]].
## In string phenomenology
In [[intersecting D-brane models]] Yukawa couplings are encoded by [[worldsheet instantons]] of open strings stretching between the [[brane intersection|intersecting]] [[D-branes]] (see [Marchesano 03, Section 7.5](#Marchesano03)). Mathematically this is encoded by [[derived hom-spaces]] in a [[Fukaya category]] (see [Marchesano 03, Section 7.5](#Marchesano03)).
\begin{center}
<img src="https://ncatlab.org/nlab/files/YukawaFukaya.jpg" width="680">
\end{center}
> table grabbed from [Marchesano 03](#Marchesano03)
## Related concepts
* [[coupling constant]]
* [[minimal coupling]]
* [[standard model of particle physics]]
* [[spinors in Yang-Mills theory]]
## References
### General
See also
* Wikipedia, _[Yukawa interactions](http://en.wikipedia.org/wiki/Yukawa_interaction)_
### In string theory
On computation of Yukawa couplings in [[heterotic string theory]]:
* Stefan Blesneag, [[Evgeny Buchbinder]], [[Andrei Constantin]], [[Andre Lukas]], [[Eran Palti]], _Matter Field Kähler Metric in Heterotic String Theory from Localisation_ ([arXiv:1801.09645](https://arxiv.org/abs/1801.09645))
* Giorgi Butbaia, Damián Mayorga Peña, Justin Tan, Per Berglund, [[Tristan Hübsch]], Vishnu Jejjala, Challenger Mishra, *Physical Yukawa Couplings in Heterotic String Compactifications* [[arXiv:2401.15078](https://arxiv.org/abs/2401.15078)]
* Andrei Constantin, Cristofero S. Fraser-Taliente, Thomas R. Harvey, [[Andre Lukas]], [[Burt Ovrut]], *Computation of Quark Masses from String Theory* [[arXiv:2402.01615](https://arxiv.org/abs/2402.01615)]
and in [[intersecting D-brane models]] of [[type II string theory]] via [[Fukaya categories]]:
* {#CremadesIbanezMarchesano03} D. Cremades, [[Luis Ibáñez]], [[Fernando Marchesano]], _Yukawa couplings in intersecting D-brane models_, JHEP 0307 (2003) 038 ([arXiv:hep-th/0302105](https://arxiv.org/abs/hep-th/0302105))
* {#Marchesano03} [[Fernando Marchesano]], section 7 of _Intersecting D-brane Models_ ([arXiv:hep-th/0307252](https://arxiv.org/abs/hep-th/0307252))
Realistic [[Yukawa couplings]] and [[fermion]] [[masses]] in an [[MSSM]] [[Pati-Salam GUT model]] with 3 [[generations of fermions]] realized on [[intersecting D-brane model|intersecting]] [[D6-branes]] [[KK-compactification|KK-compactified]] on a [[toroidal orbifold]] $T^6\sslash (\mathbb{Z}_2 \times \mathbb{Z}_2)$ are claimed in
* {#ChenLiMayesNanopoulos07a} Ching-Ming Chen, Tianjun Li, [[Van Eric Mayes]], [[Dimitri Nanopoulos]], _A Realistic World from Intersecting D6-Branes_, Phys.Lett.B665:267-270, 2008 ([arXiv:hep-th/0703280](https://arxiv.org/abs/hep-th/0703280), [doi:10.1016/j.physletb.2008.06.024](https://doi.org/10.1016/j.physletb.2008.06.024))
* {#ChenLiMayesNanopoulos07b} Ching-Ming Chen, Tianjun Li, [[Van Eric Mayes]], [[Dimitri Nanopoulos]], _Realistic Yukawa Textures and SUSY Spectra from Intersecting Branes_, Phys. Rev. D77:125023, 2008 ([arXiv:0711.0396](https://arxiv.org/abs/0711.0396))
* {#Mayes19} [[Van Eric Mayes]], _All Fermion Masses and Mixings in an Intersecting D-brane World_ ([arXiv:1902.00983](https://arxiv.org/abs/1902.00983))
* {#GemillHowingtonMayes19} Jordan Gemmill, Evan Howington, Van E. Mayes, _One String to Rule Them All: Neutrino Masses and Mixing Angles_ ([arXiv:1907.07106](https://arxiv.org/abs/1907.07106))
* Tianjun Li, Adeel Mansha, Rui Sun, _Revisiting the Supersymmetric Pati-Salam Models from Intersecting D6-branes_ ([arXiv:1910.04530](https://arxiv.org/abs/1910.04530))
[[!redirects Yukawa couplings]]
[[!redirects Yukawa coupling constant]]
[[!redirects Yukawa coupling constants]]
[[!redirects Yukawa interaction]]
[[!redirects Yukawa interactions]] |
Yuki Maehara | https://ncatlab.org/nlab/source/Yuki+Maehara |
* [website](https://yukimaehara.github.io/)
## Publications
* [[Yuki Maehara]], *Inner horns for 2-quasi-categories*, Advances in Mathematics, Volume 363, 25 March 2020, 107003 ([doi:10.1016/j.aim.2020.107003](https://doi.org/10.1016/j.aim.2020.107003)), [arXiv:1902.08720](https://arxiv.org/abs/1902.08720))
## Talks
* [[Yuki Maehara]], *A cubical model for weak omega-categories*, [[Homotopy Type Theory Electronic Seminar Talks]], 8 October 2020 ([video](https://www.youtube.com/watch?v=AIBTh7MIbew), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Maehara-2020-10-08-HoTTEST.pdf))
category: people |
Yuli Rudyak | https://ncatlab.org/nlab/source/Yuli+Rudyak |
* [webpage](http://www.math.ufl.edu/~rudyak/)
category: people
|
Yunlong Jiao | https://ncatlab.org/nlab/source/Yunlong+Jiao |
* [webpage](https://yunlongjiao.github.io/)
## Selected writings
Proof that the [[Mallows kernel]] and [[Kendall kernel]] are [[positive definite bilinear form|positive definite]]:
* [[Yunlong Jiao]], [[Jean-Philippe Vert]], *The Kendall and Mallows Kernels for Permutations*, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 40, no. 7, pp. 1755-1769, 1 July 2018 ([doi:10.1109/TPAMI.2017.2719680](https://doi.org/10.1109/TPAMI.2017.2719680), [hal:01279273](https://hal.archives-ouvertes.fr/hal-01279273/document))
Discussion of weighted variants:
* [[Yunlong Jiao]], [[Jean-Philippe Vert]], *The Weighted Kendall and High-order Kernels for Permutations*, Proceedings of the 35th International Conference on Machine Learning, PMLR 80:2314-2322, 2018 ([arXiv:1802.08526](https://arxiv.org/abs/1802.08526), [mlr:v80/jiao18a](http://proceedings.mlr.press/v80/jiao18a.html), [hal:hal-01717385](https://hal.inria.fr/hal-01717385))
category: people |
Yunze Lu | https://ncatlab.org/nlab/source/Yunze+Lu |
* [Institute page](https://lsa.umich.edu/math/people/phd-students/yunze.html)
## Selected writings
Discussion of [[equivariant ordinary cohomology]] ([[Bredon cohomology]]) over the point but in arbitrary [[RO(G)-degree]]:
for [[equivariance group]] a [[dihedral group]] of [[order of a group|order]] $2p$:
* [[Igor Kriz]], [[Yunze Lu]], *On the $RO(G)$-graded coefficients of dihedral equivariant cohomology*, Mathematical Research Letters **27** 4 (2020) ([arXiv:2005.01225](https://arxiv.org/abs/2005.01225), [doi:10.4310/MRL.2020.v27.n4.a7](https://dx.doi.org/10.4310/MRL.2020.v27.n4.a7))
for [[equivariance group]] the [[quaternion group]]:
* [[Yunze Lu]], *On the $RO(G)$-graded coefficients of $Q_8$ equivariant cohomology*, Topology and its Applications, 2021 ([doi:10.1016/j.topol.2021.107921](https://doi.org/10.1016/j.topol.2021.107921))
category: people
|
Yuri Berest | https://ncatlab.org/nlab/source/Yuri+Berest | Yuri Berest is a professor at Cornell, mathematics department: [web](http://www.math.cornell.edu/People/Faculty/berest.html)
> My research interests include mathematical physics, algebraic geometry and representation theory. I am particularly interested in various interactions between these fields. Some of my recent work is related to noncommutative geometry, integrable systems, differential operators on algebraic varieties, representation theory of Cherednik algebras, and the theory of invariants of finite reflection groups.
* Yuri Berest, Oleg Chalykh, _$A_\infty$-modules and Calogero-Moser spaces_,
J. Reine Angew. Math. __607__ (2007), 69–112, [MR2009f:16019](http://www.ams.org/mathscinet-getitem?mr=2338121), [doi](http://dx.doi.org/10.1515/CRELLE.2007.046)
* Yu. Berest, P. Etingof, V. Ginzburg, _Finite-dimensional representations of rational Cherednik algebras_, Int. Math. Res. Not. 2003, no. 19, 1053-1088.
* Yu. Berest, G. Wilson, _Automorphisms and ideals of the Weyl algebra_, Math Ann 318, 127–147 (2000) [doi](https://doi.org/10.1007/s002080000115)
* Yu. Berest, G. Wilson, _Ideal classes of the Weyl algebra and noncommutative projective geometry_ (with an Appendix by M. Van den Bergh), Internat. Math. Res. Notices __26__ (2002) 1347–1396
* Yu. Berest, G. Wilson, _Mad subalgebras of rings of differential operators on curves_, Advances in Math. __212__ no. 1 (2007), 163–190.
* Yuri Berest, _Calogero-Moser spaces over algebraic curves_, Sel. math., New ser. 14, 373–396 (2009) [doi](https://doi.org/10.1007/s00029-009-0518-9) [arXiv:0809.4521](https://arxiv.org/abs/0809.4521)
* Yuri Berest, _The problem of [[lacuna]]s and analysis on root systems_, Trans. Amer. Math. Soc. __352__ (2000), 3743–3776.
* Yuri Berest, _The theory of lacunas and quantum integrable systems_, Calogero-Moser-Sutherland models (Montréal, QC, 1997), 53–64, CRM Ser. Math. Phys., Springer 2000.
* [[Yuri Berest]], Oleg Chalykh, Farkhod Eshmatov, _Recollement of deformed preprojective algebras and the Calogero-Moser correspondence_, Mosc. Math. J. __8__:1 (2008) 21--37
[arXiv:0706.3006](https://arxiv.org/abs/0706.3006) [doi](https://doi.org/10.17323/1609-4514-2008-8-1-21-37) [MR2422265](https://www.ams.org/mathscinet-getitem?mr=2422265)
mathnet.ru/[mmj2](http://mi.mathnet.ru/mmj2)
* Yu. Berest, X. Chen, F. Eshmatov, A. Ramadoss, _Noncommutative Poisson structures, derived representation schemes and Calabi-Yau algebras_, Contemp. Math. __583__ (2012) 219--246 [arXiv:1202.2717](https://arxiv.org/pdf/1202.2717)
* Yu. Berest, [[G. Felder]], A. Ramadoss, _Derived representation schemes and noncommutative geometry_, [arXiv:1304.5314](https://arxiv.org/abs/1304.5314)
* George Khachatryan, _Derived representation schemes and non-commutative geometry_, Cornell PhD thesis under guidance of Yuri Berest [online](https://hdl.handle.net/1813/29138]
* Yuri Berest, George Khachatryan, Ajay Ramadoss, _Derived representation schemes and cyclic homology_, Adv. Math.
__245,__ (2013) 625--689 [arXiv:1112.1449](https://arxiv.org/abs/1112.1449)
* [[Yuri Berest]], [[Giovanni Felder]], Sasha Patotski, Ajay C. Ramadoss, [[Thomas Willwacher]], _Representation homology, Lie algebra cohomology and derived Harish-Chandra homomorphism_, J. Eur. Math. Soc. 19:9 (2017) 2811--2893 [arXiv:1410.0043](https://arxiv.org/abs/1410.0043)
category: people
[[!redirects Yu. Berest]] |
Yuri Golfand | https://ncatlab.org/nlab/source/Yuri+Golfand |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Yuri_Golfand)
See the introduction of
* [[Mikhail Shifman]] (ed.)
_[[The Many Faces of the Superworld]]_
[[Yuri Golfand]] memorial volume
World Scientific, 2000
[doi:10.1142/4332](https://doi.org/10.1142/4332)
## Selected writings
Introducing the [[super Poincaré Lie algebra]] ("[[supersymmetry]]"):
* {#GolfandLikhtman72} [[Yuri Golfand]], [[Evgeny Likhtman]],_On the Extensions of the Algebra of the Generators of the Poincaré Group by the Bispinor Generators_, in: [[Victor Ginzburg]] et al. (eds.) _I. E. Tamm Memorial Volume Problems of Theoretical Physics_, (Nauka, Moscow 1972), page 37,
translated and reprinted in: [[Mikhail Shifman]] (ed.) _[[The Many Faces of the Superworld]]_ pp. 44-53, World Scientific (2000) ([doi:10.1142/9789812793850_0006](https://doi.org/10.1142/9789812793850_0006))
category: people |
Yuri Gurevich | https://ncatlab.org/nlab/source/Yuri+Gurevich |
* [personal page](https://web.eecs.umich.edu/~gurevich/)
* [Wikipedia entry](https://en.wikipedia.org/wiki/Yuri_Gurevich)
## Selected writings
On [[set theory]] with commentary on [[ETCC]]:
* [[Andreas Blass]], [[Yuri Gurevich]], *Why Sets?*, Bull. Europ. Assoc. Theoret. Comp. Sci. **84** (2004) 139-156. [[doi:10.1007/978-3-540-78127-1_11](https://doi.org/10.1007/978-3-540-78127-1_11), [pdf](https://web.eecs.umich.edu/~gurevich/Opera/172.pdf), [pdf](http://www.math.lsa.umich.edu/~ablass/set.pdf)]
On formal statement and [[proof]] of the [[deferred measurement principle]] of [[quantum circuits]]:
* {#GurevichBlass21} [[Yuri Gurevich]], [[Andreas Blass]], *Quantum circuits with classical channels and the principle of deferred measurements*, Theoretical Computer Science **920** (2022) 21–32 [[arXiv:2107.08324](https://arxiv.org/abs/2107.08324), [doi:10.1016/j.tcs.2022.02.002](https://doi.org/10.1016/j.tcs.2022.02.002)]
On a formal [[quantum programming language]]-perspective on [[quantum circuits]]:
* [[Yuri Gurevich]], [[Andreas Blass]], *Software science view on quantum circuit algorithms* [[arXiv:2209.13731](https://arxiv.org/abs/2209.13731)]
category: people
|
Yuri Makeenko | https://ncatlab.org/nlab/source/Yuri+Makeenko |
* [InSpire page](https://inspirehep.net/authors/999145)
## Selected writings:
Early discussion of [[flux tubes]]/[[Wilson lines]] as effective [[strings]] in [[Yang-Mills theory]] ([Gauge/String duality](AdS-CFT+correspondence#PolyakovGaugeStringDualityReferences)):
* {#MakeenkoMigdal81} [[Yuri Makeenko]], [[Alexander A. Migdal]], *Quantum chromodynamics as dynamics of loops*, Nuclear Physics B **188** 2 (1981) 269-316 [<a href="https://doi.org/10.1016/0550-3213(81)90258-3">doi:10.1016/0550-3213(81)90258-3</a>]
> "So the [[world sheet]] of [[string]] should be interpreted as the color magnetic dipole sheet. The string itself should be interpreted as the electric [[flux tube]] in the [[monopole]] plasma."
On [[gauge theory]] ([[Yang-Mills theory]]):
* [[Yuri Makeenko]], *Methods of contemporary gauge theory*, Cambridge Monographs on Math. Physics, Cambridge University Press (2002) [[doi:10.1017/CBO9780511535147]( https://doi.org/10.1017/CBO9780511535147), [gBooks](http://books.google.com/books?id=9W-W2w75ulAC)]
category: people
[[!redirects Yuri M. Makeenko]]
|
Yuri Manin | https://ncatlab.org/nlab/source/Yuri+Manin | Yuri Ivanovich Manin (1937-2023, Russian: Юрий Иванович Манин) was a Russian-born mathematician of polymath broadness, with main works in [[number theory]] and [[arithmetic geometry]], [[noncommutative geometry]], [[algebraic geometry]] and [[mathematical physics]].
* [Wikipedia entry](http://en.wikipedia.org/wiki/Yuri_I._Manin)
* [MPI obituary](https://www.mpim-bonn.mpg.de/node/11864)
His diverse work includes a classification theorem in the theory of commutative [[formal group]], early study of monoidal transformations and exposition on [[motive]]s in 1960-s, a fundamental starting work in [[quantum information theory]], proposals on [[quantum logic]]s, an approach to [[quantum group]]s, ADHM construction in [[soliton]] theory, work with [[Maxim Kontsevich]] on [[Gromov-Witten invariants]], work on [[Frobenius manifold]]s (and introduced more general "F-manifolds" with Claus Hertling). He published a number of influential monographs including on [[noncommutative geometry]], [[quantum group]]s, [[complex geometry]] and [[gauge theory|gauge theories]], introduction to [[scheme]]s, [[Frobenius manifold]]s, mathematical [[logic]]s...
Manin's students include:
* [[A. Beilinson]],
* [[V. Drinfel'd]]
* Vera Serganova
* Ivan Penkov
* [[M. Kapranov]]
* [[Yuri Tschinkel]]
* [[Ralph Kaufmann]]
* ...
## Selected writings
* _[[New Dimensions in Geometry]]_
* _[[Gauge Field Theory and Complex Geometry]]_
Introducing what came to be called the *[[Gauss-Manin connection]]*:
* {#Manin58} [[Yuri Manin]], _Algebraic curves over fields with differentiation_, Izv. Akad. Nauk SSSR Ser. Mat. **22** 6 (1958) 737-756 $[$[mathnet:izv3998](http://mi.mathnet.ru/izv3998), [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=im&paperid=3998&volume=22&year=1958&issue=6&fpage=737&what=fullt&option_lang=eng)$]$ (in Russian)
Introducing the notion of [[quantum computation]]:
* {#Manin1980} [[Yuri I. Manin]], Introduction to: *Computable and Uncomputable*, Sov. Radio (1980) [Russian original: [[Manin-1980.pdf:file]]], English translation on p. 69-77 of *Mathematics as Metaphor: Selected essays of Yuri I. Manin*, Collected Works **20**, AMS (2007) [[ISBN:978-0-8218-4331-4](https://bookstore.ams.org/cworks-20/)]
> Perhaps, for a better understanding of [molecular biology], we need a mathematical theory of quantum automata.
and review of [[Shor's algorithm]]:
* [[Yuri I. Manin]], *Classical computing, quantum computing, and Shor's factoring algorithm*, Astérisque, **266** Séminaire Bourbaki 862 (2000) 375-404 [[arXiv:quant-ph/9903008](https://arxiv.org/abs/quant-ph/9903008), [numdam:SB_1998-1999__41__375_0](http://www.numdam.org/item/?id=SB_1998-1999__41__375_0)]
Introducing the [[ADHM construction]] for [[Yang-Mills instantons]]:
* [[Michael Atiyah]], [[Nigel Hitchin]], [[Vladimir Drinfeld]], [[Yuri Manin]], *Construction of instantons*, Physics Letters A **65** 3 (1978) 185-187 [<a href="https://doi.org/10.1016/0375-9601(78)90141-X">doi:10.1016/0375-9601(78)90141-X</a>]
On [[homological algebra]] and [[homotopical algebra]] (via [[triangulated categories]] and including the [[model structure on dgc-algebras]] for [[rational homotopy theory]]):
* [[Sergei Gelfand]], [[Yuri Manin]], _[[Methods of homological algebra]]_, transl. from the 1988 Russian (Nauka Publ.) original, Springer 1996. xviii+372 pp. 2nd corrected ed. 2002 ([doi:10.1007/978-3-662-12492-5](https://doi.org/10.1007/978-3-662-12492-5))
Introduced [[quantum linear group]]s as universal coacting bialgebras (and their quotient Hopf algebras) in
* [[Yu. I. Manin]], _Quantum groups and non-commutative geometry_, CRM, Montreal 1988.
Some of these structures have repercussion on the study of [[quadratic operad]]s, as in
* [[Yuri Ivanovich Manin]], [[Bruno Vallette]], _Monoidal structures on the categories of quadratic data_, Documenta Mathematica __25__, 1727--1786
On relations of [[AdS3/CFT2]] to [[hyperbolic geometry]] and [[Arakelov geometry]] of [[algebraic curves]]:
* [[Yuri Manin]], [[Matilde Marcolli]], _Holography principle and arithmetic of algebraic curves_, Adv. Theor. Math. Phys. 5 (2002) 617-650 ([arXiv:hep-th/0201036](https://arxiv.org/abs/hep-th/0201036)
On [[quantum cohomology]] and [[Gromov-Witten invariants]]
* [[Maxim Kontsevich]], [[Yuri Manin]], _Gromov-Witten classes, quantum cohomology, and enumerative geometry_, Commun. Math. Physics __164__ (1994) 525-562 [doi](https://doi.org/10.1007/BF02101490) arXiv:[hep-th/9402147](https://arxiv.org/pdf/hep-th/9402147)
* [[Yuri Manin]], _Frobenius manifolds, quantum cohomology, and moduli spaces_, Amer. Math. Soc. Colloqium Publications __47__, 1999
* [[Maxim Kontsevich]], [[Yuri Manin]], [[Ralph Kaufmann]], _Quantum cohomology of a product_, Invent. Math. __124__ (1996) 313-339 [doi](https://doi.org/10.1007/s002220050055) arXiv:[q-alg/9502009](https://arxiv.org/abs/q-alg/9502009)
## Quotes
> _What binds us to space-time is our rest mass, which prevents us from flying at the speed of light, when time stops and space loses meaning. In a world of light there are neither points nor moments of time; beings woven from light would live "nowhere" and "nowhen"; only poetry and mathematics are capable of speaking meaningfully about such things_
_Mathematics as Metaphor_: Selected Essays of Yuri I. Manin (ed. 2007) ([libquotes](https://libquotes.com/yuri-manin/quote/lbj0h0e))
category: people
[[!redirects Yuri I. Manin]]
[[!redirects Yuri Ivanovich Manin]]
[[!redirects Yu. Manin]]
[[!redirects Yu. I. Manin]] |
Yuri Matiyasevich | https://ncatlab.org/nlab/source/Yuri+Matiyasevich |
__Yuri Vladimirovič Matiyasevich__ (also spelled Jurij/Yurij/Yurii Matijasevič/Matijasevich/Matiasevich, Russian Ю́рий Влади́мирович Матиясе́вич) is a Russian
mathematician who obtained a negative solution to [[Hilbert's problems|Hilbert's tenth problem]].
By a seminal theorem of Matiyasevich, for every statement in mathematics (say [[ZFC|ZF set theory]]) there is a [[Diophantine equation]] whose solvability is equivalent to the validity of the statement. This is not of much practical importance, but of large theoretical importance.
* wikipedia [Yuri Matiyasevich](http://en.wikipedia.org/wiki/Yuri_Matiyasevich), [Russian version](http://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B8%D1%8F%D1%81%D0%B5%D0%B2%D0%B8%D1%87,_%D0%AE%D1%80%D0%B8%D0%B9_%D0%92%D0%BB%D0%B0%D0%B4%D0%B8%D0%BC%D0%B8%D1%80%D0%BE%D0%B2%D0%B8%D1%87)
An accessible presentation of Matiyasevich's theorem is in
* [[Yuri Manin]], _Introduction to mathematical logic_, Springer
[[!redirects Matiyasevich]]
[[!redirects Yu. Matiyasevich]]
[[!redirects Yuri Matiyasevich]]
[[!redirects Yuri Vladimirovič Matiyasevich]]
[[!redirects Ю́рий Влади́мирович Матиясе́вич]]
[[!redirects Юрий Владимирович Матиясевич]]
|
Yuri N. Obukhov | https://ncatlab.org/nlab/source/Yuri+N.+Obukhov |
* [InSpire page](https://inspirehep.net/authors/995226)
* [GoogleScholar page](https://scholar.google.com/citations?user=8wD8DMoAAAAJ&hl=en)
## Selected writings
On [[pre-metric electromagnetism]]:
* [[Friedrich W. Hehl]], [[Yuri N. Obukhov]], *Foundations of Classical Electrodynamics -- Charge, Flux, and Metric*, Progress in Mathematical Physics **33**, Springer (2003) [[doi:10.1007/978-1-4612-0051-2](https://doi.org/10.1007/978-1-4612-0051-2)]
* {#HehlItinObukhov16} [[Friedrich W. Hehl]], [[Yakov Itin]], [[Yuri N. Obukhov]], *On Kottler's path: origin and evolution of the premetric program in gravity and in electrodynamics*, International Journal of Modern Physics D **25** 11 (2016) 1640016 [[arXiv:1607.06159](https://arxiv.org/abs/1607.06159), [doi:10.1142/S0218271816400162](https://doi.org/10.1142/S0218271816400162)]
* [[Yuri N. Obukhov]], *Premetric approach in gravity and electrodynamics*, in: *The Fifteenth Marcel Grossmann Meeting* (2022) 654-659 [[doi:10.1142/9789811258251_0087](https://doi.org/10.1142/9789811258251_0087), [arXiv:1904.00180](https://arxiv.org/abs/1904.00180)]
## Related entries
* [[electromagnetism]]
[[!redirects Yuri Obukhov ]]
|
Yuri Tschinkel | https://ncatlab.org/nlab/source/Yuri+Tschinkel |
* [webpage](http://www.cims.nyu.edu/~tschinke/)
## Selected writings
On [[elliptic fibrations]]:
* [[Fedor Bogomolov]], [[Yuri Tschinkel]], *Monodromy of elliptic surfaces* [[[MonodromyOfEllipticSurfaces.pdf:file]], [arXiv:math/0002168](https://arxiv.org/abs/math/0002168)]
On [[anabelian geometry]]:
* [[Fedor Bogomolov]], [[Yuri Tschinkel]], *Introduction to birational anabelian geometry* [[pdf](http://www.math.nyu.edu/~tschinke/papers/yuri/10msri/msri7.pdf), [arXiv:1011.0883](https://arxiv.org/abs/1011.0883)]
* {#Tschinkel14} [[Yuri Tschinkel]], _Introduction to anabelian geometry_, talk at _[Symmetries and correspondences in number theory, geometry, algebra, physics: intra-disciplinary trends](https://www.maths.nottingham.ac.uk/personal/ibf/files/sc3.html)_, Oxford 2014 ([slides pdf](http://www.cims.nyu.edu/~tschinke/.talks/oxford14/oxford14.pdf))
## Related entries
* [[anabelian geometry]]
category: people
|
Yuri Ximenes Martins | https://ncatlab.org/nlab/source/Yuri+Ximenes+Martins | I am a Brazilian researcher and software developer working on abstraction processes applied to Mathematics, Physics and Philosophy. As developer I like to follow the functional paradigm.
- [webpage](https://yxm.me)
- [arXiv](https://arxiv.org/a/martins_y_1.html), [ORCID](https://orcid.org/0000-0003-3717-6774), [HAL](https://hal.archives-ouvertes.fr/search/index/?q=yuri-ximenes-martins&submit=)
- [github](https://github.com/yxm-dev)
I am co-founder of the [[Math-Phys-Cat Group]], creator and contributor of the [Math-Phys-Cat Wiki](https://wiki.math-phys.group/).
- Contact: `yxm@gmx.ie`.
category: people
[[!redirects Yuri Martins]] |
Yurii Antonovich Simonov | https://ncatlab.org/nlab/source/Yurii+Antonovich+Simonov |
* [spire entry](https://inspirehep.net/authors/988597)
* [Math-Net page](http://www.mathnet.ru/eng/person24572)
## Selected writings
On [[confinement]]:
* {#Simonov18} [[Yuri A. Simonov]], _Field Correlator Method for the confinement in QCD_, Phys. Rev. D 99, 056012 (2019) ([arXiv:1804.08946](https://arxiv.org/abs/1804.08946), [doi:10.1103/PhysRevD.99.056012](https://doi.org/10.1103/PhysRevD.99.056012))
* [[Yuri A. Simonov]], *The fundamental scale of QCD* ([arXiv:2103.08223](https://arxiv.org/abs/2103.08223))
category: people
[[!redirects Yuri A. Simonov]]
[[!redirects Yuri Simonov]] |
Yuta Kusakabe | https://ncatlab.org/nlab/source/Yuta+Kusakabe |
* [webpage](https://kusakabe.github.io)
## Selected writings
On [[Oka manifolds]]:
* {#Kusakabe20} [[Yuta Kusakabe]], _Oka properties of complements of holomorphically convex sets_ ([arXiv:2005.08247](https://arxiv.org/abs/2005.08247)).
category: people
|
Yutaka Hosotani | https://ncatlab.org/nlab/source/Yutaka+Hosotani |
* [webpage](http://kabuto.phys.sci.osaka-u.ac.jp/~hosotani/hosotani-eng.html)
## Selected writings
On [[gauge-Higgs unification]] ([[Hosotani mechanism]]):
* {#Hosotani83} [[Yutaka Hosotani]], _Dynamical Mass Generation by Compact Extra Dimensions_, Phys. Lett. 126B (1983) 309-313 ([spire:188768](http://inspirehep.net/record/188768), <a href="https://doi.org/10.1016/0370-2693(83)90170-3">doi:10.1016/0370-2693(83)90170-3</a>)
* {#Hosotani89} [[Yutaka Hosotani]], _Dynamics of non-integrable phases and gauge symmetry breaking_, Annals of Physics Volume 190, Issue 2, March 1989, Pages 233-253 (<a href="https://doi.org/10.1016/0003-4916(89)90015-8">doi:10.1016/0003-4916(89)90015-8</a>)
* {#Hosotani12} [[Yutaka Hosotani]], _Gauge-Higgs Unification Approach_, AIP Conference Proceedings 1467, 208 (2012) ([arXiv:1206.0552](https://arxiv.org/abs/1206.0552))
Specifically in [[Spin(11)]]- ("[[SO(11)]]"-) [[GUT]]-models:
* {#HosotaniYamatsu15} [[Yutaka Hosotani]], Naoki Yamatsu, _Gauge–Higgs grand unification_, Progress of Theoretical and Experimental Physics, Volume 2015, Issue 11, November 2015 ([arXiv:1504.03817](https://arxiv.org/abs/1504.03817), [doi:10.1093/ptep/ptw116](https://doi.org/10.1093/ptep/ptw116))
* {#FuruiHosotaniYamatsu16} Atsushi Furui, [[Yutaka Hosotani]], Naoki Yamatsu, _Toward Realistic Gauge-Higgs Grand Unification_, Progress of Theoretical and Experimental Physics, Volume 2016, Issue 9, September 2016, 093B01 ([arXiv:1606.07222](https://arxiv.org/abs/1606.07222))
* {#Hosotoni16} [[Yutaka Hosotani]], _Gauge-Higgs EW and Grand Unification_, International Journal of Modern Physics AVol. 31, No. 20n21, 1630031 (2016) ([arXiv:1606.08108](https://arxiv.org/abs/1606.08108))
* {#Hosotani17} [[Yutaka Hosotani]], _New dimensions from gauge-Higgs unification_ ([arXiv:1702.08161](https://arxiv.org/abs/1702.08161))
* {#HosotaniYamatsu17} [[Yutaka Hosotani]], Naoki Yamatsu, _Electroweak Symmetry Breaking and Mass Spectra in Six-Dimensional Gauge-Higgs Grand Unification_ ([arXiv:1710.04811](https://arxiv.org/abs/1710.04811))
category: people
|
Yuval Grossman | https://ncatlab.org/nlab/source/Yuval+Grossman |
* [webpage](https://physics.cornell.edu/yuval-grossman)
## Selected writings
On [[flavor physics]]:
* [[Yuval Grossman]], _Introduction to flavor physics_, CERN Yellow Report CERN-2010-002, pp. 111-144 ([arXiv:1006.3534](https://arxiv.org/abs/1006.3534))
* [[Yuval Grossman]], Philip Tanedo, _Just a Taste: Lectures on Flavor Physics_, Chapter 4 in: _Anticipating the Next Discoveries in Particle Physics (TASI 2016)_ ([arXiv:1711.03624](https://arxiv.org/abs/1711.03624), [doi:10.1142/9789813233348_0004](https://doi.org/10.1142/9789813233348_0004))
category: people |
Yuval Ne'eman | https://ncatlab.org/nlab/source/Yuval+Ne%27eman |
Yuval Ne'eman (1925-2006)
* [Wikipedia entry](https://en.wikipedia.org/wiki/Yuval_Ne%27eman)
* [Obituary](https://www.science.co.il/people/Yuval-Neeman/) at *Israel Science and Technology Directory*
## Selected writings
Precursor discussion to the [[D'Auria-Fré formulation of supergravity]]:
* [[Yuval Ne'eman]], [[Tullio Regge]], *Gravity and supergravity as gauge theories on a group manifold*, Physics Letters B **74** 1–2 (1978) 54-56 [<a href="https://doi.org/10.1016/0370-2693(78)90058-8">doi:10.1016/0370-2693(78)90058-8</a>, [spire:6328](https://inspirehep.net/literature/6328)]
also: Rivista del Nuovo Cimento **1** 5 (1978) 1–43
## Related entries
* [[hadron]]$\;$[[flavour symmetry]]
category: people
|
Yuval Roichman | https://ncatlab.org/nlab/source/Yuval+Roichman |
* [personal page](https://u.cs.biu.ac.il/~yuvalr/)
## Selected writings
On the [[combinatorics]] of [[standard Young tableaux]]:
* [[Ron M. Adin]], [[Yuval Roichman]], *Enumeration of Standard Young Tableaux*, Chapter 14 in: Miklós Bóna, *Handbook of Enumerative Combinatorics*, CRC Press 2015 ([arXiv:1408.4497](https://arxiv.org/abs/1408.4497), [ISBN:9781482220858](https://www.routledge.com/Handbook-of-Enumerative-Combinatorics/Bona/p/book/9781482220858))
category: people
|
Yuxi Liu | https://ncatlab.org/nlab/source/Yuxi+Liu | I'm a mathematician trying to make the world more harmonious, friendly, and utopic. I aim to do so by pushing forwards artificial intelligence using formal mathematics.
I am the bearer of the Element of Silence. |
Yves Andre | https://ncatlab.org/nlab/source/Yves+Andre | **Yves André** is a mathematician at École Normale Supérieure of Paris.
* [editor's page](http://versita.com/science/mathematics/cejm/editors/yves_andre) at Central European Math. J., [editor's page](http://rendiconti.math.unipd.it/board/andre.php?lan=english) at Rendiconti del Seminario Matematico della Università di Padova
Related items: [[slope filtration]].
[[!redirects Yves André]] |
Yves Diers | https://ncatlab.org/nlab/source/Yves+Diers |
* [Prabook entry](https://prabook.com/web/yves.diers/290800)
## Selected writings
On the notion of [[spectrum (geometry)|spectrum in geometry]] and introducing the notion of [[multi-adjoints]]:
* [[Yves Diers]], _Une construction universelle des spectres, topologies spectrales et faisceaux structuraux_, Communication in Algebra Volume 12, Issue 17-18 (1984) ([doi:10.1080/00927878408823101](https://doi.org/10.1080/00927878408823101))
See also:
* [[Yves Diers]], _Catégories localisables_, PhD thesis. Paris 6 et Centre universitaire de Valenciennes et du Hainaut Cambrésis (1977) [[[Categories localisables.pdf:file]]]
* [[Yves Diers]], _Catégories localement multiprésentables_, Archiv der Mathematik 34.1 (1980), pp. 344–356.
* Yves Diers, _Categories of Boolean Sheaves of Simple Algebras_, Springer, Lecture Notes in Mathematics, vol. 1187, (1986).
* Yves Diers, _Familles universelles de morphismes_, Ann. Soc. Sci. Bruxelles, 93 (1979), 175-195.
* Yves Diers, _Multimonads and multimonadic categories_, J. Pure Appl. Algebra 17 (1980), 153-170.
category: people
|
Yves Félix | https://ncatlab.org/nlab/source/Yves+F%C3%A9lix | [[!redirects Yves Felix]]
* [webpage](http://uclouvain.be/yves.felix)
## Selected writings
On [[rational homotopy theory]]:
* {#FelixHalperinThomas00} [[Yves Félix]], [[Stephen Halperin]], [[Jean-Claude Thomas]], _Rational Homotopy Theory_, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000 ([doi:10.1007/978-1-4613-0105-9](https://link.springer.com/book/10.1007/978-1-4613-0105-9))
* [[Yves Félix]], [[John Oprea]], [[Daniel Tanré]], _Algebraic models in geometry_, Oxford University Press 2008 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/tanre.pdf), [ISBN:9780199206520](https://global.oup.com/academic/product/algebraic-models-in-geometry-9780199206520))
* {#FelixHalperin17} [[Yves Félix]], [[Steve Halperin]], _Rational homotopy theory via Sullivan models: a survey_, Notices of the International Congress of Chinese Mathematicians Volume 5 (2017) Number 2 ([arXiv:1708.05245](https://arxiv.org/abs/1708.05245), [doi:10.4310/ICCM.2017.v5.n2.a3](https://dx.doi.org/10.4310/ICCM.2017.v5.n2.a3))
On [[rational homotopy theory]] with general [[fundamental groups]]:
* [[Urtzi Buijs]], [[Yves Félix]], [[Aniceto Murillo]], [[Daniel Tanré]], _Homotopy theory of complete Lie algebras and Lie models of simplicial sets_, Journal of Topology (2018) 799-825 ([arXiv:1601.05331](https://arxiv.org/abs/1601.05331), [doi:10.1112/topo.12073](https://doi.org/10.1112/topo.12073))
* {#FelixHalperinThomas15} [[Yves Félix]], [[Steve Halperin]], [[Jean-Claude Thomas]], _Rational Homotopy Theory II_, World Scientific 2015 ([doi:10.1142/9473](https://doi.org/10.1142/9473))
On rational models for [[spherical fibrations]] and [[Thom spaces]]:
* {#FelixOpreaTanre16} [[Yves Félix]], John Oprea, [[Daniel Tanré]], _Lie-model for Thom spaces of tangent bundles_, Proc. Amer. Math. Soc. 144 (2016), 1829-1840 ([pdf](http://www.ams.org/journals/proc/2016-144-04/S0002-9939-2015-12829-8/S0002-9939-2015-12829-8.pdf), [doi:10.1090/proc/12829](https://doi.org/10.1090/proc/12829))
On [[rational parametrized stable homotopy theory]]:
* {#FelixMurilloTanre10} [[Yves Félix]], Aniceto Murillo [[Daniel Tanré]], _Fibrewise stable rational homotopy_, Journal of Topology, Volume 3, Issue 4, 2010, Pages 743–758 ([doi:10.1112/jtopol/jtq023](https://doi.org/10.1112/jtopol/jtq023))
On [[rational models of mapping spaces]]:
* {#BuijsFelixMurillo12} [[Urtzi Buijs]], [[Yves Félix]], [[Aniceto Murillo]], _$L_\infty$-rational homotopy of mapping spaces_, published as _$L_\infty$-models of based mapping spaces_, J. Math. Soc. Japan Volume 63, Number 2 (2011), 503-524 ([arXiv:1209.4756](https://arxiv.org/abs/1209.4756), [doi:10.2969/jmsj/06320503](https://doi.org/10.2969/jmsj/06320503))
On the [[ordinary cohomology]] of [[configuration spaces of points]]:
* [[Yves Félix]], [[Daniel Tanré]], _The cohomology algebra of unordered configuration spaces_, Journal of the LMS, Vol 72, Issue 2 ([arxiv:math/0311323](https://arxiv.org/abs/math/0311323), [doi:10.1112/S0024610705006794](https://doi.org/10.1112/S0024610705006794))
On the [[cobar construction]]:
* [[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))
## Related $n$Lab entries
* [[rational homotopy theory]]
* [[Sullivan model of free loop space]]
category: people |
Yves Guiraud | https://ncatlab.org/nlab/source/Yves+Guiraud | Yves Guiraud is a French mathematician and theoretical computer scientist. He is Chargé de recherche INRIA attached to [INRIA Rocquencourt](http://www.inria.fr/centre/paris-rocquencourt), but based in Irif [[PPS]], at Paris 7. His main research is in rewriting systems, especially in higher dimensions.
His webpage is [here](http://www.pps.univ-paris-diderot.fr/~guiraud/).
The list of his preprints on the Arxiv is [here](http://arxiv.org/find/math/1/au:+Guiraud_Y/0/1/0/all/0/1)
His thesis (2004) was
* [Présentations d'opérades et systèmes de réécriture](http://tel.archives-ouvertes.fr/docs/00/04/70/96/PDF/tel-00006863.pdf)
####Thesis abstract
_This thesis studies the computational properties of operad presentations, or Penrose diagrams rewrite systems, together with their links with classical types of rewrite systems. With new criteria for termination and confluence, the convergence of the presentation $L(\mathbb{Z}_2)$ of $\mathbb{Z}/2\mathbb{Z}$-vector spaces, a commutative equational theory, is proved. Operad presentations are shown to be generalizations of both word rewrite systems and Petri nets; furthermore, they provide explicit resource management calculi for left-linear term rewrite systems. This work is concluded by the description of obstructions for proving the same result for the lambda-calculus. Two appendices present the links between operads and other structures from universal algebra, together with a calculus of explicit substitutions._
category : people |
Yves Lafont | https://ncatlab.org/nlab/source/Yves+Lafont | Yves Lafont is a Professor at the _Université de la Méditerranée (Aix-Marseille 2)_ and a researcher at the _Institut de Mathématiques de Luminy_. He is well known for work in Linear Logic, and more recently on polygraphic resolutions.
* [website](https://www.i2m.univ-amu.fr/perso/yves.lafont/)
## Selected writings
* Doctoral thesis: [Logiques, catégories & machines : implantation de langages de programmation guidée par la logique catégorique](https://www-apr.lip6.fr/~tasson/doc/reading/Yves_Laffont_phd.pdf)
On [[formal logic|form]] [[logic]] and [[formal proof|formal]] [[proof]] [[proof theory|theory]] ([[type theory|typed]] [[lambda-calculus|$\lambda$-calculus]], [[linear logic]], [[coherence spaces]], ...)
* {#Girard89} [[Jean-Yves Girard]] (translated and with appendiced by [[Paul Taylor]] and [[Yves Lafont]]), *Proofs and Types*, Cambridge University Press (1989) [[ISBN:978-0-521-37181-0](), [webpage](http://www.paultaylor.eu/stable/Proofs+Types.html), [pdf](https://www.paultaylor.eu/stable/prot.pdf)]
category: people |
Yves Laszlo | https://ncatlab.org/nlab/source/Yves+Laszlo |
* [webpage](http://www.math.u-psud.fr/~laszlo/)
category: people
|
Yvette Kosmann-Schwarzbach | https://ncatlab.org/nlab/source/Yvette+Kosmann-Schwarzbach |
* [webage](http://www.math.polytechnique.fr/~kosmann/)
category: people |
Yvonne Choquet-Bruhat | https://ncatlab.org/nlab/source/Yvonne+Choquet-Bruhat |
* [wikipedia entry](http://en.wikipedia.org/wiki/Yvonne_Choquet-Bruhat)
## Selected wrirings
On [[analysis]] and [[differential geometry]] on [[smooth manifolds]] with application to [[mathematical physics]]:
* [[Yvonne Choquet-Bruhat]], [[Cécile DeWitt-Morette]], *Analysis, manifolds and physics*, North Holland (1982, 2001) $[$[ISBN:9780444860170](https://www.elsevier.com/books/analysis-manifolds-and-physics-revised-edition/choquet-bruhat/978-0-444-86017-0)$]$
## Related entries
* [[Einstein's equations]]
* [[The Cauchy Problem in Classical Supergravity]]
category: people |
Z | https://ncatlab.org/nlab/source/Z |
The [[integers]] $\mathbb{Z}$.
## Related entries
* [[S]] |
Z'-boson | https://ncatlab.org/nlab/source/Z%27-boson |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[particle physics]] the term _Z'-boson_ refers to a hypothetical higher-[[mass]] cousin of the [[Z-boson]] which is predicted by some [[model (in theoretical physics)|models]] that extend the [[standard model of particle physics]], such as [[grand unified theories]] (e.g. [Sahoo 06](#Sahoo06)).
The existence of the $Z'$ has been proposed as one possible explanation of the apparently observed [[flavour anomalies]] ([Gauld-Goertz-Haisch 13](#GauldGoertzHaisch13), [D'Ambrosio-Iver-Piccinini-Polosa 19](#DAmbrosioIverPiccininiPolosa19))
## References
### General
* Paul Langacker, _The Physics of Heavy $Z'$ Gauge Bosons_, Rev.Mod.Phys.81:1199-1228, 2009 ([arXiv:0801.1345](https://arxiv.org/abs/0801.1345))
* [The Review of Particle Physics 2017](http://pdg.lbl.gov/2017/), _59. $Z'$-boson searches_ ([pdf](http://pdg.lbl.gov/2017/reviews/rpp2017-rev-zprime-searches.pdf))
* [kjende.web.cern.ch/kjende/de/zpath_zprime.htm](https://kjende.web.cern.ch/kjende/de/zpath_zprime.htm)
* _[The Z' Hunter's Guide](https://sites.google.com/site/zprimeguide/)_
Experimental constraints:
* [[CMS collaboration]], _Search for an $L_\mu - L_\tau$ gauge boson using $Z \to 4\mu$ events in proton-proton collisions at $\sqrt{s} = 13TeV$_ ([arXiv:1808.03684](https://arxiv.org/abs/1808.03684))
As arising in [[GUTs]]:
* {#Sahoo06} S. Sahoo, _The prediction of mass of $Z'$-boson in an $SO(10)$-based model_, Indian J. Phys. 80 (2), 191-194 (2006) ([pdf](http://arxiv.iacs.res.in:8080/jspui/bitstream/10821/6019/1/The%20Prediction%20of%20Mass%20of%20Z%27-Boson_By%20S%20Sahoo.pdf))
### In relation to flavour anomalies
As a possible solution to the [[flavour anomalies]]:
* {#GauldGoertzHaisch13} Rhorry Gauld, Florian Goertz, Ulrich Haisch, _An explicit Z'-boson explanation of the $B \to K^\ast \mu^+ \mu^-$ anomaly_, JHEP01(2014)069 ([arXiv:1310.1082](https://arxiv.org/abs/1310.1082))
* Richard H. Benavides, Luis Muñoz, William A. Ponce, Oscar Rodríguez, Eduardo Rojas, _Minimal $Z^\prime$ models for flavor anomalies_ ([arXiv:1812.05077](https://arxiv.org/abs/1812.05077))
* {#DAmbrosioIverPiccininiPolosa19} Giancarlo D'Ambrosio, A. M. Iyer, F. Piccinini, A.D. Polosa, _Confronting $B$ anomalies with atomic physics_ ([arXiv:1902.00893](https://arxiv.org/abs/1902.00893))
* P. Ko, Takaaki Nomura, Chaehyun Yu, _$b \to s \mu^+ \mu^-$ anomalies and related phenomenology in $U(1)_{B_{3-x_\mu L_\mu - x_\tau L_\tau}}$ flavor gauge models_ ([arXiv:1902.06107](https://arxiv.org/abs/1902.06107))
* Joe Davighi, _Connecting neutral current $B$ anomalies with the heaviness of the third family_, Contribution to the [2019 QCD session](http://moriond.in2p3.fr/2019/QCD/) of the [54th Rencontres de Moriond](http://moriond.in2p3.fr/2019/) ([arXiv:1905.06073](https://arxiv.org/abs/1905.06073))
* [[Wolfgang Altmannshofer]], Joe Davighi, Marco Nardecchia, _Gauging the accidental symmetries of the Standard Model, and implications for the flavour anomalies_ ([arXiv:1909.02021](https://arxiv.org/abs/1909.02021))
* Rigo Bause, [[Gudrun Hiller]], Tim Höhne, Daniel F. Litim, Tom Steudtner, *B-Anomalies from flavorful $U(1)'$ extensions, safely* ([arXiv:2109.06201](https://arxiv.org/abs/2109.06201))
> (in view of [[Higgs field]] [metastability](Higgs+field#MassAndVacuumInstability))
* Disha Bhatia, Nishita Desai, Amol Dighe, Frugal $U(1)_X$ models with non-minimal flavor violation for $b \to s \ell \ell$ anomalies and neutrino mixing* ([arXiv:2109.07093](https://arxiv.org/abs/2109.07093))
Realization in [[F-theory]] of [[GUT]]-models with [[Z'-bosons]] and/or [leptoquarks]] addressing the [[flavour anomalies]] and the [(g-2) anomalies](anomalous+magnetic+moment#Anomalies):
* Miguel Crispim Romao, Stephen F. King, George K. Leontaris, _Non-universal $Z'$ from Fluxed GUTs_, Physics Letters B Volume 782, 10 July 2018, Pages 353-361 ([arXiv:1710.02349](https://arxiv.org/abs/1710.02349))
* A. Karozas, G. K. Leontaris, I. Tavellaris, N. D. Vlachos, _On the LHC signatures of $SU(5) \times U(1)'$ F-theory motivated models_ ([arXiv:2007.05936](https://arxiv.org/abs/2007.05936))
* [[Ben Allanach]], _$U(1)_{B_3-L_2}$ Explanation of the Neutral Current $B$−Anomalies_ ([arXiv:2009.02197](https://arxiv.org/abs/2009.02197))
* [[Andreas Crivellin]], Claudio Andreas Manzari, Marcel Alguero, Joaquim Matias, _Combined Explanation of the $Z \to b \bar b$ Forward-Backward Asymmetry, the Cabibbo Angle Anomaly, $\tau \to \mu \nu \nu$ and $b \to s \ell^+ \ell^-$ Data_ ([arXiv:2010.14504](https://arxiv.org/abs/2010.14504))
* {#ACMM22} Marcel Algueró, [[Andreas Crivellin]], Claudio Andrea Manzari, Joaquim Matias, *Unified Explanation of the Anomalies in Semi-Leptonic $B$ decays and the $W$ Mass*, Physical Review D (2022) [[arXiv:2201.08170](https://arxiv.org/abs/2201.08170)]
[[!redirects Z'-bosons]]
[[!redirects Z' boson]]
[[!redirects Z' bosons]]
[[!redirects Z']]
|
Z+ ring | https://ncatlab.org/nlab/source/Z%2B+ring | #Contents#
* table of contents
{:toc}
## Idea
$\mathbb{Z}_+$ rings (also known as $\mathbb{N}$-rings or fusion rings) are a sort of generalization of finite groups. A $\mathbb{Z}_+$ ring consists of a finite set of objects, and a set "fusion rules" between them. This generalizes groups in the sense that two objects don't necessarily fuse to create a third object from the set. Instead, they will fuse into a "direct sum" of the other elements. The compatibility conditions on these fusion rules can be concisely phrased as saying ring-theoretically. Namely, extending the fusion rules $\mathbb{Z}$-linearly one arrives at a ring whose underlying abelian group is a freely generated $\mathbb{Z}$-module.
$\mathbb{Z}_+$ rings naturally appear in the context of [[fusion categories]]. This is because in a fusion categories there is a notion of [[tensor product]], [[direct sum]], and [[simple object]]. The tensor product of simple objects will be the direct sum of other simple objects. This gives a "fusion rule" on the finite set of isomorphism classes of simple objects, which in turn induces the structure of a $\mathbb{Z}_+$.
Fusion rings appear very naturally in the algebraic theory of non-abelian [[anyon | anyons]]. Namely, the fusion of two non-abelian anyons has a non-deterministic result. The spectrum of possible results of the fusion and their relative probabilities gives a fusion rule.
## Definition
There are various definitions of $\mathbb{Z}_+$ ring. Our treatment here follows closely [EGNO15](#EGNO15).
\begin{definition}Let $A$ be a ring whose underlying abelian group is a finitely generated free $\mathbb{Z}$-module. A __$\mathbb{Z}_+$ basis__ for $A$ is a basis for $B=\{b_i\}_{i\in I}$ for $A$ as a $\mathbb{Z}$-module such that $b_ib_j$ is a non-negative linear combination of elements of $B$ for all $i,j\in I$. That is, there exists $c^{k}_{i,j}\in \mathbb{Z}_+$ such that
$$b_i b_j=\sum_{k\in I}c^{k}_{i,j}\cdot b_k.$$
\end{definition}
Simple put, a __$\mathbb{Z}_+$-ring__ is a ring equipped with a $\mathbb{Z}_+$-basis, with the extra condition that the multiplicative identity element $1$ be a non-negative linear combination of basis elements.
The relation of the multiplicative identity to the $\mathbb{Z}_+$-basis is a very subtle one. In particular, the fact that the the identity is not part of the basis can cause troubles. For this reason, we define a __unital $\mathbb{Z}_+$-ring__ to be a $\mathbb{Z}_+$ ring whose identity element is part of the distinguished $\mathbb{Z}_+$-basis.
A $\mathbb{Z}_+$ ring is uniquely defined by its fusion rules. Hence, the definition of $\mathbb{Z}_+$-ring can be restated in terms of a collection of explicit properties that the fusion rules must satisfy. Seeing as the fusion rules are often what is of most interest, this restatement can be very useful.
\begin{proposition} Let $B=\{b_i\}_{i\in I}$ be a finite set, equipped with integers $(c^{k}_{i,j})_{i,j,k\in I}$. Let $A$ be the free $\mathbb{Z}$-module generated by $B$. Define a binary operation $A\times A\to A$ by linearly extending the rule
$$b_i\cdot b_j=\sum_{k\in I}c^{k}_{i,j}\cdot b_k,$$
for all $i,j\in I$.
* (Associativity) Given $i,j,k\in I$, $b_i \cdot (b_j\cdot b_k)=(b_i\cdot b_j)\cdot b_k$ if and only if
$$\sum_{s\in I}c^{s}_{i,j}c^t_{s,k}=\sum_{s\in I}c^{s}_{j,k}c^{t}_{i,s}$$
for all $t\in I$.
* (Identity) Given $i \in I$, we have that $b_1\cdot b_{i}=b_{i}\cdot b_1=b_{i}$ if and only if
$$c^{j}_{1,i}=c^{j}_{i,1}=
\begin{cases}
1 & i=j\\
0 & \text{otherwise}
\end{cases}$$
for all $j\in I$.
\end{proposition}
Thus, the fusion rule $(c^{k}_{i,j})$ induces a $\mathbb{Z}_+$ ring structure if and only if the properties listed are satisfied.
## Fusion rings
Most of the subtlety and power in the theory of finite groups comes from the existence of inverses. Similarly, a good theory of $\mathbb{Z}_+$-rings should incorporate some idea of inverses. On the level of fusion categories this corresponds to [[rigid monoidal category | rigidity]], and on the level of anyons this corresponds to the existence of [[antiparticles]].
\begin{definition} Let $A$ be a unital $\mathbb{Z}_+$-ring with $\mathbb{Z}_+$-basis $B$. We call $A$ a __fusion ring__ if for all $i\in I$ there exists a unique $i^*\in I$ such that
$$c^{1}_{i,j}=
\begin{cases}
1 & j=i^*\\
0 & \text{otherwise}
\end{cases}$$
and the quantity $c^{k^*}_{i,j}$ is invariant under cyclic permutations of $i,j,k$.
\end{definition}
In this sense, $b_{i^*}$ is the "dual" or "inverse" of $b_i$. This induces an involution $*: A\to A$ by sending
$$a:=\sum_{i\in I}a_i\cdot b_i\mapsto a^*:=\sum_{i\in I}a_i\cdot b_{i^*}.$$
## References
* {#EGNO15} [[Pavel Etingof]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], section 3.1 in _Tensor categories_, Mathematical Surveys and Monographs, Volume 205, American Mathematical Society, 2015 ([pdf](http://www-math.mit.edu/~etingof/egnobookfinal.pdf
)) |
Z-infinity-module | https://ncatlab.org/nlab/source/Z-infinity-module |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The idea is that there should be some kind of "complete [[local ring]]" $\mathbb{Z}_\infty$ corresponding to the archimedean [[valuation]] on $\mathbb{Q}$, by analogy with the (genuine!) complete local rings $\mathbb{Z}_p$ corresponding to the (non-archimedean) $p$-adic valuations on $\mathbb{Q}$: the [[p-adic integers]]. However, the naïve approach taking
$$\mathbb{Z}_\infty = \lbrace x \in \mathbb{R} : -1 \le x \le 1 \rbrace$$
fails because this is not a subring of $\mathbb{R}$!
[[Nikolai Durov]]'s definition of $\mathbb{Z}_\infty$ is inspired by classical [[Arakelov geometry]] and starts with the observation that any $\mathbb{Z}_p$-lattice $\Lambda$ in a finite-dimensional $\mathbb{Q}_p$-[[vector space]] $E$ defines a maximal compact subgroup submonoid $M_\Lambda$ of $End(E)$, and $M_\Lambda = M_{\Lambda'}$ if and only if $\Lambda$ and $\Lambda'$ are similar lattices; accordingly, a $\mathbb{Z}_\infty$-lattice up-to-similarity should correspond to maximal compact submonoids of $End(E)$ for a finite-dimensional $\mathbb{R}$-vector space $E$, i.e. the monoid of $\mathbb{R}$-linear endomorphisms of $E$ that are [[short map|short]] with respect to some [[norm]] on $E$. Thus, Durov defines a $\mathbb{Z}_\infty$-lattice to be a finite-dimensional _normed_ $\mathbb{R}$-vector space, and a morphism of $\mathbb{Z}_\infty$ lattices is defined to be a [[short map|short]] $\mathbb{R}$-[[linear map]]. This gives a category $\mathbb{Z}_{\infty}\mathbf{\text{-Lat}}$ with finite limits and colimits. (Note, however, that it is not an additive category!)
Now, notice that every [[flat module|flat]] $\mathbb{Z}_p$-module is the [[filtered colimit]] of its finite-dimensional $\mathbb{Z}_p$-submodules (which are necessarily free because $\mathbb{Z}_p$ is a local ring), and in fact the category of flat $\mathbb{Z}_p$-modules is equivalent to the category of [[ind-object|ind-objects]] of $\mathbb{Z}_{p}\mathbf{\text{-Lat}}$. So we may define a flat $\mathbb{Z}_\infty$-module to be an ind-object of $\mathbb{Z}_{\infty}\mathbf{\text{-Lat}}$. One may also give the following explicit description: a flat $\mathbb{Z}_\infty$-module $E$ is a (possibly infinite-dimensional) $\mathbb{R}$-vector space $E_{\mathbb{R}}$ together with a symmetric convex body $E_{\mathbb{Z}_\infty}$, and a morphism $E \to E'$ is an $\mathbb{R}$-linear map $E_{\mathbb{R}} \to E'_{\mathbb{R}}$ that restricts to a map $E_{\mathbb{Z}_\infty} \to E'_{\mathbb{Z}_\infty}$. This defines a category $\mathbb{Z}_{\infty}\mathbf{\text{-FlMod}}$.
Finally, noting that the forgetful functor $U : \mathbb{Z}_{\infty}\mathbf{\text{-FlMod}} \to \mathbf{\text{Set}}$ taking a flat $\mathbb{Z}_\infty$-module $E$ to its underlying symmetric convex body $E_{\mathbb{Z}_\infty}$ has a [[left adjoint]] $F$, Durov defines a (not necessarily flat) $\mathbb{Z}_\infty$-module to be a [[module over a monad|module]] for the induced [[monad]] $\Sigma_\infty = U F$. The comparison functor embeds $\mathbb{Z}_{\infty}\mathbf{\text{-FlMod}}$ as a full subcategory of $\mathbb{Z}_{\infty}\mathbf{\text{-Mod}}$.
## Definition
Let $\Sigma_\infty : \mathbf{\text{Set}} \to \mathbf{\text{Set}}$ be the functor sending a set $S$ to the set
$$\left\lbrace \vec{v} \in \mathbb{R}^{(S)} : \left\| \vec{v} \right\|_1 = \sum_{s \in S} \left| v_s \right| \le 1 \right\rbrace$$
i.e. the solid regular cross-polytope with $S$-many vertices. The action of $\Sigma_\infty$ on maps of sets is the obvious one. Let $\eta : id \Rightarrow \Sigma_\infty$ be the natural transformation given by insertion of generators, and let $\mu : \Sigma_\infty \Sigma_\infty \Rightarrow \Sigma_\infty$ be the natural transformation given by "evaluation" of "octahedral" combinations:
$$\sum_i \alpha_i \eta \left( \sum_j \beta_{i,j} \eta (s_j) \right) \mapsto \sum_{i, j} \alpha_i \beta_{i, j} \eta (s_j)$$
One may verify that this defines a [[monad]] $(\Sigma_\infty, \eta, \mu)$ on $\mathbf{\text{Set}}$. A **$\mathbb{Z}_\infty$-module** is defined to be a [[module over a monad|module]] for this monad.
## Properties
The $\Sigma_\infty$ monad is a [[monad with arities]]: the category of arities may be taken to be $\mathbf{\text{FinSet}}$.
The $\mathbb{Z}_\infty$-module structure on a set $M$ is entirely determined by the map $\alpha_2 : \Sigma_\infty (2) \times M^2 \to M$ given by $((\lambda_1, \lambda_2), (x_1, x_2)) \mapsto \lambda_1 x_1 + \lambda_2 x_2$. Conversely, a set $M$ together with an element $\alpha_0$ and a map $\alpha_2 : \Sigma_\infty (2) \times M^2 \to M$ satisfying certain equations is a $\mathbb{Z}_\infty$-module. A map commuting with $\alpha_0$ and $\alpha_2$ is a homomorphism of $\mathbb{Z}_\infty$-modules, thus the theory of $\mathbb{Z}_\infty$-modules is a finitary [[algebraic theory]], with all that this implies.
The category $\mathbb{Z}_{\infty}\mathbf{\text{-Mod}}$ has the following properties:
* It is a [[complete category]] (with the forgetful functor creating all limits).
* It is a [[cocomplete category]] (with the forgetful functor creating all filtered colimits).
* It has a [[zero object]].
* It has a [[tensor product]] and an [[internal hom]], making it into a [[symmetric monoidal category|symmetric]] [[monoidal closed category]].
## Examples
Every normed $\mathbb{R}$-vector space $V$ induces a flat $\mathbb{Z}_\infty$-module $E$ where $E_{\mathbb{R}} = V$ and $E_{\mathbb{Z}_\infty} = \left\lbrace \vec{v} \in V : \left\| \vec{v} \right\| \le 1 \right\rbrace$. (In the other direction, every flat $\mathbb{Z}_\infty$-module induces a _seminorm_ on the underlying $\mathbb{R}$-vector space.)
A finitely-generated flat $\mathbb{Z}$-module is the symmetric convex hull of a finite set of vectors.
## References
* [[Nikolai Durov]], _New approach to Arakelov theory_. Ph.D. thesis (2007) [arXiv:0704.2030](http://arxiv.org/abs/0704.2030/). |
Z-theory | https://ncatlab.org/nlab/source/Z-theory |
#Contents#
* table of contents
{:toc}
## Idea
A term used for aspects of [[topological M-theory]].
## References
* [[Nikita Nekrasov]], _Z Theory_, ([arXiv:hep-th/0412021](http://arxiv.org/abs/hep-th/0412021))
[[!redirects Z theory]]
|
Zach Goldthorpe | https://ncatlab.org/nlab/source/Zach+Goldthorpe |
* [blog](https://leftadjoint.wordpress.com/about/)
* [institute page](https://apps.ualberta.ca/directory/person/zgoldtho)
## Selected writings
An [[(∞,1)-category]] of [[(∞,∞)-categories]] in terms of [[induction|inductively]] and [[coinduction|coinductively]] defined [[equivalences]]:
* [[Zach Goldthorpe]], _Homotopy theories of $(\infty,\infty)$-categories as universal fixed points with respect to enrichment_, International Mathematics Research Notices **2023** 22 (2023) 19592–19640 [[arXiv:2307.00442](https://arxiv.org/abs/2307.00442), [doi:10.1093/imrn/rnad196](https://doi.org/10.1093/imrn/rnad196)]
category: people
|
Zachary Murray | https://ncatlab.org/nlab/source/Zachary+Murray |
* [LinkedIn page](https://ca.linkedin.com/in/zachary-murray-dal)
## Selected writings
On [[constructive analysis]] with the [[Cauchy real numbers]] according to [Bishop (1967)](#Bishop67), formalized in the [[Agda]] [[proof assistant]]:
* [[Zachary Murray]], *Constructive Analysis in the Agda Proof Assistant* [[arXiv:2205.08354](https://arxiv.org/abs/2205.08354), [github](https://github.com/z-murray/honours-project-constructive-analysis-in-agda)]
review:
* [[Zachary Murray]], *Constructive Real Numbers in the Agda Proof Assistant*, [talk at *CQTS*](Center+for+Quantum+and+Topological+Systems#MurrayFeb2023), (Feb 2023) [video:[YT](https://www.youtube.com/watch?v=7Q_sjfyPJqU)]
category: people |
Zachery Lindsey | https://ncatlab.org/nlab/source/Zachery+Lindsey |
* [Mathematics Genealogy page](https://www.mathgenealogy.org/id.php?id=262620)
## Selected writings
On the [[model structure for quasi-categories|Joyal-type]] [[model structure for cubical quasi-categories]] on [[cubical sets]] [[connection on a cubical set|with connections]]:
* [[Brandon Doherty]], [[Chris Kapulkin]], [[Zachery Lindsey]], [[Christian Sattler]], _Cubical models of (∞,1)-categories_, Memoirs of the AMS ([accepted 2022](https://www.ams.org/cgi-bin/mstrack/accepted_papers/memo)) [[arXiv:2005.04853](https://arxiv.org/abs/2005.04853)]
On a [[calculus of fractions]] generalized from categories to [[quasi-categories]] ("$(\infty,1)$-calculus of fractions", for *[[localization of an (infinity,1)-category|localization of $(\infty,1)$-categories]]*):
* [[Daniel Carranza]], [[Chris Kapulkin]], [[Zachery Lindsey]], *Calculus of Fractions for Quasicategories* [[arXiv:2306.02218](https://arxiv.org/abs/2306.02218)]
category: people
|
Zamolodchikov equation | https://ncatlab.org/nlab/source/Zamolodchikov+equation |
see _[[Knizhnik-Zamolodchikov equation]]_
[[!redirects Zamolodchikov equation -- history]] |
Zariski geometry | https://ncatlab.org/nlab/source/Zariski+geometry | Zariski geometry is a structure defined by Boris Zilber. See his book
* [[Boris Zilber]], _Zariski Geometries_, Cambridge University Press, 2010 (publisher's [book page](http://www.cambridge.org/gb/knowledge/isbn/item2713251/?site_locale=en_GB))
> Zariski Geometries are abstract structures in which a suitable generalisation of Zariski topology makes sense. Algebraic varieties over an algebraically closed field and compact complex spaces in a natural language are examples of Zariski geometries. The main theorem by Hrushovski and the lecturer states that under certain non-degeneracy conditions a 1-dimensional Zariski geometry can be identified as an algebraic curve over an algebraically closed field. The proof of the theorem exhibits, as a matter of fact, a way to develop algebraic geometry from purely geometric abstract assumptions not involving any algebra at all. Recent works in model theory of complex manifolds, differential fields and non-commutative geometry point to exciting perspectives for the theory.
[[!redirects Zariski geometries]]
[[!redirects Zariski geometries]] |
Zariski site | https://ncatlab.org/nlab/source/Zariski+site |
#Contents#
* table of contents
{:toc}
## Idea
There is a [[little site]] notion of the [[Zariski topology]], and a [[big site]] notion. As for the little site notion: the Zariski topology on the set of prime ideals of a commutative ring $R$ is the smallest topology that contains, as open sets, sets of the form $\{p\; \text{prime}: a \notin p\}$ where $a$ ranges over elements of $R$.
As for the big site notion, the _Zariski topology_ is a [[coverage]] on the [[opposite category]] [[CRing]]${}^{op}$ of commutative rings. This article is mainly about the big site notion.
## Definition
For $R$ a commutative [[ring]], write $Spec R \in CRing^{op}$ for its [[spectrum of a commutative ring]], hence equivalently for its incarnation in the [[opposite category]].
+-- {: .num_defn #StandardOpenImmersion}
###### Definition
For $S \subset R$ a [[multiplicative subset]], write $R[S^{-1}]$ for the corresponding [[localization of a ring|localization]] and
$$
Spec(R[S^{-1}]) \longrightarrow Spec(R)
$$
for the dual of the canonical ring homomorphism $R \to R[S^{-1}]$.
=--
+-- {: .num_remark}
###### Remark
The maps as in def. \ref{StandardOpenImmersion} are not [[open immersion of schemes|open immersions]] for arbitrary multiplicative subsets $S$ (see [a MathOverflow discussion](http://mathoverflow.net/questions/20782/ring-theoretic-characterization-of-open-affines)). They are for subsets of the form $S = \{ f^0, f^1, f^2, \ldots \}$, in which case they are called the _standard opens_ of $Spec(R)$.
=--
+-- {: .num_defn}
###### Definition
A family of [[morphisms]] $\{Spec A_i \to Spec R\}$ in $CRing^{op}$ is a Zariski-[[covering]] precisely if
* each ring $A_i$ is the [[localization]]
$$
A_i = R[r_i^{-1}]
$$
of $R$ at a single element $r_i \in R$
* $Spec A_i \to Spec R$ is the canonical inclusion, dual to the canonical ring homomorphism $R \to R[r_i^{-1}]$;
* There exists $\{f_i \in R\}$ such that
$$
\sum_i f_i r_i = 1
\,.
$$
=--
+-- {: .num_remark}
###### Remark
Geometrically, one may think of
* $r_i$ as a function on the [[space]] $Spec R$;
* $R[r_i^{-1}]$ as the [[open subset]] of points in this space on which the function is not 0;
* the covering condition as saying that the functions form a [[partition of unity]] on $Spec R$.
=--
+-- {: .num_defn #ZariskiTopos}
###### Definition
Let $CRing_{fp} \hookrightarrow CRing$ be the [[full subcategory]] on [[finitely presented object]]s. This inherits the Zariski [[coverage]].
The [[sheaf topos]] over this [[site]] is the [[big topos]] version of the **Zariski topos**.
=--
## Properties
### Points
The [[maximal ideal]] in $R$ correspond precisely to the [[closed subset|closed]] points of the [[prime spectrum]] $Spec(R)$ in the Zariski topology.
### As a site
+-- {: .num_prop}
###### Proposition
The Zariski coverage is [[subcanonical coverage|subcanonical]].
=--
+-- {: .num_prop}
###### Proposition
* $CRing_{fp}^{op}$ is the [[syntactic category]] of the [[cartesian theory]] of commutative rings;
* $CRing_{fp}^{op}$ equipped with the Zariski topology is the [[syntactic site]] of the [[geometric theory]] of [[local ring]]s.
Hence
* the big Zariski topos, def. \ref{ZariskiTopos}, is the [[classifying topos]] for [[local ring]]s.
* a [[locally ringed topos]] is equivalently a topos $\mathcal{E}$ equipped with a [[geometric morphism]] into the big Zariski topos.
=--
See [[classifying topos]] and [[locally ringed topos]] for more details on this.
### Sheafification
If $F$ is a presheaf on $CRing^{op}$ and $F^{++}$ denotes its [[sheafification]], then the canonical morphism $F(R) \to F^{++}(R)$ is an isomorphism for all [[local ring|local rings]] $R$. This follows from the explicit description of the [[plus construction]] and the fact that a local ring admits only the trivial covering.
## Kripke–Joyal semantics
{#KripkeJoyal}
Writing $R \models \varphi$ for the interpretation of a formula $\varphi$ of the [[internal language]] of the big Zariski topos over $Spec(R)$ with the [[Kripke–Joyal semantics]], the forcing relation can be expressed as follows.
$$
\begin{array}{lcl}
R \models x = y : F &\Longleftrightarrow&
x = y \in F(R). \\
R \models \top &\Longleftrightarrow&
1 = 1 \in R. \\
R \models \bot &\Longleftrightarrow&
1 = 0 \in R. \\
R \models \phi \wedge \psi &\Longleftrightarrow&
R \models \phi \,\text{and}\, R \models \psi. \\
R \models \phi \vee \psi &\Longleftrightarrow&
\text{there is a partition}\, \sum_i f_i = 1 \in R \,\text{such that for all}\, i,
R[f_i^{-1}] \models \phi \,or\, R[f_i^{-1}] \models \psi. \\
R \models \phi \Rightarrow \psi &\Longleftrightarrow&
\text{for any}\, R\text{-algebra}\, S \,\text{it holds that}\, (S \models \phi) \,implies\, (S \models \psi). \\
R \models \forall x:F. \phi &\Longleftrightarrow&
\text{for any}\, R\text{-algebra}\, S \,\text{and any element}\, x \in F(S) \,\text{it holds that}\, S \models \phi[x]. \\
R \models \exists x.F. \phi &\Longleftrightarrow&
\text{there is a partition}\, \sum_i f_i = 1 \in R \,\text{such that for all}\, i,
\text{there exists an element}\, x_i \in F(R[f_i^{-1}]) \,\text{such that}\,
R[f_i^{-1}] \models \phi[x_i].
\end{array}
$$
The only difference to the Kripke–Joyal semantics of the _little_ Zariski topos is that in the clauses for $\Rightarrow$ and $\forall$, one has to restrict to $R$-algebras $S$ of the form $S = R[f^{-1}]$.
## Related concepts
[[fpqc-site]] $\to$ [[fppf-site]] $\to$ [[syntomic site]] $\to$ [[étale site]] $\to$ [[Nisnevich site]] $\to$ **Zariski site**
* [[spectrum of a commutative ring]]
## References
Examples A2.1.11(f) and D3.1.11 in
* [[Peter Johnstone]], _[[Sketches of an Elephant]]_
{#Johnstone}
Section VIII.6 of
* [[Saunders MacLane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_
{#MacLaneMoerdijk}
* [[The Stacks Project]], chapter 33 _Topologies on Schemes_
* Nick Duncan, _Gros and Petit Toposes_, talk notes, [88th Peripatetic Seminar on Sheaves and Logic](http://cheng.staff.shef.ac.uk/pssl88/), [pdf](http://cheng.staff.shef.ac.uk/pssl88/pssl88-duncan.pdf).
* Daniel Murfet, [The Zariski Site](http://therisingsea.org/notes/ZariskiTopology.pdf)
category: algebraic geometry
[[!redirects Zariski topos]]
[[!redirects big Zariski topos]]
[[!redirects Zariski sheaf]]
[[!redirects Zariski sheaves]]
[[!redirects Zariski descent]]
[[!redirects theory of local rings]]
[[!redirects geometric theory of local rings]] |
Zariski topology | https://ncatlab.org/nlab/source/Zariski+topology |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _Zariski topology_ is a [[topological space|topology]] on the [[prime spectrum of a commutative ring]]. It serves as the basis for much of [[algebraic geometry]].
{#Outline} We consider the definition in increasing generality and sophistication:
1. First we discuss the naive Zariski topology [on affine spaces](#OnAffineSpace) $k^n$, consider the classical proofs and discover thereby the special role of prime ideals and maximal ideals;
1. then we turn to the modern definition of the Zariski topology [on affine varieties](#OnAffineVarieties) $Spec(R)$ which takes the concept of prime (and maximal) ideals as primary, and again we provide the classical arguments;
1. finally we discuss the abstract [[category theory|category theoretic]] perspective on these matters [in terms of Galois connections](#InTermsOfGaloisConnections) and obtain slick category theoretic proofs of all the previous statements.
Starting with [[affine space]] $k^n$, then the idea of the Zariski topology is to take as the _[[closed subsets]]_ those defined by the vanishing of any set of [[polynomials]] over $k$ in $n$ [[variables]], hence the solution sets to [[equations]] of the form
$$
\underset{i \in I}{\forall}
\left(
f_i(x_1, \cdots, x_n) = 0
\right)
$$
for $f_i \in k[X_1, \cdots, X_n]$ [[polynomials]]. The [[open subsets]] of the topology are the [[complements]] of these _vanishing sets_.
It is clear that the vanishing set such a set of polynomials depends only on the [[ideal]] in the [[polynomial ring]] which is generated by them. Under this translation then forming the intersection of closed subsets corresponds to forming the sum of these ideals, and forming the union of closed subsets corresponds to forming the product of the corresponding ideals. This way the Zariski topology establishes a dictionary between topological concepts of the affine space $k^n$, and algebra inside the polynomial ring.
In particular one finds that the [[irreducible closed subsets]] of the Zariski topology correspond to the [[prime ideals]] in the polynomial ring (prop. \ref{PrimeVanishingIdealOfIrreducibleZariskiClosed} and prop. \ref{PrimeIdealClosedsubspaceBijection} below), and that the [[closed points]] correspond to the [[maximal ideals]] among these (prop. \ref{MaximalIdealsAreClosedPoints}).
This motivates the modern refinement of the concept of the Zariski topology, where one considers any [[commutative ring]] $R$ and equips its set of [[prime ideals]] with a topology, by direct analogy with the previously naive affine space $k^n$, which is recovered with $R$ a polynomial ring and restricting attention to the maximal ideals (example \ref{AffinSpaceAsPrimeSpectrum} below).
These sets of prime ideals of a ring $R$ equipped with the Zariski topology are called the (topological spaces underlying) the _[[prime spectrum of a commutative ring]]_, denoted $Spec(R)$.
The Zariski topology is in general not [[Hausdorff topological space|Hausdorff]] (example \ref{AffineSpaceOverInfiniteFieldNotHausdorff} below) which makes it sometimes be regarded as an "exotic" type of topology. But it is in fact [[sober topological space|sober]] (prop. \ref{ZariskiTopologyIsSober} below) and hence as well-behaved in this respect as general [[locales]] are.
## On affine space
{#OnAffineSpace}
We consider here, for $k$ a [[field]], the [[vector space]] $k^n$ equipped with a Zariski topology. This is the original definition of Zariski topology, and serves well to motivate the concept, but eventually it was superceded by a more refined concept of Zariski topologies of prime spectra, discussed in the next subsection [below](#OnAffineVarieties). In example \ref{AffinSpaceAsPrimeSpectrum} below we reconsider the naive case of interest in this subsection here from that more refined perspective.
### Definition
+-- {: .num_defn #ZariskiOpenSubsetsOnAffineSpace}
###### Definition
**(Zariski topology on affine space)**
Let $k$ be a [[field]], let $n \in \mathbb{N}$, and write $k[X_1, \cdots, X_n]$ for the [[set]] of [[polynomials]] in $n$ [[variables]] over $k$.
For $\mathcal{F} \subset k[X_1, \cdots, X_n]$ a subset of polynomials, let the subset $V(\mathcal{F}) \subset k^n$ of the $n$-fold [[Cartesian product]] of the underlying set of $k$ (the _vanishing set_ of $\mathcal{F}$) be the subset of points on which all these polynomials jointly vanish:
$$
V(\mathcal{F})
\coloneqq
\left\{
(a_1, \cdots, a_n) \in k^n
\,\vert\,
\underset{f \in \mathcal{F}}{\forall} f(a_1, \cdots, a_n) = 0
\right\}
\,.
$$
These subsets are called the _Zariski [[closed subsets]]_.
Write
$$
\tau_{\mathbb{A}^n_k}
\;\coloneqq\;
\left\{
k^n \backslash V(\mathcal{F}) \subset k^n
\,\vert\,
\mathcal{F} \subset k[X_1, \cdots, X_n]
\right\}
$$
for the set of [[complements]] of the Zariski closed subsets. These are called the _Zariski [[open subsets]]_ of $k^n$.
=--
+-- {: .num_prop #VerifyingZariskiTopologyOnAffineSpace}
###### Proposition
**(Zariski topology is well defined)**
Assuming [[excluded middle]], then:
For $k$ a [[field]] and $n \in \mathbb{N}$, then the Zariski open subsets of $k^n$ (def. \ref{ZariskiOpenSubsetsOnAffineSpace}) form a [[topological space|topology]]. The resulting [[topological space]]
$$
\mathbb{A}^n_k
\;\coloneqq\;
\left(
k^n, \tau_{\mathbb{A}^n_k}
\right)
$$
is also called the $n$-dimensional _[[affine space]]_ over $k$.
=--
+-- {: .proof}
###### Proof
We need to show for $\{\mathcal{F}_i \subset k[X_1, \cdots, X_n]\}_{i \in I}$ a set of subsets of polynomials that
1. $\underset{i \in I}{\cup} \left(k^n \backslash V(\mathcal{F}_i)\right) = k^n \backslash V(\mathcal{F}_\cup)$ for some $\mathcal{F}_\cup \subset k[X_1, \cdots, X_n]$;
1. if $I$ is [[finite set|finite]] then $\underset{i \in I}{\cap} \left( k^n \backslash V(\mathcal{F}_{\cap})\right) = k^n \backslash \mathcal{F}_\cap$ for some $\mathcal{F}_{\cap} \subset k[X_1, \cdots, X_n]$.
By [[de Morgan's law]] for [[complements]] (and using [[excluded middle]]) this is equivalent to
1. $\underset{i \in I}{\cap} V(\mathcal{F}_i) = V(\mathcal{F}_\cup)$ for some $\mathcal{F}_\cup \subset k[X_1, \cdots, X_n]$;
1. if $I$ is [[finite set|finite]] then $\underset{i \in I}{\cup} V(\mathcal{F}_i) = V(\mathcal{F}_{\cap})$ for some $\mathcal{F}_{\cap} \subset k[X_1, \cdots, X_n]$.
We claim that we may take
1. $\mathcal{F}_\cup = \underset{i \in I}{\cup} \mathcal{F}_i$
1. $\mathcal{F}_{\cap} = \underset{i \in I}{\prod} \mathcal{F}_i \coloneqq \left\{ \underset{i \in I}{\prod} f_i \,\vert\, f_i \in \mathcal{F}_i \right\}$.
(In the second line we have the set of all those polynomials which arise as products of polynomials with one factor from each of the $\mathcal{F}_i$.)
Regarding the first point:
$$
\begin{aligned}
& (a_1, \cdots, a_n) \in \underset{i \in I}{\cap} V(\mathcal{F}_i)
\\
\Leftrightarrow\;
&
\underset{i \in I}{\forall}
\left(
(a_1, \cdots, a_n) \in V(\mathcal{F}_i)
\right)
\\
\Leftrightarrow\;
&
\underset{i \in I}{\forall}
\left(
\underset{f \in \mathcal{F}_i}{\forall}
\left(
f(a_1, \cdots, a_n) = 0
\right)
\right)
\\
\Leftrightarrow\;
&
\underset{f \in \underset{i \in I}{\cup} \mathcal{F}_i}{\forall}
\left(
f(a_1, \cdots, a_n) = 0
\right)
\\
\Leftrightarrow\;
&
(a_1, \cdots, a_n)
\in
V\left(
\underset{i \in I}{\cup} \mathcal{F}_i
\right)
\end{aligned}
$$
Regarding the second point, in one direction we have the immediate implication
$$
\begin{aligned}
&
(a_1, \cdots, a_n) \in \underset{i \in I}{\cup} V(\mathcal{F}_i)
\\
\Leftrightarrow\;
&
\underset{i \in I}{\exists}
\left(
\underset{f \in \mathcal{F}_i}{\forall}
\left(
f(a_1, \cdots, a_n) = 0
\right)
\right)
\\
\Rightarrow \;
&
\underset{(f_i) \in \underset{i \in I}{\prod} \mathcal{F}_i}{\forall}
\left(
\underset{i \in I}{\prod} f_i(a_1, \cdots, a_n) = 0
\right)
\\
\Leftrightarrow\;
&
(a_1, \cdots, a_n)
\in
V\left(
\underset{i \in I}{\prod} \mathcal{F}_i
\right)
\,.
\end{aligned}
$$
For the converse direction we need to show that
$$
\left(
(a_1 , \cdots , a_n)
\in
V\left(
\underset{i \in I}{\prod} \mathcal{F}_i
\right)
\right)
\;\Rightarrow\;
\left(
(a_1, \cdots, a_n)
\in
\underset{i \in I}{\cup} V(\mathcal{F}_1)
\right)
\,.
$$
hence that
$$
\left(
\underset{(f_i) \in \underset{i \in I}{\prod} \mathcal{F}_i }{\forall}
\left(
\underset{i \in I}{\prod} f_i(a_1, \cdots, a_n) = 0
\right)
\right)
\;\Rightarrow\;
\left(
\underset{i \in I}{\exists}
\left(
\underset{f_i \in \mathcal{F}_i}{\forall}
\left(
f_i(a_1, \cdots, a_n) = 0
\right)
\right)
\right)
\,.
$$
By [[excluded middle]], this is equivalent to its [[contraposition]], which by [[de Morgan's law]] is
$$
\left(
\underset{i \in I}{\forall}
\left(
\underset{f_i \in \mathcal{F}_i}{\exists}
\left(
f_i(a_1, \cdots, a_n) \neq 0
\right)
\right)
\right)
\;\Rightarrow\;
\left(
\underset{(f_i) \in \underset{i \in I}{\prod} \mathcal{F}_i }{\exists}
\left(
\underset{i \in I}{\prod} f_i(a_1, \cdots, a_n) \neq 0
\right)
\right)
\,.
$$
This now is true by the assumption that $k$ is a [[field]]: If all factors $f_i(a_1, \dots a_n) \in k$ are non-zero, then their product $\underset{i \in I}{\prod} f_i(a_1, \cdots, a_n) \in k$ is non-zero.
=--
### Properties
#### Topological closures
+-- {: .num_prop}
###### Proposition
For $k$ a [[field]] and $n \in \mathbb{N}$, consider a [[subset]]
$$
S \subset k^n
$$
of the underlying set of the $n$-fold [[Cartesian product]] of $k$ with itself. Then the [[topological closure]] $Cl(S)$ of this subset with respect to the Zariski topology $\tau_{\mathbb{A}^n_k}$ (def. \ref{ZariskiOpenSubsetsOnAffineSpace}) is the vanishing set of all those polynomials that vanish on $S$:
$$
Cl(S)
\;=\;
V \left(
\left\{
f \in k[X_1, \cdots, X_n]
\,\vert\,
\underset{(a_1, \cdots, a_n) \in S}{\forall} f(a_1, \cdots, a_n) = 0
\right\}
\right)
\,.
$$
=--
+-- {: .proof}
###### Proof
We compute as follows:
$$
\begin{aligned}
Cl(S)
& \coloneqq
\underset{ {C \subset k^n \, \text{closed}} \atop {C \supset S} }{\cap}
C
\\
& =
\underset{
{ \mathcal{F} \subset k[X_1, \cdots, X_n] }
\atop
{ S \subset V(\mathcal{F}) }
}{\cap} V(\mathcal{F})
\\
& =
V\left(
\underset{
{ \mathcal{F} \subset k[X_1, \cdots, X_n] }
\atop
{ S \subset V(\mathcal{F}) }
}{\cup} \mathcal{F}
\right)
\\
& =
V \left(
\left\{
f \in k[X_1, \cdots, X_n]
\,\vert\,
\underset{(a_1, \cdots, a_n) \in S}{\forall} f(a_1, \cdots, a_n) = 0
\right\}
\right)
\,.
\end{aligned}
$$
Here the first equality is the definition of [[topological closure]], the second is the definition of closed subsets in the Zariski topology (def. \ref{ZariskiOpenSubsetsOnAffineSpace}), the third is the expression of intersections of these in terms of unions of polynomials as in the proof of prop. \ref{VerifyingZariskiTopologyOnAffineSpace}, and then the last one is immediate.
=--
#### Irreducible closed subsets as prime ideals
In every [[topological space]] the [[irreducible closed subsets]] play a special role, as being precisely the points in the space as seen in its incarnation as a [[locale]] ([this prop.](irreducible%20closed%20subspace#FrameHomomorphismsToPointAreIrrClSub)). The following shows that in the Zariski topology the irreducible closed subsets all come from [[prime ideals]] in the corresponding [[polynomial ring]], and that when the [[ground field]] is [[algebraically closed field|algebraically closed]], then they are in fact in bijection to the prime ideals. See also at _[[schemes are sober]]_.
+-- {: .num_defn #VanishingIdeal}
###### Definition
**([[vanishing ideal]] of Zariski closed subset)**#
Let $k$ be a [[field]], and let $n \in \mathbb{N}$. Then for $V(\mathcal{F}) \subset k^n$ a Zariski closed subset, according to def. \ref{ZariskiOpenSubsetsOnAffineSpace}, hence for $\mathcal{F} \subset k[X_1, \cdots, X_n]$ a set of polynomials, write
$$
I(V(\mathcal{F}))
\subset
k[X_1, \cdots, X_n]
$$
for the maximal subset of polynomials that still has the same joint vanishing set:
$$
I(V(\mathcal{F}))
\;\coloneqq\;
\left\{
f \in k[X_1, \cdots, X_n]
\,\vert\,
\underset{(a_1, \cdots, a_n) \in V(\mathcal{F})}{\forall} f(a_1, \cdots, a_n) = 0
\right\}
\,.
$$
This set is clearly an [[ideal]] in the [[polynomial]] [[ring]] $k[X_1, \cdots, X_n]$, called the _[[vanishing ideal]]_ of $V(\mathcal{F})$.
=--
+-- {: .num_prop #PrimeVanishingIdealOfIrreducibleZariskiClosed}
###### Proposition
With [[excluded middle]] then:
Let $k$ be a [[field]], let $n \in \mathbb{N}$, and let $V(\mathcal{F}) \subset k^n$ be a Zariski closed subset (def. \ref{ZariskiOpenSubsetsOnAffineSpace}). Then the following are equivalent:
1. $V(\mathcal{F})$ is an [[irreducible closed subset]];
1. The [[vanishing ideal]] $I(V(\mathcal{F}))$ (def. \ref{VanishingIdeal}) is a [[prime ideal]].
=--
+-- {: .proof}
###### Proof
In one direction, assume that $V(\mathcal{F})$ is irreducible and consider $f,g \in k[X_1, \cdots, X_n]$ with $f \cdot g \in I(V(\mathcal{F}))$. We need to show that then already $f \in I(V(\mathcal{F}))$ or $g \in I(V(\mathcal{F}))$.
Now since $k$ is a field, we have
$$
\left(
f(a_1, \cdots a_n) \cdot g(a_1, \cdots, a_n) = 0
\right)
\Rightarrow
\left(
\left(
f(a_1, \cdots, a_n) = 0
\,\text{or}\,
g(a_1, \cdots, a_n) = 0
\right)
\right)
\,.
$$
This implies that
$$
V(\mathcal{F}) \subset V(\{f\}) \cup V(\{g\})
$$
and hence that
$$
V(\mathcal{F})
=
(V(\mathcal{F}) \cap F(\{f\}))
\,\,\cup\,\,
(V(\mathcal{F}) \cap F(\{g\}) )
\,.
$$
But since $V(\{f\})$, $V(\{g\})$ and $V(\mathcal{F})$ are all closed, by construction, their intersections are closed and hence this is a decomposition of $V(\mathcal{F})$ as a union of closed subsets. Therefore now the assumption that $V(\mathcal{F})$ is [[irreducible closed subset|irreducible]] implies that
$$
\begin{aligned}
&
\left(
\,
V(\mathcal{F}) = V(\mathcal{F}) \cap V(\{f\})
\,
\right)
\,\text{or}\,
\left(
\,
V(\mathcal{F}) = V(\mathcal{F}) \cap V(\{g\})
\,
\right)
\\
\Leftrightarrow \;
&
\left(
\left(
\,
V(\mathcal{F}) \subset V(\{f\})
\,
\right)
\,\text{or}\,
\left(
\,
V(\mathcal{F}) \subset V(\{g\})
\,
\right)
\right)
\\
\Leftrightarrow \,
&
\left(
\left(
\,
f \in I(X)
\,
\right)
\,\text{or}\,
\left(
\,
g \in I(X)
\,
\right)
\right)
\end{aligned}
\,.
$$
Now for the converse, assume that $I(V(\mathcal{F}))$ is a prime ideal, and that $V(\mathcal{F}) = V(\mathcal{F}_1) \cup V(\mathcal{F}_2)$. We need to show that $V(\mathcal{F}) = V(\mathcal{F}_1)$ or that $V(\mathcal{F}) = V(\mathcal{F}_2)$.
Assume on the contrary, that there existed elements
$$
(a_1, \cdots, a_n) \in V(\mathcal{F}_1) \backslash V(\mathcal{F}_2)
\;\text{and}\;
(b_1, \cdots, b_n) \in V(\mathcal{F}_2) \backslash V(\mathcal{F}_1)
\,
$$
Then in particular the vanishing ideals would not contain each other
$$
\not\left(
I(V(\mathcal{F}_1))
\subset
I(V(\mathcal{F}_2))
\right)
\,\,\,\text{and}\,\,\,
\not\left(
I(V(\mathcal{F}_2))
\subset
I(V(\mathcal{F}_1))
\right)
$$
and hence there were polynomials
$$
f\in I(V(\mathcal{F}_1)) \backslash I(V(\mathcal{F}_2))
\,\,\,\text{and}\,\,\,
g \in I(V(\mathcal{F}_2)) \backslash I(V(\mathcal{F}_1))
\,.
$$
But since a product of polynomials vanishes at some point once one of the factors vanishes at that point, it would follows that
$$
f \cdot g \in I(V(\mathcal{F}_1)) \cap I(V(\mathcal{F}_2)) = I(V(\mathcal{F}))
\,,
$$
which were in contradiction to the assumption that $I(V(\mathcal{F}))$ is a prime ideal. Hence we have a [[proof by contradiction]].
=--
Proposition \ref{PrimeVanishingIdealOfIrreducibleZariskiClosed} gives an [[injection]]
$$
\left\{
\array{
\text{irreducible Zariski closed}
\\
V(\mathcal{F}) \subset k^n
}
\right\}
\overset{\phantom{AAA}}{\hookrightarrow}
\left\{
\array{
\text{prime ideals}
\\
I \in k[X_1, \cdots, X_n]
}
\right\}
\,.
$$
The following says that for [[algebraically closed fields]] then this is in fact a [[bijection]]:
+-- {: .num_prop #PrimeIdealClosedsubspaceBijection}
###### Proposition
Let $k = \overline{k}$ be an [[algebraically closed field]] and let $n \in \mathbb{N}$. Then the function
$$
\array{
IrrClSub(\mathbb{A}^n_k)
&\overset{}{\longrightarrow}&
PrimeIdl(k[X_1, \cdots, X_n])
\\
V(\mathcal{F})
&\overset{\phantom{AAA}}{\mapsto}&
I(V(\mathcal{F}))
}
$$
from prop. \ref{PrimeVanishingIdealOfIrreducibleZariskiClosed} is a [[bijection]].
=--
The **proof** uses [[Hilbert's Nullstellensatz]].
+-- {: .num_remark}
###### Remark
**(generalization to affine varieties)**
Prop \ref{ZariskiClosedSubsetsInSpecR} suggests to consider the set of [[prime ideals]] of a [[polynomial ring]] $k[X_1, \cdots, X_n]$ for general $k$ as more fundamental, in some sense, than the set $k^n$. Morover, the set of prime ideals makes sense for every [[commutative ring]] $R$, not just $R = k[X_1, \cdots, X_n]$, and hence this suggests to consider a Zariski topology on sets of prime ideals. This leads to the more general concept of Zariski topologies for [[affine varieties]], def. \ref{ZariskiClosedSubsetsInSpecR} below.
=--
### Examples
+-- {: .num_example #AffineSpaceOverInfiniteFieldNotHausdorff}
###### Example
If the [[field]] $k$ is _not_ a [[finite field]], then the
Zariski topology on the [[affine space]] (def. \ref{ZariskiOpenSubsetsOnAffineSpace}) is _not_ [[Hausdorff topological space|Hausdorff]].
This is because the solution set to a system of [[polynomials]] over an infinite polynomial is always a [[finite set]]. This means that in this case all the Zariski closed subsets $V(\mathcal{F})$ are [[finite sets]]. This in turn implies that the [[intersection]] of _every_ pair of [[inhabited set|non-empty]] Zariski open subsets is [[inhabited|non-empty]].
But the Zariski topology is always [[sober topological space|sober]], see prop. \ref{ZariskiTopologyIsSober} below.
=--
## On affine varieties
{#OnAffineVarieties}
### Definition
+-- {: .num_defn #ZariskiClosedSubsetsInSpecR}
###### Definition
**(Zariski topology on set of prime ideals)**
Let $R$ be a [[commutative ring]]. Write $PrimeIdl(R)$ for its set of [[prime ideals]]. For $\mathcal{F} \subset R$ any subset of elements of the ring, consider the subsets of those prime ideals that contain $\mathcal{F}$:
$$
V(\mathcal{F})
\;\coloneqq\;
\left\{
p \in PrimeIdl(R) \,\vert\, \mathcal{F} \subset p
\right\}
\,.
$$
These are called the _Zariski [[closed subsets]]_ of $PrimeIdl(R)$. Their [[complements]] are called the _Zariski open subsets_.
=--
+-- {: .num_prop #WellDefinedZariskiTopologyOnSpecR}
###### Proposition
**(Zariski topology well defined)**
Assuming [[excluded middle]], then:
Let $R$ be a [[commutative ring]]. Then the collection of Zariski open subsets (def. \ref{ZariskiClosedSubsetsInSpecR}) in its set of [[prime ideals]]
$$
\tau_{Spec(R)} \subset P(PrimeIdl(R))
$$
satisfies the axioms of a [[topological space|topology]], the _Zariski topology_.
This [[topological space]]
$$
Spec(R) \coloneqq (PrimeIdl(R), \tau_{Spec(R)})
$$
is called (the space underlying) the _[[prime spectrum of a commutative ring|prime spectrum of the commutative ring]]_.
=--
+-- {: .proof #WellDefinedZariskiTopologyOnSpecRProof}
###### Proof
For $\mathcal{F} \subset R$ write $\mathcal{F} \subset I(\mathcal{F})$ for the [[ideal]] which is generated by $\mathcal{F}$. Evidently the Zariski closed subsets depend only on this ideal
$$
V(I(\mathcal{F})) = V(\mathcal{F})
$$
and therefore it is sufficient to consider the $V(\mathcal{F})$ for the case that $\mathcal{F} \subset R$ is not just a subset, but an ideal.
So let $\{F_i \in Idl(R)\}_{i \in I}$ be a set of ideals in $R$ and let $\{V(\mathcal{F}_i) \subset PrimeIdl(R)\}_{i \in I}$ be the corresponding set of Zariski closed subsets. We need to show that there exists $\mathcal{F}_\cup, \mathcal{F}_\cap \subset R$ such that
1. $\underset{i \in I}{\cap} V(\mathcal{F}_i) = V(\mathcal{F}_\cup)$;
1. if $I$ is [[finite set]] then $\underset{i \in I}{\cup} V(\mathcal{F}_i) = V(\mathcal{F}_\cap)$.
We claim that
* $\mathcal{F}_{\cup} = \underset{i \in I}{\sum} \mathcal{F}_i \coloneqq \left\{ \underset{i \in I}{\sum} f_i \,, \in R\;\vert\; f_i \in \mathcal{F}_i \right\}$
* $\mathcal{F}_{\cap} = \underset{i \in I}{\prod} \mathcal{F}_i \coloneqq \left\{ \underset{i \in I}{\prod} f_i \, \in R \;\vert\; f_i \in \mathcal{F}_i \right\}$,
Regarding the first point:
By using the various definitions, we get the following chain of logical equivalences:
$$
\begin{aligned}
& p \in \underset{i \in I}{\cap} V(\mathcal{F}_i)
\\
\Leftrightarrow\;
&
\underset{i \in I}{\forall}
\left(
p \in V(\mathcal{F}_i)
\right)
\\
\Leftrightarrow\;
&
\underset{i \in I}{\forall}
\left(
\mathcal{F}_i \subset p
\right)
\\
\Leftrightarrow\;
&
\left(\underset{i \in I}{\sum} \mathcal{F}_i\right)
\subset p
\\
\Leftrightarrow\;
&
p \in V\left(
\underset{i \in I}{\sum} \mathcal{F}_i
\right)
\,.
\end{aligned}
$$
Regarding the second point, in one direction we have the immediate implication
$$
\begin{aligned}
&
p \in \underset{i \in I}{\cup} V(\mathcal{F}_i)
\\
\Leftrightarrow\;
&
\underset{i \in I}{\exists}
\left(
\mathcal{F}_i \subset p
\right)
\\
\Rightarrow \;
&
\underset{(f_i) \in \underset{i \in I}{\prod} \mathcal{F}_i}{\forall}
\left(
\underset{i \in I}{\prod} f_i \in p
\right)
\\
\Leftrightarrow\;
&
p
\in
V\left(
\underset{i \in I}{\prod} \mathcal{F}_i
\right)
\,.
\end{aligned}
$$
For the converse direction we need to show that
$$
\left(
p
\in
V\left(
\underset{i \in I}{\prod} \mathcal{F}_i
\right)
\right)
\;\Rightarrow\;
\left(
p
\in
\underset{i \in I}{\cup} V(\mathcal{F}_1)
\right)
\,.
$$
hence that
$$
\left(
\underset{(f_i) \in \underset{i \in I}{\prod} \mathcal{F}_i }{\forall}
\left(
\underset{i \in I}{\prod} f_i \in p
\right)
\right)
\;\Rightarrow\;
\left(
\underset{i \in I}{\exists}
\left(
\underset{f_i \in \mathcal{F}_i}{\forall}
\left(
f_i \in p
\right)
\right)
\right)
\,.
$$
By [[excluded middle]], this is equivalent to its [[contraposition]], which by [[de Morgan's law]] is
$$
\left(
\underset{i \in I}{\forall}
\left(
\underset{f_i \in \mathcal{F}_i}{\exists}
\not
\left(
f_i \in p
\right)
\right)
\right)
\;\Rightarrow\;
\left(
\underset{(f_i) \in \underset{i \in I}{\prod} \mathcal{F}_i }{\exists}
\not
\left(
\underset{i \in I}{\prod} f_i \in p
\right)
\right)
\,.
$$
This holds by the assumption that $p$ is a [[prime ideal]].
=--
### Properties
{#PropertiesOnAffineVarieties}
We discuss some properties of the Zariski topology on [[prime spectra of commutative rings]].
#### Topological closures
+-- {: .num_lemma #ZariskiClorsuredOfPont}
###### Lemma
**([[topological closure]] of points)**
Let $R$ be a [[commutative ring]] and consider $Spec(R) = (PrimeIdl(R), \tau_{Spec(R)})$ its [[prime spectrum of a commutative ring|prime spectrum]] equipped with the Zariski topology (def. \ref{ZariskiClosedSubsetsInSpecR}).
Then the [[topological closure]] of a point $p \in PrimeIdl(R)$ is $V(p) \subset PrimeIdl(R)$ (def. \ref{ZariskiClosedSubsetsInSpecR}).
=--
+-- {: .proof}
###### Proof
By definition the topological closure of $\{p\}$ is
$$
Cl(\{p\})=\underset{ {I \in Idl(R) } \atop { p \in V(I) } }{\cap} V(I)
\,.
$$
Hence unwinding the definitions, we have the following sequence of logical equivalences:
$$
\begin{aligned}
& q \in Cl(\{q\})
\\
\Leftrightarrow\; &
q \in
\underset{ {I \in Idl(R)} \atop { p \in V(I) } }{\cap} V(I)
\\
\Leftrightarrow\;
&
\underset{ { I \in Idl(R) } \atop { I \subset p } }{\forall} (q \in V(I))
\\
\Leftrightarrow\;
&
\underset{ { I \in Idl(R) } \atop { I \subset p } }{\forall} (I \subset q)
\\
\Leftrightarrow\;
&
p \subset q
\\
\Leftrightarrow\;
&
q \in V(p)
\end{aligned}
$$
=--
Recall:
+-- {: .num_lemma #PrimeIdealTheorem}
###### Lemma
**([[prime ideal theorem]])
Assuming the [[axiom of choice]] or at least the [[ultrafilter principle]] then:
For $R$ a [[commutative ring]] and $I \subset R$ a [[proper ideal]], then $I$ is contained in some [[prime ideal]].
=--
The [[axiom of choice]] even implies that every proper ideal is contained in a [[maximal ideal]] (by [this prop.](maximal+ideal#EveryProperIdealisContainedInAMaximalOne)).
+-- {: .num_prop #MaximalIdealsAreClosedPoints}
###### Proposition
**([[maximal ideals]] are [[closed points]])
Let $R$ be a [[commutative ring]], consider the [[topological space]] $Spec(R) = (PrimeIdl(R),\tau_{Spec(R)})$, i.e. its [[prime spectrum of a commutative ring|prime spectrum]] equipped with the Zariski topology from def. \ref{ZariskiClosedSubsetsInSpecR}.
Then the [[maximal ideals]] inside the prime ideals constitute [[closed points]].
Assuming the [[axiom of choice]] or at least the [[ultrafilter principle]] then also the converse is true:
Then the inclusion of [[maximal ideals]] $\mathfrak{m} \in MaxIdl(R) \subset PrimeIdl(R)$ into all [[prime ideals]] is precisely the inclusion of the subset of [[closed points]] into all points of $Spec(R)$.
$$
ClosedPoints(Spec(R)) \simeq MaxIdl(R) \subset PrimeIdl(R)
\,.
$$
=--
+-- {: .proof}
###### Proof
By lemma \ref{ZariskiClorsuredOfPont} we have
$$
Cl(\{p\}) = V(p)
$$
and hence we need to show that
$$
\{\mathfrak{m}\} = V(\mathfrak{m})
$$
precisely if $\mathfrak{m}$ is maximal.
In one direction, assume that $\mathfrak{m}$ is maximal. By definition $V(\mathfrak{m})$ contains all the prime ideals $p$ such that $\mathfrak{m} \subset p$. That $\mathfrak{m}$ is maximal means that it is not contained in a larger proper ideal, in particular not in any larger prime ideal, and hence $V(\mathfrak{m}) = \{\mathfrak{m}\}$.
In the other direction, assume that $\mathfrak{m}$ is a prime ideal such that $V(\mathfrak{m}) = \{\mathfrak{m}\}$. By definition this means equivalently that the only prime ideal $p$ with $\mathfrak{m} \subset p$ is $\mathfrak{m}$ itself. We need to show that more generally $\mathfrak{m} \subset I$ for $I$ any [[proper ideal]] implies that $\mathfrak{m} = I$.
But the [[axiom of choice]]/[[ultrafilter principle]] imply the [[prime ideal theorem]] (lemma \ref{PrimeIdealTheorem}), which says that there is a prime ideal $p$ with $I \subset p$, hence a sequence of inclusions $\mathfrak{m} \subset I \subset p$. Since this implies $\mathfrak{m} \subset p$, we have $\mathfrak{m} = p$, hence $I = \mathfrak{m}$.
=--
#### Irreducible closed subsets as prime ideals
+-- {: .num_prop #ZariskiIrreducibleClosedSubsetsArePreciselyPrimeIdeals}
###### Proposition
**([[irreducible closed subsets]] correspond to [[prime ideals]])**
With [[excluded middle]] then:
Let $R$ be a [[commutative ring]], and let $\mathcal{F} \subset R$ be an ideal in $R$, hence $V(\mathcal{F}) \subset Spec(R)$ is a Zariski closed subset in the [[prime spectrum of a commutative ring|prime spectrum]] of $R$. Then the following are equivalent:
1. $V(\mathcal{F})$ is an [[irreducible closed subset]];
1. $\mathcal{F}$ is a [[prime ideal]].
=--
+-- {: .proof}
###### Proof
In one direction, assume that $V(\mathcal{F})$ is irreducible, and that $f,g \in R$ with $f \cdot g \in \mathcal{F}$. We need to show that then already $f \in \mathcal{F}$ or $g \in \mathcal{F}$.
To this end, first observe that
$$
V(\mathcal{F}) \subset V((f)) \cup V((g))
\,.
$$
This is because
$$
\begin{aligned}
& p \in V(\mathcal{F})
\\
\Leftrightarrow\;
&
\mathcal{F} \subset p
\\
\Rightarrow \;
&
f \cdot g \in p
\\
\Rightarrow\;
&
\left(
f \in p
\right)
\,\text{or}\,
\left(
g \in p
\right)
\\
\Leftrightarrow\;
&
\left(
p \in V(g)
\right)
\,\text{or}\,
\left(
p \in V(f)
\right)
\\
\Leftrightarrow\;
&
p \in V(f) \cup V(g)
\,,
\end{aligned}
$$
where the implication in the middle uses that $p$ is a prime ideal.
It follows that
$$
V(\mathcal{F})
\;=\;
\left(
V(f) \cap V(\mathcal{F})
\right)
\cup
\left(
V(g) \cap V(\mathcal{F})
\right)
\,.
$$
This is a decomposition of $V(\mathcal{F})$ as a union of closed subsets,
hence the assumption that $V(\mathcal{F})$ is irreducible implies that
$$
\begin{aligned}
&
\left(
V(\mathcal{F}) = V(f) \cap V(\mathcal{F})
\right)
\,\text{or}\,
\left(
V(\mathcal{F}) = V(g) \cap V(\mathcal{F})
\right)
\\
\Leftrightarrow
&
\left(
V(\mathcal{F})
\subset
V(f)
\right)
\,\text{or}\,
\left(
V(\mathcal{F}) \subset V(g)
\right)
\\
\Leftrightarrow\,
&
\left(
f \in \mathcal{F}
\right)
\,\text{or}\,
\left(
g \in \mathcal{F}
\right)
\,.
\end{aligned}
$$
Now for the converse. Assume that $\mathcal{F}$ is a prime ideal and that $V(\mathcal{F}) = V(\mathcal{F}_1) \cup V(\mathcal{F}_2)$. Observe (as in the [proof](#WellDefinedZariskiTopologyOnSpecRProof) of prop. \ref{WellDefinedZariskiTopologyOnSpecR}) that this means equivalently that $\mathcal{F} = \mathcal{F}_1 \cdot \mathcal{F}_2$.
We need to show
that then $V(\mathcal{F}) = V(\mathcal{F}_1)$ or that $V(\mathcal{F} = V(\mathcal{F}_2))$.
Suppose on the contrary that neither $\mathcal{F}_1$ nor $\mathcal{F}_2$ coincided with $\mathcal{F}$. This means that there were elements $f \in \mathcal{F}_1 \backslash \mathcal{F}$ and $g \in \mathcal{F}_2 \backslash \mathcal{F}$ such that still $f \cdot g \in \mathcal{F}$, in contradiction to the assumption. Hence we have a [[proof by contradiction]].
=--
As a corollary:
+-- {: .num_prop #ZariskiTopologyIsSober}
###### Proposition
**([[schemes are sober|Zariski topology on prime spectra is sober]])**
With [[excluded middle]] and [[axiom of choice]] (or at least the [[ultrafilter principle]]) then:
Let $R$ be a [[commutative ring]]. Then $Spec(R)$ (its [[prime spectrum of a commutative ring|prime spectr]] equipped with the Zariski topology of def. \ref{ZariskiClosedSubsetsInSpecR}) is a [[sober topological space]].
=--
+-- {: .proof}
###### Proof
We need to show that the function
$$
Cl(\{-\})
\;\colon\;
PrimeIdl(R)
\longrightarrow
IrrClSub(Spec(R))
$$
which sends a point to its topological closure, is a [[bijection]].
By lemma \ref{ZariskiClorsuredOfPont} this function is given by sending a [[prime ideal]] $p \in PrimeIdl(R)$ to the Zariski closed subset $V(p)$. That this is a bijection is the statement of prop. \ref{ZariskiIrreducibleClosedSubsetsArePreciselyPrimeIdeals}.
=--
### Examples
+-- {: .num_example #AffinSpaceAsPrimeSpectrum}
###### Example
**(affine space as prime spectrum)**
Reconsider the case where $R = k[X_1,\cdots, X_n]$
is a [[polynomial ring]], for $k$ a [[field]], as in the discussion of the naive affine space $k^n$
[above](#OnAffineSpace).
Observe that, by \ref{MaximalIdealsAreClosedPoints}, the [[closed points]] in the [[prime spectrum of a commutative ring|prime spectrum]] $Spec(k[X_1, \cdots, X_n])$ correspond to the [[maximal ideals]] in the [[polynomial ring]]. These are of the form
$$
(a_1, \cdots, a_n)
\coloneqq
\left(
(X_1 - a_1) \cdot (X_2 - a_2) \cdots (X_n - a_n)
\right)
$$
and hence are in bijection with the points of the naive affine space
$$
k^n \simeq MaxIdl(k[X_1, \cdots, X_n])
\,.
$$
There is however also [[prime ideals]] in $k[X_1, \cdots, X_n]$ which are not maximal. In particular there is the 0-ideal $(0)$.
=--
+-- {: .num_prop #SpecZ}
###### Proposition
**([[Spec(Z)]])**
Let $R = \mathbb{Z}$ be the [[commutative ring]] of [[integers]]. Consider the corresponding Zariski [[prime spectrum of a commutative ring|prime spectrum]] (prop. \ref{WellDefinedZariskiTopologyOnSpecR}) [[Spec(Z)]].
The [[prime ideals]] of the ring of integers are
1. the ideals $(p)$ generated by [[prime numbers]] $p$ (this special case is what motivates the terminology "prime ideal");
1. the ideal $(0) = \{0\}$.
$$
PrimeIdl(\mathbb{Z})
=
\left\{
0, \; 2, 3, 5, 7, 11, \cdots
\right\}
\,.
$$
All the prime ideals $p \geq 2$ are [[maximal ideals]]. Hence by prop. \ref{MaximalIdealsAreClosedPoints} these are [[closed points]] of $Spec(\mathbb{Z})$.
Only the prime ideal $(0)$ is not maximal, hence the point $(0)$ is not closed.
Its closure is the entire space
$$
Cl(\{0\}) = Spec(\mathbb{Z})
\,.
$$
To see this, notice that in fact $Spec(\mathbb{Z})$ is the only closed subset containing the point $(0)$. This is because
$$
\begin{aligned}
& (0) \in V(I)
\\
\Leftrightarrow\;
&
I \subset (0)
\\
\Leftrightarrow\;
&
I = (0)
\end{aligned}
$$
and $V(0) = Spec(\mathbb{Z})$, because
$$
(p \in V(0))
\Leftrightarrow
(0 \subset p)
\Leftrightarrow
true
\,.
$$
=--
## In terms of Galois connections
{#InTermsOfGaloisConnections}
We now discuss how all of the above constructions and statements, and a bit more, follows immediately as a special case of the general theory of what is called _[[Galois connections]]_ or _[[adjoint functors]] between [[posets]]_.
### Background on Galois connections
{#BackgroundOnGaloisConnections}
+-- {: .num_defn #GaloisConnection}
###### Definition
**([[Galois connection]] induced from a [[relation]])**
Consider two [[sets]] $X,Y \in Set$ and a [[relation]]
$$
E \hookrightarrow X \times Y
\,.
$$
Define two [[functions]] between their [[power sets]] $P(X), P(Y)$, as follows.
(In the following we write $E(x, y)$ to abbreviate the formula $(x, y) \in E$.)
1. Define
$$
V_E
\;\colon\;
P(X)
\longrightarrow
P(Y)
$$
by
$$
V_E(S)
\coloneqq
\left\{
y \in Y \vert \underset{x \in X}{\forall} \left( \left(x \in S\right) \Rightarrow E(x, y)
\right) \right\}
$$
1. Define
$$
I_E
\;\colon\;
P(Y)
\longrightarrow
P(X)
$$
by
$$
I_E(T) \coloneqq \left\{x \in X \vert \underset{y \in Y}{\forall} \left( \left(y \in T \right) \Rightarrow E(x, y) \right)\right\}
$$
=--
+-- {: .num_prop #GaloisConnectionAsAdjunction}
###### Proposition
The construction in def. \ref{GaloisConnection} has the following properties:
1. $V_E$ and $I_E$ are [[contravariant functor|contravariant]] order-preserving in that
1. if $S \subset S'$, then $V_E(S') \subset V_E(S)$;
1. if $T \subset T'$, then $I_E(T') \subset I_E(T)$
1. The _[[adjunction]] law_ holds:
$
\left(
T \subset V_E(S)
\right)
\,\LeftRightarrow\,
\left(
S \subset I_E(T)
\right)
$
which we denote by writing
$$
P(X)
\underoverset{\underset{V_E}{\longrightarrow}}{\overset{I_E}{\longleftarrow}}{\bot}
P(Y)^{op}
$$
1. both $V_E$ as well as $I_E$ take [[unions]] to [[intersections]].
=--
+-- {: .proof}
###### Proof
Regarding the first point: the larger $S$ is, the more conditions that are placed on $y$ in order to belong to $V_E(S)$, and so the smaller $V_E(S)$ will be.
Regarding the second point: This is because both these conditions are equivalent to the condition $S \times T \subset E$.
Regarding the third point: Observe that in a poset such as $P(Y)$, we have that $A = B$ iff for all $C$, $C \leq A$ iff $C \leq B$ (this is the [[Yoneda lemma]] applied to posets). It follows that
$$
\array{
T \subset V_E(\bigcup_{i \in I} S_i) & iff & \bigcup_{i: I} S_i \subset I_E(T) \\
& iff & \forall_{i: I} S_i \subset I_E(T) \\
& iff & \forall_{i: I} T \subset V_E(S_i) \\
& iff & T \subset \bigcap_{i: I} V_E(S_i)
}
$$
and we conclude $V_E(\bigcup_{i: I} S_i) = \bigcap_{i: I} V_E(S_i)$ by the [[Yoneda lemma]].
=--
+-- {: .num_prop #GaloisClosureOperator}
###### Proposition
**([[closure operators]] from [[Galois connection]])**
Given a [[Galois connection]] as in def. \ref{GaloisConnection}, consider the [[composition|composites]]
$$
I_E \circ V_E \;\colon\; P(X) \longrightarrow P(X)
$$
and
$$
V_E \circ I_E \;\colon\; P(Y) \longrightarrow P(Y)
\,.
$$
These satisfy:
1. For all $S \in P(X)$ then $S \subset I_E \circ V_E(S)$.
1. For all $S \in P(X)$ then $V_E \circ I_E \circ V_E (S) = V_E(S)$.
1. $I_E \circ V_E$ is [[idempotent]] and [[covariant functor|covariant]].
and
1. For all $T \in P(Y)$ then $T \subset V_E \circ I_E(T)$.
1. For all $T \in P(Y)$ then $I_E \circ V_E \circ I_E (T) = I_E(T)$.
1. $V_E \circ I_E$ is [[idempotent]] and [[covariant functor|covariant]].
This is summarized by saying that
$I_E \circ V_E$ and $V_E \circ I_E$ are _[[closure operators]]_ ([[idempotent monads]]).
=--
+-- {: .proof}
###### Proof
The first statement is immediate from the adjunction law (prop. \ref{GaloisConnectionAsAdjunction}).
Regarding the second statement:
This holds because applied to sets $S$ of the form $I_E(T)$, we see $I_E(T) \subset I_E \circ V_E \circ I_E(T)$. But applying the contravariant map $I_E$ to the inclusion $T \subset V_E \circ I_E(T)$, we also have $I_E \circ V_E \circ I_E(T) \subset I_E(T)$.
This directly implies that the function $I_E \circ V_E$. is idempotent, hence the third statement.
The argument for $V_E \circ I_E$ is directly analogous.
=--
In view of prop. \ref{GaloisClosureOperator} we say that:
+-- {: .num_defn #GaloisClosedElements}
###### Definition
**(closed elements)**
Given a [[Galois connection]] as in def. \ref{GaloisConnection}, then
1. $S \in P(X)$ is called *closed* if $I_E \circ V_E(S) = S$;
1. the _closure_ of $S \in P(X)$ is $Cl(S) \coloneqq I_E \circ V_E(S)$
and similarly
1. $T \in P(Y)$ is called *closed* if $V_E \circ I_E(T) = T$;
1. the _closure_ of $T \in P(Y)$ is $Cl(T) \coloneqq V_E \circ I_E(T)$.
=--
It follows from the properties of [[closure operators]], hence form prop. \ref{GaloisClosureOperator}:
+-- {: .num_prop #GaloisFixedPoints}
###### Proposition
**([[fixed point of an adjunction|fixed points]] of a [[Galois connection]])**
Given a [[Galois connection]] as in def. \ref{GaloisConnection},
then
1. the closed elements of $P(X)$ are precisely those in the [[image]] $im(I_E)$ of $I_E$;
1. the closed elements of $P(Y)$ are precisely those in the [[image]] $im(V_E)$ of $V_E$.
We says these are the _[[fixed point of an adjunction|fixed points]]_ of the Galois connection. Therefore the restriction of the Galois connection
$$
P(X)
\underoverset{\underset{V_E}{\longrightarrow}}{\overset{I_E}{\longleftarrow}}{\bot}
P(Y)^{op}
$$
to these fixed points yields an [[equivalence of categories|equivalence]]
$$
im(I_E)
\underoverset{\underset{V_E}{\longrightarrow}}{\overset{I_E}{\longleftarrow}}{\simeq}
im(V_E)^{op}
$$
now called a _[[Galois correspondence]]_.
=--
+-- {: .num_prop}
###### Proposition
Given a [[Galois connection]] as in def. \ref{GaloisConnection}, then the sets of closed elements according to def. \ref{GaloisClosedElements} are closed under forming [[intersections]].
=--
+-- {: .proof}
###### Proof
If $\{T_i \in P(Y)\}_{i: I}$ is a collection of elements closed under the operator $K = V_E \circ I_E$, then by the first item in prop. \ref{GaloisClosureOperator} it is automatic that $\bigcap_{i: I} T_i \subset K(\bigcap_{i: I} T_i)$, so it suffices to prove the reverse inclusion. But since $\bigcap_{i: I} T_i \subset T_i$ for all $i$ and $K$ is covariant and $T_i$ is closed, we have $K(\bigcap_{i: I} T_i) \subset K(T_i) \subset T_i$ for all $i$, and $K(\bigcap_{i: I} T_i) \subset \bigcap_{i: I} T_i$ follows.
=--
### Applied to affine space
{#GaloisConnectionAppliedToAffineSpace}
We now redo the discussion of the Zariski topology on the affine space $k^n$ from [above](#OnAffineSpace) as a special case of the general considerations of [[Galois connections]].
+-- {: .num_example #ZariskiClosedSubsetsInaffineViaGalois}
###### Example
**(Zariski closed subsets in affine space via Galois connection)**
Let $k$ be a [[field]] and let $n \in \mathbb{N}$, and write $k[X_1, \cdots, X_n]$ for the [[polynomial ring]] over $k$ in $n$ [[variables]]. Define a [[relation]]
$$
E \hookrightarrow k[x_1, \ldots, x_n] \times k^n
$$
by
$$
E(f, x)\coloneqq \left( f(x) = 0\right)
\,.
$$
By def. \ref{GaloisConnection} and prop. \ref{GaloisConnectionAsAdjunction} we obtain the corresponding [[Galois connection]] of the form
$$
P(k[X_1, \cdots, X_n])
\underoverset{\underset{V_E}{\longrightarrow}}{\overset{I_E}{\longleftarrow}}{\bot}
P(k^n)^{op}
$$
(where now $k[X_1, \cdots, X_n]$ and $k^n$ denote their underlying sets).
Here by def. \ref{GaloisConnection} the function
$$
V_E \;\colon\; P(k[x_1, \ldots, x_n]) \longrightarrow P(k^n)
$$
sends a set $\mathcal{F}$ of [[polynomials]] to its corresponding *[[variety]]*,
$$
V_E(\mathcal{F}) = \{\vec x \in k^n \,\vert\, \forall_{f \in k[x_1, \ldots, x_n]} \; (f \in \mathcal{F}) \Rightarrow (f(x) = 0)\}
\,.
$$
These are just the Zariski closed subsets from def. \ref{ZariskiOpenSubsetsOnAffineSpace}.
In the other direction,
$$
I_E \;\colon\; P(k^n) \longrightarrow P(k[x_1, \ldots, x_n])
$$
sends a set of points $T \subseteq k^n$ to its corresponding *[[vanishing ideal]]*
$$
I_E(T) = \{f \in k[x_1, \ldots, x_n] \,\vert\, \forall_{x: k^n} \; x \in T \Rightarrow f(x) = 0\}
$$
which we considered earlier in def. \ref{VanishingIdeal}.
=--
We may now use the abstract theory of Galois connections to verify that Zariski closed subsets form a [[topological space|topology]]:
+-- {: .num_prop #ZariskiTopologyOnAffineSpaceViaGaloisConnectionWellDefined}
###### Proposition
**(Zariski topology is well defined)**
Using [[excluded middle]], then:
The set of Zariski closed subsets of $k^n$ from example \ref{ZariskiClosedSubsetsInaffineViaGalois} constitutes a [[topological space|topology]] in that it is closed under
1. arbitrary intersections;
1. finite untions.
=--
+-- {: .proof}
###### Proof
Regarding the first point: From prop. \ref{GaloisConnectionAsAdjunction} we know that $V_E$ takes unions to intersections, hence that
$$
\underset{i \in I}{\cap} V_E(\mathcal{F}_i)
\;=\;
V_E\left(
\underset{i \in I}{\cup} \mathcal{F}_i
\right)
\,.
$$
Regarding the second point, we exploit the [[commutative ring]] structure of $k[x_1, \ldots, x_n]$. It is sufficient to show that the set of Zariski closed sets is closed under the empty union and under binary unions.
The empty union is the entire space $k^n$, which is $V(1)$ (the variety associated with the constant polynomial $1$),
Hence it only remains to see closure under binary unions.
To this end, recall from prop. \ref{GaloisClosureOperator} that we may replace $\mathcal{F}$ with the corresponding ideal
$$
I \coloneqq I_E \circ V_E(\mathcal{F})
$$
without changing the variety:
$$
V_E(I) = V_E(\mathcal{F})
\,.
$$
With this it is sufficient to show that
$$
V_E(I) \cup V_E(I') = V(I \cdot I')
$$
where $I \cdot I'$ is the ideal consisting of finite sums of elements of the form $f g$ with $f \in I$ and $g \in I'$.
We conclude by proving this statement:
Applying the contravariant operator $V_E$ to the inclusions $I \cdot I' \subseteq I$ and $I \cdot I' subseteq I'$ (which are clear since $I, I'$ are ideals), we derive $V_E(I) \subseteq V_E(I \cdot I')$ and $V_E(I') \subseteq V(I \cdot I')$, so the inclusion $V_E(I) \cup V_E(I') \subseteq V(I \cdot I')$ is automatic.
In the other direction, to prove $V(I \cdot I') \subseteq V_E(I) \cup V(I')$, suppose $x \in V(I \cdot I')$ and that $x$ *doesn't* belong to $V(I)$. Then $f(x) \neq 0$ for some $f \in I$. For every $g \in I'$, we have $f(x)g(x) = (f \cdot g)(x) = 0$ since $f \cdot g \in I \cdot I'$ and $x \in V_E(I \cdot I')$. Now divide by $f(x)$ to get $g(x) = 0$ for every $g \in I'$, so that $x \in V_E(I')$.
=--
+-- {: .num_example #ZariskiTopologyOnMaximalIdealsOfPolynomialRingViaGaloisConnection}
###### Example
Let $k$ be a [[field]], let $n \in \mathbb{N}$ and write $k[X_1, \cdots, X_n]$ for the [[polynomial ring]] over $k$ in $n$ [[variables]], and $MaxIdl(k[X_1, \cdots, X_n])$ for the [[set]] of [[maximal ideals]] in this ring.
Define then a [[relation]]
$$
E \hookrightarrow k[x_1, \ldots, x_n] \times MaxIdeal(k[x_1, \ldots, x_n])
$$
by
$$
E(f, M) \Leftrightarrow (f \in M)
\,.
$$
For a subset $T \subseteq MaxIdl(k[x_1, \ldots, x_n])$ we calculate
$$
I_E(T) = \{f \in k[x_1, \ldots, x_n]: \forall_{\mathfrak{m} \in MaxIdl} M \in S \Rightarrow f \in \mathfrak{m}\} = \bigcap_{\mathfrak{m} \in S} \mathfrak{m}
$$
which is an ideal, since the intersection of any collection of ideals is again an ideal. (However, not all ideals are given as intersections of maximal ideals, a point to which we will return in a moment.)
=--
+-- {: .num_remark}
###### Remark
This is a slight generalization of example \ref{ZariskiClosedSubsetsInaffineViaGalois} since each point $a = (a_1, \ldots, a_n)$ induces a maximal ideal
$$
\mathfrak{m}_a \coloneqq \langle x_1 - a_1, \ldots, x_n - a_n \rangle
\,,
$$
i.e. the [[kernel]] of the function
$$
\array{
k[x_1, \ldots, x_n] &\longrightarrow& k
\\
f &\mapsto& f(a)
}
$$
which evaluates polynomials $f$ at the point $a$, where we have $f(a) = 0$ iff $f \in \mathfrak{m}_a$.
Of course it need not be the case that all maximal ideals $\mathfrak{m}$ are given by points in this way; for example, the ideal $(x^2 + 1)$ is maximal in $\mathbb{R}[x]$ but is not given by evaluation at a point because $x^2 + 1$ does not vanish at any real point. However, if the [[ground field]] $k$ is [[algebraically closed field|algebraically closed]], then every maximal ideal of $k[x_1, \ldots, x_n]$ is given by evaluation at a point $a = (a_1, \ldots, a_n)$. This result is not completely obvious; it is sometimes called the _weak [[Nullstellensatz]]_.
=--
+-- {: .num_prop}
###### Proposition
The set $S \subseteq k^n$ that are closed under the operator $V_E \circ I_E: P(k^n) \to P(k^n)$ in example \ref{ZariskiTopologyOnMaximalIdealsOfPolynomialRingViaGaloisConnection} form a [[topology]].
=--
+-- {: .proof}
###### Proof
The proof is virtually the same as in the proof of prop. \ref{ZariskiTopologyOnAffineSpaceViaGaloisConnectionWellDefined}: they are closed under arbitrary intersections by our earlier generalities, and they are closed under finite unions by the similar reasoning: $V_E(S) = V_E(I)$ where $I = I_E \circ V_E(S)$ is an ideal, so there is no loss of generality in considering $V_E(I)$ for ideals $I$, and $V_E(I) \cup V_E(I') = V_E(I \cdot I')$. If $\mathfrak{m} \in V_E(I \cdot I')$ (meaning $I \cdot I' \subseteq M$) but $\mathfrak{m}$ *doesn't* belong to $V_E(I)$, i.e., $f \notin \mathfrak{m}$ for some $f \in I$, then for every $g \in I'$ we have $f \cdot g \in \mathfrak{m}$. Taking the [[quotient]] map $\pi: R \to R/\mathfrak{m}$ to the field $R/\mathfrak{m}$, we have $\pi(f \cdot g) = \pi(f)\cdot \pi(g) = 0$, and since $\pi(f) \neq 0$ we have $\pi(g) = 0$ for every $g \in I'$, hence $\mathfrak{m} \in V_E(I')$.
=--
Thus the fixed elements of $V_E \circ I_E$ on one side of the Galois correspondence are the closed sets of a topology. The fixed elements of $I_E \circ V_E$ on the other side are a matter of interest; in the case where $k$ is [[algebraically closed field|algebraically closed]], they are the *[[radical ideals]]* of $k[X_1, \ldots, X_n]$ according to the "strong" [[Nullstellensatz]].
### Applied to affine schemes
{#GaloisAppliedToAffineSchemes}
We now redo the discussion of the Zariski topology on the [[prime spectrum of a commutative ring]] from [above](#OnAffineVarieties) as a special case of the general considerations of [[Galois connections]].
## Related concepts
* [[Zariski site]]
* [[schemes are sober]]
## References
Lecture notes include
* Jim Carrell, _Zariski topology_ [pdf](https://personal.math.ubc.ca/~carrell/423.pdf)
See also
* Wikipedia, _[Zariski topology](https://en.wikipedia.org/wiki/Zariski_topology)_
[[!redirects Zariski topologies]]
|
Zdenek Dolezal | https://ncatlab.org/nlab/source/Zdenek+Dolezal |
* [personal page](http://ipnp.cz/~dolezal/)
## Selected writings
On [[B meson]]-[[decays]] and [[flavour anomalies]]:
* [[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))
category: people
|
zekeriya arvasi | https://ncatlab.org/nlab/source/zekeriya+arvasi | Zekeriya Arvasi is a Turkish mathematician with expertise in category theory, especially simplicial ojects, crossed modules, algebraic models for homotopy connected types. In homotopy theory he collaborated with E. Ulualan.
[Home Page](http://fef.ogu.edu.tr/zarvasi/) |
Zena M. Ariola | https://ncatlab.org/nlab/source/Zena+M.+Ariola |
* [personal page](http://ix.cs.uoregon.edu/~ariola/)
* [institute page](https://cas.uoregon.edu/directory/profiles/all/ariola)
## Selected writings
Introduction to [[proof theory]] via [[natural deduction]], [[sequent calculus]]:
* [[Paul Downen]], [[Zena M. Ariola]], *A tutorial on computational classical logic and the sequent calculus*, Journal of Functional Programming **28** (2018) e3 [[doi:10.1017/S0956796818000023](https://doi.org/10.1017/S0956796818000023), [pdf](https://www.pauldownen.com/publications/sequent-intro.pdf)]
category: people |
Zeno | https://ncatlab.org/nlab/source/Zeno |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
=--
=--
An ancient [[philosophy|philosopher]] known from [[Plato]]'s [[Parmenides dialogue]], renowned for his [[paradoxes]] (which [[Hegel]] claims to be the origin of [[dialectics]] ) such as [[Zeno's paradox of motion]].
## As seen by Hegel
[[Georg Hegel]], in _[Lectures on the History of Philosophy -- Zeno](Lectures+on+the+History+of+Philosophy)_ writes this:
> {#HegelHistory1} What specially characterizes Zeno is the [[dialectic]] which, properly speaking, begins with him;
> he is the master of the Eleatic school in whom its pure thought arrives at the movement of the Notion in itself and becomes the pure soul of science. That is to say, in the Eleatics hitherto considered, we only have the proposition: "The nothing has no reality and is not at all, and thus what is called origin and decease disappears." With Zeno, on the contrary, we certainly see just such an assertion of the one and removal of what contradicts it, but we also see that this assertion is not made the starting point; for reason begins by calmly demonstrating in that which is established as existent, its negation. Parmenides asserts that "The all is immutable, for, in change, the non-being of that which is would be asserted, but Being only is; in saying that "non-being is, the subject and the predicate contradict themselves." Zeno, on the other hand, says: "Assert your change; in it as change there is the negation to it, or it is nothing." To the former change existed as motion, definite and complete. Zeno protested against motion as such, or pure motion.
> Pure Being is not motion; it is rather the negation of motion." We find it specially interesting that there is in Zeno the higher consciousness, the consciousness that when one determination is denied, this negation is itself again a determination, and then in the absolute negation not one determination, but both the opposites must be negated. Zeno anticipated this, and because he foresaw that Being is the opposite of nothing, he denied of the One what must be said of the nothing. But the same thing must occur with all the rest. We find this higher dialectic in Plato's Parmenides; here it only breaks forth in respect to some determinations, and not to the determination of the One and of Being. The higher consciousness is the consciousness of the nullity of Being as of what is determined as against the nothing, partly found in Heraclitus and then in the Sophists; with them it never has any truth, it has no existence in itself, but is only the for-another, or the assurance of the individual consciousness, and assurance as refutation, i.e. the negative side, of dialectic.
> According to Diogenes Laërtius, (IX. 25) Zeno was like wise an Eleat; he is the youngest, and lived most in company with Parmenides. The latter became very fond of him and adopted him as a son; his own father was called Telentagoras. Not in his State alone was his conduct held in high respect, for his fame was universal, and he was esteemed particularly as a teacher. Plato mentions that men came to him from Athens and other places, in order to profit from his learning. Proud self-sufficiency is ascribed to him by Diogenes (IX. 28) because he — with the exception of a journey made to Athens — continued to reside in Elea, and did not stay a longer time in the great, mighty Athens, and there attain to fame. In very various narratives his death was made for ever celebrated for the strength of his mind evinced in it; it was said that he freed a State (whether his own home at Elea or in Sicily, is not known) from its Tyrant (the name is given differently, but an exact historical account has not been recorded) in the following way, and by the sacrifice of his life. He entered into a plot to overthrow the Tyrant, but this was betrayed. When the Tyrant now, in face of the people, caused him to be tortured in every possible way to get from him an avowal of his confederates, and when he questioned him about the enemies of the State, Zeno first named to the Tyrant all his friends as participators in the plot, and then spoke of the Tyrant himself as the pest of the State. The powerful remonstrances or the horrible tortures and death of Zeno aroused the citizens, inspired them with courage to fall upon the Tyrant, kill him, and liberate themselves. The manner of the end, and his violent and furious state of mind, is very variously depicted. He is said to have pretended to wish to say something into the Tyrant's ear, and then to have bitten his ear, and thus held him fast until lie was slain by the others. Others say that he seized him by the nose between his teeth; others that as on his reply great tortures were applied, he bit off his tongue and spat it into the Tyrant's face, to show him that he could get nothing from him, and that he then was pounded in a mortar.
> It has just been noticed that Zeno had the very important character of being the originator of the true objective dialectic. Xenophanes, Parmenides, and Melisibus, start with the proposition: "Nothing is nothing; the nothing does not exist at all, or the like is real existence," that is, they make one of the opposed predicates to be existence. Now when they encounter the opposite in a determination, they demolish this determination, but it is only demolished through another, through my assertion, through the distinction that I form, by which one side is made to be the true, and the other the null. We have proceeded from a definite proposition; the nullity of the opposite does not appear in itself; it is not that it abrogates itself, i.e. that it contains a contradiction in itself. For instance, I assert of something that it is the null; then I show this by hypothesis in motion, and it follows that it is the null. But another consciousness does not assert this I declare one thing to be directly true; another has the right of asserting something else as directly true, that is to say, motion. Similarly what seems to be the case when one philosophic system contradicts another, is that the first is pre-established, and that men starting from this point of view, combat the other. The matter is thus easily settled by saying: "The other has no truth, because it does not agree with me," and the other bas the right to say the same. It does not help if I prove my system or my proposition and then conclude that thus the opposite is false; to this other proposition the first always seems to be foreign and external. Falsity must not be demonstrated through another, and as untrue because the opposite is true, but in itself; we find this rational perception in Zeno.
> In Plato's [[Parmenides]] (pp. 127, 128, Steph., pp. 6, 7, Bekk.) this dialectic is very well described, for Plato makes Socrates say of it: "Zeno in his writings asserts fundamentally the same as does Parmenides, that All is One, but he would feign delude us into believing that he was telling something new. Parmenides thus shows in his poems that All is One; Zeno, on the contrary, shows that the Many cannot be." Zeno replies, that "He wrote thus really against those who try to make Parmenides' position ridiculous, for they try to show what absurdities and self-contradictions can be derived from his statements; he thus combats those who deduce Being from the many, in order to show that far more absurdities arise from this than from the statements of Parmenides." That is the special aim of objective dialectic, in which we no longer maintain simple thought for itself, but see the battle fought with new vigour within the enemy's camp. Dialectic has in Zeno this negative side, but it bas also to be considered from its positive side.
> According to the ordinary ideas of science, where propositions result from proof, proof is the movement of intelligence, a connection brought about by mediation. Dialectic is either (a) external dialectic, in which this movement is different from the comprehension of the movement, or (b) not a movement of our intelligence only, but what proceeds from the nature of the thing itself, i.e. from the pure Notion of the content. The former is a manner of regarding. objects in such a way that reasons are revealed and new light thrown, by means of which all that was supposed to be firmly fixed, is made to totter; there may be reasons which are altogether external too, and we shall speak further of this dialectic when dealing with the Sophists. The other dialectic, however, is the immanent contemplation of the object; it is taken for itself, without previous hypothesis, idea or obligation, not under any outward conditions, laws or causes; we have to put ourselves right into the thing, to consider the object in itself, and to take it in the determinations which it has. In regarding it thus, it shows from itself that it contains opposed determinations, and thus breaks up; this dialectic we more especially find in the ancients. The subjective dialectic, which reasons from external grounds, is moderate, for it grants that: "In the right there is what is not right, and in the false the true." True dialectic leaves nothing whatever to its object, as if the latter were deficient on one side only; for it disintegrates itself in the entirety of its nature. The result of this dialectic is null, the negative; the affirmative in it does not yet appear. This true dialectic may be associated with the work of the Eleatics. But in their case the real meaning and quality of philosophic understanding was not great, for they got no further than the fact that through contradiction the object is a nothing.
> Zeno's dialectic of matter has not been refuted to the present day; even now we have not got beyond it, and the matter is left in uncertainty. Simplicius, writing on the Physics of [[Aristotle]] (p. 30), says: "Zeno proves that if the many is, it must be great and small; if great, the many must be infinite in number" (it must have gone beyond the manifold, as indifferent limit, into the infinite; but what is infinite is no longer large. and no longer many, for it is the negation of the many). "If small, it must be so small as to have no size," like atoms. "Here he shows that what has neither size, thickness nor mass, cannot be. For if it were added to another, it would not cause its increase; were it, that is to say, to have no size and be added thereto, it could not supplement the size of the other and consequently that which is added is nothing. Similarly were it taken away, the other would not be made less, and thus it is nothing. If what has being is, each existence necessarily has size and thickness, is outside of one another, and one is separate from the other; the same applies to all else (peri tou proucontoς), for it, too, has size, and in it there is what mutually differs (proexei autou ti). But it is the same thing to say something once and to say it over and over again; in it nothing can be a last, nor will there not be another to the other. Thus if many are, they are small and great; small, so that they have no size; great, so that they are infinite."
> Aristotle (Phys. VI. 9) explains this dialectic further; Zeno's treatment of motion was above all objectively dialectical. But the particulars which we find in the [[Parmenides]] of [[Plato]] are not his. For Zeno's consciousness we see simple unmoved thought disappear, but become thinking movement; in that he combats sensuous movement, he concedes it. The reason that dialectic first fell on movement is that the dialectic is itself this movement, or movement itself the dialectic of all that is. The thing, as self-moving, has its dialectic in itself, and movement is the becoming another, self-abrogation. If Aristotle says that Zeno denied movement because it contains an inner contradiction, it is not to be understood to mean that movement did. not exist at all. The point is not that there is movement and that this phenomenon exists; the fact that there is movement is as sensuously certain as that there are elephants; it is not in this sense that Zeno meant to deny movement. The point in question concerns its truth. Movement, however, is held to be untrue, because the conception of it involves a contradiction; by that he meant to say that no true Being eau be predicated of it.
> Zeno's utterances are to be looked at from this point of view, not as being directed against the reality of motion, as would at first appear, but as pointing out how movement must necessarily be determined, and showing the course which must be taken. Zeno now brings forward four different arguments against motion; the proofs rest on the infinite divisibility of space and time.
> (a) This is his first form of argument: — "Movement has no truth, because what is in motion must first reach the middle of the space before arriving at the end." Aristotle expresses this thus shortly, because he had earlier treated of and worked out the subject at length. This is to be taken as indicating generally that the continuity of space is pre-supposed. What moves itself must reach a certain, end, this way is a whole. In order to traverse the whole, what is in motion must first pass over the half, and now the end of this half is considered as being the end; but this half of space is again a whole, that which also has a half, and the half of this half must first have been reached, and so on into infinity. Zeno here arrives at the infinite divisibility of space; because space and time are absolutely continuous, there is no point at which the division can stop. Every dimension (and every time and space always have a dimension) is again divisible into two halves, which must be measured off; and however small a space we have, the same conditions reappear. Movement would be the act of passing through these infinite moments, and would therefore never end 4 thus what is in motion cannot reach its end. It is known how Diogenes of Sinope, the Cynic, quite simply refuted these arguments against movement; without speaking he rose and walked about, contradicting them by action. But when reasons are disputed, the only valid refutation is one derived from reasons; men have not merely to satisfy themselves by sensuous assurance, but also to understand. To refute objections is to prove their non-existence, as when they are made to fall away and can hence be adduced no longer; but it is necessary to think of motion as Zeno thought of it, and yet to carry this theory of motion further still.
> We have here the spurious infinite or pure appearance, whose simple principle Philosophy demonstrates as universal Notion, for the first time making its appearance as developed in its contradiction; in the history of Philosophy a consciousness of this contradiction is also attained. Movement, this pure phenomenon, appears as something thought and shown forth in its real being — that is, in its distinction of pure self-identity and pure negativity, the point as distinguished from continuity. To us there is no contradiction in the idea that the here of space and the now of time are considered as a continuity and length; but their Notion is self-contradictory. Self-identity or continuity is absolute cohesion, the destruction of all difference, of all negation, of being for self; the point, on the contrary, is pure being-for-self, absolute self-distinction and the destruction of all identity and all connection with what is different. Both of these, however, are, in space and time, placed in one; space and time are thus the contradiction; it is necessary, first of all, to show the contradiction in movement, for in movement that which is opposed is, to ordinary conceptions, inevitably manifested. Movement is just the reality of time and space, and because this appears and is made manifest, the apparent contradiction is demonstrated, a and it is this contradiction that Zeno notices. The limitation of bisection which is involved in the continuity of space, is not absolute limitation, for that which is limited is again continuity; however, this continuity is again not absolute, for the opposite has to be exhibited in it, the limitation of bisection; but the limitation of continuity is still not thereby established, the half is still continuous, and so on into infinity. In that we say "into infinity," we place before ourselves a beyond, outside of the ordinary conception, which cannot reach so far. It is certainly an endless going forth, but in the Notion it is present, it is a progression from one opposed determination to others, from continuity to negativity, from negativity to continuity; but both of these are before us. Of these moments one in the process may be called the true one; Zeno first asserts continuous progression in such a way that no limited space can be arrived at as ultimate, or Zeno upholds progression in this limitation.
> The general explanation which Aristotle gives to this contradiction, is that space and time are not infinitely divided, but are only divisible. But it now appears that, because they are divisible — that is, in potentiality — they must actually be infinitely divided, for else they could not be divided into infinity. That is the general answer of the ordinary man in endeavouring to refute the explanation of Aristotle. Bayle (Tom. IV. art. Zénon, not. E.) hence says of Aristotle's answer that it is "pitoyable: C'est se moquer du monde que de se servir de cette doctrine; car si Ia matière est divisible à l'infini, elle contient un nombre infini de parties. Ce n'est done point un infini en puissance, c'est un infini, qui existe réellement, actuellement. Mais quandmême on accorderait cet infini en puissance, qui deviendrait un infini par Ia division actuelle de ses parties, on ne perdrait pas ses avantages; car le mouvement est une chose, qui a la même vertu, que la division. Il touche une partie de l'espace sans toucher l'autre, et il les touche toutes les unes après les autres. N'est-ce pas les distinguer actuellement? N'est-ce pas faire ce que ferait un géomètre sur une table en tirant des lignes, qui désignassent tous les demiponces? II ne brise pas Ia table en demi-pouces, mais il y fait néanmoins une division, qui marque Ia distinction actuelle des parties; et ie ne crois pas qu'Aristote ent voulu nier, que si l'on tirait une infinité de lignes sur un pouce de matière, on n'y introduisit une division, qui reduirait en infini actuel ce qui n'etait selon lui qu'un infini virtuel." This si is good! Divisibility is, as potentiality, the universal; there is continuity as well as negativity or the point posited in it — but posited as moment, and not as existent in and for itself. I can divide matter into infinitude, but I only can do so; I do not really divide it into infinitude. This is the infinite, that no one of its moments has reality. It never does happen that, in itself, one or other — that absolute limitation or absolute continuity — actually comes into existence in such a way that the other moment disappears. There are two absolute opposites, but they are moments, i.e. in the simple Notion or in the universal, in thought, if you will; for in thought, in ordinary conception, what is set forth both is and is not at the same time. What is represented either as such. or as an image of the conception, is not a thing; it has no Being, and yet it is not nothing.
> Space and time furthermore, as quantum, form a limited extension, and thus can be measured off; just as I do not actually divide space, neither does the body which is in motion. The partition of space as divided, is not absolute discontinuity [Punktualität], nor is pure continuity the undivided and indivisible; likewise time is not pure negativity or discontinuity, but also continuity. Both are manifested in motion, in which the Notions have their reality for ordinary conception — pure negativity as time, continuity as space. Motion itself is just this actual unity in the opposition, and the sequence of both moments in this unity. To comprehend motion is to express its essence in the form of Notion, i.e., as unity of negativity and continuity; but in them neither continuity nor discreteness can be exhibited as the true existence. If we represent space or time to ourselves as infinitely divided, we have an infinitude of points, but continuity is present therein as a space which comprehends them; as Notion, however, continuity is the fact that all these are alike, and thus in reality they do not appear one out of the other like points. But both these moments make their appearance as existent; if they are manifested indifferently, their Notion is no longer posited, but their existence. In.them as existent, negativity is a limited size, and they exist as limited space and time; actual motion is progression through a limited space — and a limited time and not through infinite space and infinite time.
> That what is in motion must reach the half is the assertion of continuity, i.e. the possibility of division as mere possibility; it is thus always possible in every space, however small. It is said that it is plain that the half must be reached, but in so saying, everything is allowed, including the fact that it never will be reached; for to say so in one case, is the same as saying it an infinite number of times. We mean, on the contrary, that in a larger space the half can be allowed, but we conceive that we must somewhere attain to a space so small that no halving is possible, or an indivisible, non-continuous space which is no space. This, however, is false, for continuity is a necessary determination; there is undoubtedly a smallest in space, i.e. a negation of continuity, but the negation is something quite abstract. Abstract adherence to the subdivision indicated, that is, to continuous bisection into infinitude, is likewise false, for in the conception of a half, the interruption of continuity is involved. We must say that there is no half of space, for space is continuous; a piece of wood may be broken into two halves, but not space, and space only exists in movement. It might equally be said that space consists of an endless number of points, i.e. of infinitely many limits and thus cannot be traversed. Men think themselves able to go from one indivisible point to another, but they do not thereby get any further, for of these there is an unlimited number. Continuity is split up into its opposite, a number which is indefinite; that is to say, if continuity is not admitted, there is no motion. It is false to assert that it is possible when one is reached, or that which is not continuous; for motion is connection. Thus when it was said that continuity is the presupposed possibility of infinite division, continuity is only the hypothesis; but what is exhibited in this continuity is the being of infinitely many, abstractly absolute limits.
> (b) The second proof, which is also the presupposition of continuity and the manifestation of division, is called "Achilles, the Swift." The ancients loved to clothe difficulties in sensuous representations. Of two bodies moving in one direction, one of which is in front and the other following at a fixed distance and moving quicker than the first, we know that the second will overtake the first. But Zeno says, "The slower can never be overtaken by the quicker." And he proves it thus: "The second one requires a certain space of time to reach the place from which the one pursued started at the beginning of the given period." Thus during the time in which the second reached the point where the first was, the latter went over a new space which the second has again to pass through in a part of this period; and in this way it goes into infinity.
c d e f g
B A
> B, for instance, traverses two miles (c d) in an hour, A in the same time, one mile (d e); if they are two miles (c d) removed from one another, B has in one hour come to where A was at the beginning of the hour. While B, in the next half hour, goes over the distance crossed by A of one mile (d e), A has got half a mile (e f) further, and so on into infinity. Quicker motion does not help the second body at all in passing over the interval of space by which he is behind: the time which he requires, the slower body always has at its avail in order to accomplish some, although an ever shorter advance; and this, because of the continual division, never quite disappears.
> Aristotle, in speaking of this, puts it shortly thus: "This proof asserts the same endless divisibility, but it is untrue, for the quick will overtake the slow body if the limits to be traversed be granted to it." This answer is correct and contains all that can be said; that is, there are in this representation two periods of time and two distances, which are separated from one another, i.e. they are limited in relation to one another; when, on the contrary, we admit that time and space are continuous, so that two periods of time or points of space are related to one another as continuous, they are, while being two, not two, but identical. In ordinary language we solve the matter in the easiest, way, for we say: "Because the second is quicker, it covers a greater distance in the same time as the slow; it can therefore come to the place from which the first started and get further still." After B, at the end of the first hour, arrives at d and A at e, A in one and the same period, that is, in the second hour, goes over the distance e g, and B the distance d g. But this period of time which should be one, is divisible into that in which B accomplishes d e and that in which B passes through e g. A has a start of the first, by which it gets over the distance e f, so that A is at f at the same period as B is at e. The limitation which, according to Aristotle, is to be overcome, which must be penetrated, is thus that of time; since it is continuous, it must, for the solution of the difficulty, be said that what is divisible into two spaces of time is to be conceived of as one, in which B gets from d to e and from e to g, while A passes over the distance c g. In motion two periods, as well as two points in space, are indeed one.
> If we wish to make motion clear to ourselves, we say that,,the body is in one place and then it goes to another; because it moves, it is no longer in the first, but yet not in the second; were it in either it would be at rest. Where then is it? If we say that it is between both, this is to convey nothing at all, for were it between both, it would be in a place, and this presents the same difficulty. But movement means to be in this place and not to be in it, and thus to be in both alike; this is the continuity of space and time which first makes motion possible. Zeno, in the deduction made by him, brought both these points into forcible opposition. The discretion of space and time we also uphold, but there must also be granted to them the over-stepping of limits, i.e. the exhibition of limits as not being, or as being divided periods of time, which are also not divided. In our ordinary ideas we find the same determinations as those on which the dialectic of Zeno rests; we arrive at saying, though unwillingly, that in one period two distances of space are traversed, but we do not say that the quicker comprehends two moments of time in one; for that we fix a definite space. But in order that the slower may lose its precedence, it must be said that it loses its advantage of a moment of time, and indirectly the moment of space.
> Zeno makes limit. division, the moment of discretion in space and time, the only element which is enforced in the whole of his conclusions, and hence results the contradiction. The difficulty is to overcome thought. for what makes the difficulty is always thought alone, since it keeps apart the moments of an object which in their separation are really united. It brought about the Fall, for man ate of the tree of the knowledge of good and evil; but it also remedies these evils.
> (c) The third form, according to Aristotle, is as follows: — Zeno says. "I The flying arrow rests, and for the reason that what is in motion is always in the self-same Now and the self-same Here, in the indistinguishable;" it is here and here and here. It can be said of the arrow that it is always the same, for it is always in the same space and the same time; it does not get beyond its space. does not take in another, that is, a greater or smaller space. That, however, is what we call rest and not motion. In the Here and Now, the becoming "other" is abrogated, limitation indeed being established, but only as moment; since in the Here and Now as such, there is no difference, continuity is here made to prevail against the mere belief in diversity. Each place is a different place, and thus the same; true, objective difference does not come forth in these sensuous relations, but in the spiritual.
> This is also apparent in mechanics; of two bodies the question as to which moves presents itself before us. It requires more than two places — three at least — to determine which of them moves. But it is correct to say this, that motion is plainly relative; whether in absolute space the eye, for instance, rests, or whether it moves, is all the same. Or, according to a proposition brought forward by Newton, if two bodies move round. one another in a circle, it may be asked whether the one rests or both move. Newton tries to decide this by means of an external circumstance, the strain on the string. When I walk on a ship in a direction opposed to the motion of the ship, this is in relation to the ship, motion, and in relation to all else, rest.
> In both the first proofs, continuity in progression has the predominance; there is no absolute limit, but an overstepping of all limits. Here the opposite is established; absolute limitation, the interruption of continuity, without however passing into something else; while discretion is pre-supposed, continuity is maintained. Aristotle says of this proof: "It arises from the fact that it is taken for granted that time consists of the Now; for if this is not conceded, the conclusions will not follow."
> (d) "The fourth proof," Aristotle continues, "is derived from similar bodies which move in opposite directions in the space beside a similar body, and with equal velocity, one from one end of the space, the other from the middle. It necessarily results from this that half the time is equal to the double of it. The fallacy rests in this, that Zeno supposes that what is beside the moving body, and what is beside the body at rest, move through an equal distance in equal time with equal velocity, which, however, is untrue."
1
E|-|-|-|-|F
k i m
C|-|-|-|-|D
g n h
A|-|-|-|-|B
> In a definite space such as a table (A B) let us suppose two bodies of equal length with it and with one another, one of which (C D) lies with one end (C) on the middle (g) of the table, and the other (B F), being in the same direction, has the point (B) only touching the end of the table (h); and supposing they move in opposite directions, and the former (C D) reaches in an hour the end (h) of the table; we have the result ensuing that the one (E F) passes in the half of the time through the same space (1 k) which the other does in the double (g h); hence the half is equal to the double. That is to say, this second passes (let us say, in the point 1) by the whole of the first C D. In the first half-hour 1 goes from m to i, while k only goes from g to n.
1
E|-|-|-|-|F
k o i m
C|-|-|-|-|D
g n h
A|-|-|-|-|B
>In the second half-hour 1 goes past o to k, and altogether passes from m to k, or the double of the distance.
1
E|-|-|-|-|F
k o i m
C|-|-|-|-|D
g n h
A|-|-|-|-|B
> This fourth form deals with the contradiction presented in opposite motion; that which is common is given entirely to one body, while it only does part for itself. Here the distance travelled by one body is the sum of the distance travelled by both, just as when I go two feet east, and from the same point another goes two feet west, we are four feet removed from one another; in the distance moved both are positive, and hence have to be added together. Or if I have gone two feet forwards and two feet backwards, although I have walked four feet, I have not moved from the spot; the motion is then nil, for by going forwards and backwards an opposition ensues which annuls itself.
> This is the dialectic of Zeno; he had a knowledge of the determinations which our ideas of space and time contain, and showed in them their contradiction; Kant's antinomies do no more than Zeno did here. The general result of the Eleatic dialectic has thus become, "the truth is the one, all else is untrue," just as the Kantian philosophy resulted in "we know appearances only." On the whole the principle is the same; "the content of knowledge is only an appearance and not truth," but there is also a great difference present. That is to say, Zeno and the Eleatics in their proposition signified "that the sensuous world, with its multitudinous forms, is in itself appearance only, and has no truth." But Kant does not mean this, for he asserts: "Because we apply the activity of our thought to the outer world, we constitute it appearance; what is without, first becomes an untruth by the fact that we put therein a mass of determinations. Only our knowledge, the spiritual, is thus appearance; the world is in itself absolute truth; it is our action alone that ruins it, our work is good for nothing." It shows excessive humility of mind to believe that knowledge has no value; but Christ says, "Are ye not better than the sparrows?" and we are so inasmuch as we are thinking; as sensuous we are as good or as bad as sparrows. Zeno's dialectic has greater objectivity than this modern dialectic.
> Zeno's dialectic is limited to Metaphysics; later, with the Sophists, it became general. We here leave the Eleatic school. which perpetuates itself in Leucippus and, on the other side, in the Sophists, in such a way that these last extended the Eleatic conceptions to all reality, and gave to it the relation of consciousness; the former, however, as one who later on worked out the Notion in its abstraction, makes a physical application of it, and one which is opposed to consciousness. There are several other Eleatics mentioned, to Tennemann's surprise, who, however, cannot interest us. "It is so unexpected," he says (Vol. I., p. 190), "that the Eleatic system should find disciples; and yet Sextus mentions a certain Xeniades."
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Zeno's paradox of motion | https://ncatlab.org/nlab/source/Zeno%27s+paradox+of+motion |
One of the kind of [[paradoxes]] considered by [[Zeno]]. Widely regarded as resolved by the modern concept of [[differentiation]] via [[limit of a sequence]] ([[analysis]], [[differential calculus]]) (see e.g. [Boyer 49, p. 267](#Boyer49)).
## References
* Wikipedia, _[Zeno's paradoxes -- Paradox of motion](http://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Paradoxes_of_motion)_
* Stanford Encyclopedia of Philosophy, _[Zeno's paradoxes -- The paradoxes of motion](http://plato.stanford.edu/entries/paradox-zeno/#ParMot)_
* {#Boyer49} Carl Benjamin Boyer, _The history of the Calculus and its conceptual development_, Dover 1949
* [[Georg Hegel]], chapter on _[[cohesion]]_ and _[[elasticity]]_ in _[[Encyclopedia of the Philosophical Sciences]]_, see also at _[Science of Logic -- Kohäsion](#Science+of+Logic#Kohaesion)_.
[[!redirects Zeno's paradox of motion]]
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Zentralblatt MATH | https://ncatlab.org/nlab/source/Zentralblatt+MATH | [Zentralblatt MATH](https://zbmath.org/) (zbMATH) is an abstracting and reviewing service in [[mathematics]].
The editorial work is done by [[FIZ Karlsruhe]], a nonprofit organization.
## See also
* [[Mathematical Reviews]] (alias [[MathSciNet]])
* [[Math-Net.Ru]]
[[!redirects zbMATH]] |
Zeph Landau | https://ncatlab.org/nlab/source/Zeph+Landau |
* [personal page](https://people.eecs.berkeley.edu/~landau/)
## Selected writings
On [[adiabatic quantum computation]]:
* [[Dorit Aharonov]], [[Wim van Dam]], [[Julia Kempe]], [[Zeph Landau]], [[Seth Lloyd]], [[Oded Regev]], *Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation*, SIAM Journal of Computing **37** 1 (2007) 166-194 [[arXiv:quant-ph/0405098](https://arxiv.org/abs/quant-ph/0405098), [jstor:20454175](https://www.jstor.org/stable/20454175), [doi:10.1109/FOCS.2004.8](https://doi.org/10.1109/FOCS.2004.8), [doi:10.1137/080734479](https://doi.org/10.1137/080734479)]
category: people |
zero | https://ncatlab.org/nlab/source/zero |
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{:toc}
## Definition
The additive [[neutral element]] in the [[natural numbers]], [[integers]], [[real numbers]] and [[complex numbers]] is called **zero** and written as $0$.
More generally, in any [[abelian group]], or even any [[commutative monoid]], the group operation is often called 'addition' and written as $+$, and then the [[neutral element]] is called **zero** and written as $0$. As a consequence, in any [[ring]], or more generally any [[rig]], the two binary operations are called 'multiplication' and 'addition', and the identity for addition is called **zero**.
## Categorification
[[categorification|Categorifying]] this idea, in any [[2-rig]] the additive identity may be called **zero**. This is especially true in the case of a [[distributive category]], that is a category with (at least finitary) [[products]] and [[coproducts]], the former distributing over the latter. In this case the [[initial object]], which serves as the identity for coproducts, is often called **zero**:
$$x + 0 \cong x$$
For example, in the category [[Set]], the [[empty set]] is often written $0$ in the category-theoretic literature.
In an [[abelian category]], the initial object is also [[terminal object|terminal]], and denoted $0$. More generally, any object with this property is called a [[zero object]].
Categorifying [[horizontal categorification|horizontally]] instead, we get the notion of [[zero morphism]].
All these ideas can be, and have been, categorified further.
## Related concepts
* [[zero morphism]]
* [[trivial group]]
* [[one]]
* [[two]]
[[!redirects zero]]
[[!redirects 0]]
|
zero bundle | https://ncatlab.org/nlab/source/zero+bundle |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
For any notion of [[bundles]] with [[fibers]] from a [[pointed category]], a *zero bundle* is a [[bundle]] all whose fibers are [[zero objects]].
The construction of zero-bundles typically (such as in the following examples) constitutes a [[bireflective subcategory]] inclusion
$$
Spaces \hookrightarrow Bundles
$$
of the category of base spaces into the given category of bundles.
For example:
* in the context of [[vector bundles]] the zero-bundle over a base space $B$ is the bundle $B \times \{0\} \xrightarrow{pr_1} X$ all whose [[fibers]] are the [[zero]]-[[dimension of a vector space|dimensional]] [[vector space]], and the induced [[bireflective subcategory]] inclusion is that of base spaces into the corresponding category [[VectBund]];
* in the context of [[retractive spaces]] the zero-bundle over a base space $B$ is the [[identity]] map on $B$, all whose fibers are (equal or, in [[homotopy theory]], [[weak homotopy equivalence|equivalent]] to) the [[point]] $\ast$ regarded as a [[pointed space]];
* in the context of [[parameterized spectra]], the zero-bundle over a base space $X$ has all fibers the [[zero]]-[[spectrum]] $0_\bullet$ (i.e. the spectrum all whose components are [[contractible homotopy type|contractible]], $0_n \simeq \ast$ for all $n$).
(Beware that a zero-bundle is generally -- such as in the above examples -- *not* an [[empty bundle]].)
## Related concepts
* [[zero object]]
* [[bireflective subcategory]]
* [[empty bundle]]
[[!redirects zero bundles]]
[[!redirects zero-bundle]]
[[!redirects zero-bundles]]
[[!redirects zero vector bundle]]
[[!redirects zero vector bundles]]
[[!redirects zero-vector bundle]]
[[!redirects zero-vector bundles]]
[[!redirects zero spectrum bundle]]
[[!redirects zero spectrum bundles]]
[[!redirects zero-spectrum bundle]]
[[!redirects zero-spectrum bundles]]
|
zero function | https://ncatlab.org/nlab/source/zero+function |
A __zero function__ is a [[constant function]] whose constant value is [[zero]].
Formally, if $X$ is a [[set]] and $Y$ is a [[pointed set]] with [[basepoint]] $0$, then the __zero function__ from $X$ to $Y$ is
$$ 0\colon X \to Y\colon t \mapsto 0 .$$
If $X$ is also pointed, then the zero function must preserve basepoints, making this a case of a [[zero morphism]]. However, $X$ need not be pointed for the zero function to make sense.
[[!redirects zero function]]
[[!redirects zero functions]]
|
zero ideal | https://ncatlab.org/nlab/source/zero+ideal |
The __zero ideal__ or __trivial ideal__ of a [[ring]] $R$ is the two-sided [[ideal]] that consists entirely of the [[zero]] element. It may be denoted $\{0\}$, $\mathbf{0}$, or simply $0$ (since it is the zero element of the [[rig of ideals]]).
We may generalize to a [[rig]], including the special case of a [[distributive lattice]] (in which the zero element is the [[bottom element]]), then generalize further to any [[poset]] with a bottom element $\bot$, in which the trivial ideal is $\{\bot\}$.
The trivial ideal of $R$ is the [[intersection]] of all of the ideals of $R$. (If $R$ is a poset *without* a bottom element, then we may still consider the intersection of all of its ideals, but I\'m not sure if this deserves the name.)
[[!redirects zero ideal]]
[[!redirects zero ideals]]
[[!redirects trivial ideal]]
[[!redirects trivial ideals]]
|
zero locus | https://ncatlab.org/nlab/source/zero+locus |
#Contents#
* table of contents
{:toc}
## Idea
The _zero locus_ or _vanishing locus_ of a [[function]] is the set of points where it vanishes, in that it takes the value [[zero]].
## Definition
For $f \colon X \to \mathbb{A}$ a [[function]], its _zero locus_ is the [[preimage]] $f^{-1}(0)$ of [[zero]], hence the [[level set]] at 0.
## Properties
### Nullstellensatz
Hilbert's [[Nullstellensatz]] (German: "zero locus theorem") characterizes joint zero loci of [[ideals]] of functions in a [[polynomial ring]].
## Related concepts
* [[quadratic formula]]
* [[critical locus]], [[derived critical locus]]
* [[moment map]], [[Hamiltonian]]
* [[zero-set structure]]
[[!redirects zero loci]]
[[!redirects zero set]]
[[!redirects zero sets]]
[[!redirects vanishing locus]]
[[!redirects vanishing loci]]
|
zero morphism | https://ncatlab.org/nlab/source/zero+morphism |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Definition
+-- {: .num_defn}
###### Definition
In a [[category]] $C$ with [[zero object]] $0$ the **zero morphism ** $0_{c,d} : c \to d$ between two [[object]]s $c, d \in C$ is the unique [[morphism]] that factors through $0$:
$$
0_{c,d} : c \to 0 \to d
\,.
$$
More generally, in any category [[enriched category|enriched]] over the [[closed monoidal category|closed monoidal]] [[category of pointed sets]] (with [[tensor product]] the [[smash product]]), the **zero morphism** $0_{c,d} : c \to d$ is the basepoint of the [[hom-object]] $[c,d]$.
=--
+-- {: .num_remark}
###### Remark
In fact, an enrichment over pointed sets consists precisely of the choice of a 'zero' morphism $0_{c,d}:c\to d$ for each pair of objects, with the property that $0_{c,d} \circ f = 0_{b,d}$ and $f\circ 0_{a,b} = 0_{a,c}$ for any morphism $f:b\to c$. Such an enrichment is unique if it exists, for if we are given a different collection of zero morphisms $0'_{c,d}$, we must have
$$0'_{c,d} = 0'_{c,d} \circ 0_{c,c} = 0_{c,d}$$
for any $c,d$. Thus, the existence of zero morphisms can be regarded as a [[stuff, structure, property|property]] of a category, rather than structure on it. (To be more precise, it is an instance of [[property-like structure]], since not every functor between categories with zero morphisms will necessarily preserve the zero morphisms, although an [[equivalence of categories]] will.)
=--
## Examples
See at _[[zero object]]_ for examples.
## Related concepts
* [[zero]]
* [[zero function]]
* [[null homotopy]]
[[!redirects zero morphism]]
[[!redirects zero morphisms]]
[[!redirects zero map]]
[[!redirects zero maps]]
|
zero object | https://ncatlab.org/nlab/source/zero+object |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Additive and abelian categories
+-- {: .hide}
[[!include additive and abelian categories - contents]]
=--
=--
=--
# Zero objects
* table of contents
{: toc}
## Definition
+-- {: .num_defn}
###### Definition
In a [[category]], an [[object]] is called
a **zero object**, **null object**, or **biterminator** if it is both an [[initial object]] and a [[terminal object]].
=--
A category with a zero object is sometimes called a _[[pointed category]]_.
+-- {: .num_remark}
###### Remark
This means that $0 \in \mathcal{C}$ is a zero object precisely if for every other object $A$ there is a unique [[morphism]] $A \to 0$ to the zero object as well as a unique morphism $0 \to A$ from the zero object.
=--
+-- {: .num_remark}
###### Remark
If $\mathcal{C}$ is a [[pointed category]], then an object $A$ of $\mathcal{C}$ is a zero object precisely when the only endomorphism of $A$ is the identity morphism.
=--
+-- {: .num_remark}
###### Remark
There is also a notion of **zero object** in [[algebra]] which does not always coincide with the category-theoretic terminology. For example the zero [[ring]] $\{0\}$ is not an [[initial object]] in the category of unital rings (this is instead the [[integers]] $\mathbb{Z}$); but it is the
[[terminal object]]. However, the zero ring *is* the zero object in the category of [[nonunital ring]]s (although it happens to be unital).
=--
## Examples
+-- {: .num_prop}
###### Proposition
* The one-point set is the zero object of the [[category of pointed sets]] (denoted $\Set_*$) and of the category of [[pointed topological spaces]] (denoted $\Top_*$), but only the [[terminal object]] of [[Set]] and [[Top]].
* The [[trivial group]] is a zero object in the category [[Grp]] of [[groups]] and in the category [[Ab]] of [[abelian groups]].
* For $R$ a [[ring]], the trivial $R$-[[module]] (that whose underlying abelian group is the [[trivial group]]) is the zero-object in $R$[[Mod]].
In particular for $R = k$ a [[field]], the $k$-[[vector space]] of [[dimension]] 0 is the zero object in [[Vect]].
* For $R$ and $S$ [[rings]], the trivial $R$-$S$-[[bimodule]] (that whose underlying abelian group is the [[trivial group]]) is the zero-object in $R$-$S$-[[Bimod]].
* However, the zero [[ring]] is not a zero object in the category of [[ring|rings]], at least as long as rings are required to have units (and ring homomorphisms to preserve them).
* For every category $C$ with a [[terminal object]] $*$ the [[under category]] $pt \downarrow C$ of [[pointed objects]] in $C$ has a zero object: the morphism $Id_{pt}$.
=--
+-- {: .num_prop #InCatsEnrichedInPointedSets}
###### Proposition
In any category $C$ [[enriched category|enriched]] over the [[category of pointed sets]] $(Set_*, \wedge)$ with [[tensor product]] the [[smash product]], any object that is either initial _or_ terminal is automatically both and hence a zero object.
=--
+-- {: .proof}
###### Proof
Write $* \in Set_*$ for the singleton pointed set. Suppose $t$ is [[terminal object|terminal]]. Then $C(x,t) = *$ for all $x$ and so in particular $C(t,t) = *$ and hence the [[identity]] morphism on $t$ is the basepoint in the pointed [[hom-set]]. By the axioms of a [[category]], every morphism $f : t \to x$ is equal to the composite
$$
f : t \stackrel{Id}{\to} t \stackrel{f}{\to} x
\,.
$$
By the axioms of an $(Set_*, \wedge)$-enriched category, since $Id_{t}$ is the basepoint in $C(t,t)$, also this composite is the basepoint in $C(t,x)$ and is hence the [[zero morphism]]. So $C(t,x) = *$ for all $x$ and therefore $t$ is also an [[initial object]].
Analogously from assuming $t$ to be initial it follows that it is also terminal.
=--
+-- {: .num_remark}
###### Remark
This is a special case of an [[absolute limit]].
=--
+-- {: .num_remark}
###### Remark
Categories enriched in $(Set_*, \wedge)$ include in particular [[Ab]]-enriched categories. So any [[additive category]], hence every [[abelian category]] has a zero object.
=--
* In the [[stable homotopy category]]: [[zero spectrum]].
## Properties
+-- {: .num_prop}
###### Proposition
A category has a zero object precisely if it has an [[initial object]] $\emptyset$ and a [[terminal object]] $*$ and the unique morphism $\emptyset \to *$ is an [[isomorphism]].
=--
+-- {: .num_remark}
###### Remark
In a category with a zero object 0, there is always a canonical morphism from any object $A$ to any other object $B$ called the _[[zero morphism]]_, given by the composite $A\to 0 \to B$.
Thus, such a category becomes [[enriched category|enriched]] over the [[category of pointed sets]], a partial converse to prop \ref{InCatsEnrichedInPointedSets}.
=--
## Related concepts
* [[pointed category]]
[[pointed (∞,1)-category]], [[pointed model category]]
* [[zero morphism]]
* [[zero object in a derivator]]
## References
* [[Saunders MacLane]], §I.5 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]
[[!redirects zero object]]
[[!redirects zero objects]]
[[!redirects 0 object]]
[[!redirects 0 objects]]
[[!redirects 0-object]]
[[!redirects null object]]
[[!redirects null objects]]
[[!redirects biterminator]]
[[!redirects biterminators]]
|
zero section | https://ncatlab.org/nlab/source/zero+section |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
For $E \to X$ [[vector bundle]], its **zero section** is the [[section]] $X \to E$ that sends every point to the 0-vector over it.
## Properties
A vector bundle minus its zero-section is a [[spherical fibration]].
## Related concepts
* [[Thom space]], [[Thom spectrum]]
* [[projective bundle]]
* [[vanishing locus]], [[critical locus]], [[derived critical locus]]
* [[Hörmander's criterion]] on [[wave front set]]
[[!redirects zero-section]]
[[!redirects zero sections]]
[[!redirects zero-sections]] |
zero spectrum | https://ncatlab.org/nlab/source/zero+spectrum |
## Idea
The [[zero object]] in the [[stable (infinity,1)-category of spectra]]/[[stable homotopy category]].
As a [[sequential spectrum]] this is represented by the sequence of [[pointed topological space]] which consists of the one-point space in each degree (an [[Omega-spectrum]]). This particular representative happens to be itself the [[zero object]] in the [[1-category]] of [[sequential spectra]]. But, as usual, there are other sequential spectra, not [[isomorphism|isomorphic]] to this one, which still represent the [[zero-spectrum]] in the [[stable homotopy theory]] of spectra (hence which are connected to the spectrum constant on the point by a sequence of [[weak equivalences]] in the stable [[model structure on topological sequential spectra]]).
[[!redirects zero spectra]]
[[!redirects zero-spectrum]]
[[!redirects zero-spectra]] |
zero-divisor | https://ncatlab.org/nlab/source/zero-divisor | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Monoid theory
+-- {: .hide}
[[!include monoid theory - contents]]
=--
=--
=--
# Zero-divisors
* table of contents
{: toc}
## Idea
A zero-divisor is something that, like [[zero]] itself, when [[multiplication|multiplied]] by something possibly nonzero still produces zero as a product.
## Definitions
Let $M$ be a [[absorption monoid]] (such as a [[commutative ring]] or any [[ring]]).
An element $x$ of $M$ is a __non-zero-divisor__ if, whenever $x \cdot y = 0$ or $y \cdot x = 0$, then $y = 0$. An element $x$ is a __zero-divisor__ if there exists $y \ne 0$ such that $x \cdot y = 0$ or $y x = 0$.
In [[constructive mathematics]], we want $\ne$ to be a [[tight apartness relation]] on $M$ in the definition of zero-divisor. We also say that $x$ is a __strong non-zero-divisor__ if, whenever $y \ne 0$, then $x y \ne 0$ and $y x \ne 0$. (The notion of (weak) non-zero-divisor makes sense even without any apartness relation.)
If $M$ is (or may be) non-commutative, then we may distinguish __left__ and __right__ (non)-zero-divisors in the usual way.
## Properties
By this definition, [[zero]] itself is a zero-divisor if and only if $M$ is non-trivial. (Some authorities will differ on this point, but if you think about it, this is clearly the correct definition, by the same principle that the trivial ring is not a field, $1$ is not a prime number, etc. See [[too simple to be simple]].)
An [[integral domain]] is precisely a commutative ring (whose multiplicative monoid is an absorption monoid by definition) in which [[zero]] is the unique zero-divisor of the multiplicative monoid of the commutative ring (or constructively, in which the strong non-zero-divisors are precisely the strong non-zero elements in the multiplicative monoid, that is those elements $x$ such that $x \ne 0$).
The non-zero-divisors of any absorption monoid $M$ form a [[monoid]] under multiplication, which may be denoted $M^{\times}$. Note that if $M$ happens to be a [[field]], then this $M^{\times}$ agrees with the usual notation $M^{\times}$ for the [[group]] of invertible elements of the multiplicative monoid $M$, but $M^{\times}$ is not a group in general. (We may use $M^{\div}$ or $M^*$ for the group of invertible elements.)
## Generalisations
If $I$ is any [[ideal of a monoid|ideal]] of $M$, then we can generalise from a zero-divisor to an $I$-[[divisor]]. In a way, this is nothing new; $x$ is an $I$-divisor in $M$ if and only if $[x]$ is a zero-divisor in $M/I$. Ultimately, this is related to the notion of [[divisor]] in [[algebraic geometry]].
## Related concepts
* [[absorption monoid]]
* [[integral domain]]
* [[cancellative element]]
[[!redirects zero-divisor]]
[[!redirects zero-divisors]]
[[!redirects zero divisor]]
[[!redirects zero divisors]]
|
zero-one law | https://ncatlab.org/nlab/source/zero-one+law | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--
## Idea
(...Work in progress...)
## See also
* [[Markov category]]
* [[probability monad]]
* [[probability theory]]
## References
* {#fritzrischel} [[Tobias Fritz]] and Eigil Fjeldgren Rischel, _Infinite products and zero-one laws in categorical probability_, Compositionality 2(3) 2020. ([arXiv:1912.02769](http://arxiv.org/abs/1912.02769))
* {#ergodic_dagger} Noé Ensarguet, [[Paolo Perrone]], Categorical probability spaces, ergodic decompositions, and transitions to equilibrium, [arXiv:2310.04267](https://arxiv.org/abs/2310.04267)
category: probability
[[!redirects zero-one laws]]
[[!redirects 0-1 law]]
[[!redirects 0-1 laws]]
[[!redirects Kolmogorov zero-one law]]
[[!redirects Hewitt-Savage zero-one law]]
[[!redirects Kolmogorov 0-1 law]]
[[!redirects Hewitt-Savage 0-1 law]] |
zero-section into Thom space of universal line bundle is weak equivalence | https://ncatlab.org/nlab/source/zero-section+into+Thom+space+of+universal+line+bundle+is+weak+equivalence |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The [[Thom space]] of the [[universal complex line bundle]] is [[weak homotopy equivalence|weakly homotopy equivalent]] to the base space of the line bundle, hence to the [[classifying space]] for the [[circle group]], and this is equivalence is exhibited by the [[zero section]] of the universal line bundle, followed by its inclusion into its [[Thom space]] ([Adams 74, Part I, Example 2.1](#Adams74), see Prop. \ref{TheStatement} below).
This statement plays a key role in the discussion of [[complex oriented cohomology]], as it implies that any choice of [[universal characteristic class|universal]] first [[Conner-Floyd Chern class]] $c^E_1$ is equivalent to a choice of universal [[Thom class]] on the [[universal complex line bundle]], and hence induces a Thom class, hence a "fiberwise complex orientation" on any [[complex line bundle]]. (See at _[Conner-Floyd E-Chern classes are E-Thom classes](universal+complex+orientation+on+MU#ConnerFloydChernClassesAreThomClasses)_ for more on this.)
In this context the statement is naturally stated in the form
$$
B \mathrm{U}(1) \overset{\simeq}{\longrightarrow} M \mathrm{U}(1)
\,,
$$
where on the right the [[Thom space]] is thought of as the component space in degree 2 (complex degree 1) of the universal [[unitary group|unitary]] [[Thom spectrum]] [[MU]].
The analogous statement is true also for the universal [[real vector bundle|real]]- and [[quaternionic line bundle|quaternionic]] [[line bundles]], and it implies the analogous consequence for [[quaternionic oriented cohomology theory]], etc., notably
$$
B Sp(1) \overset{\simeq}{\longrightarrow} M Sp(1)
\,,
$$
where on the right the [[Thom space]] is thought of as the component space in degree 4 (quaternionic degree 1) of the universal [[quaternionic unitary group|quaternionic unitary]] [[Thom spectrum]] [[MSp]].
## Preliminaries
### The universal line bundle
Let $\mathbb{K} \,\in\, \{\mathbb{R}, \mathbb{C}, \mathbb{H}\}$
be the [[real numbers]] or [[complex numbers]] or [[quaternions]].
Write
\[
\label{GroupOfUnitNormElements}
S(\mathbb{K})
\;\coloneqq\;
\big\{
q \in \mathbb{K}
\;\big\vert\;
q q^\ast =1
\big\}
\;\;
\in
\;
Groups
\]
for its [[multiplicative group]] of unit-[[norm]] elements. Specifically this is the [[cyclic group of order 2]], the [[circle group]] or the [[quaternionic unitary group]]/[[SU(2)]]:
$$
S(\mathbb{R})
\,\simeq\,
\mathbb{Z}/2
\,,
\phantom{AA}
S(\mathbb{C})
\,\simeq\,
\mathrm{U}(1)
\,,
\phantom{AA}
S(\mathbb{H})
\,\simeq\,
Sp(1) \,\simeq\, SU(2)
$$
By either left or right multiplication in $\mathbb{K}$ this group [[action|acts]] on $\mathbb{K}$, $\mathbb{R}$-[[linear map|linearly]], making $\mathbb{K}$ a [[linear representation]]
\[
\label{KAsSKRepresentation}
\mathbb{K}
\,\in\,
S(\mathbb{K})
Representations_{\mathbb{R}}
\,.
\]
Hence with
$$
E
\big(
S(\mathbb{K})
\big)
\overset{\;\;\;}{\longrightarrow}
B
\big(
S(\mathbb{K})
\big)
$$
denoting the $S(\mathbb{K})$-[[universal principal bundle]] over the [[classifying space]] for the group (eq:GroupOfUnitNormElements), the [[real vector bundle]] underlying the [[universal complex line bundle|universal K-line bundle]] is the corresponding [[associated bundle]] via the above action (eq:KAsSKRepresentation):
\[
\label{UniversalKLineBundle}
\array{
E
\big(
S(\mathbb{K})
\big)
\underset{
S(\mathbb{K})
}{\times}
\mathbb{K}
\\
\big\downarrow
\\
B
\big(
S(\mathbb{K})
\big)
}
\]
### Thom spaces
The [[Thom space]] of a [[real vector bundle|real]] [[topological vector bundle]] $\mathcal{V}_X$ over some base space $X$ is the [[homotopy cofiber]] of its associated [[spherical fibration]]:
\[
\label{ThomSpaceAsHomotopyCofiber}
S_X(\mathcal{V}_X)
\overset{ p_{S(\mathcal{V}_X)} }{\longrightarrow}
X
\overset{ hocofib }{\longrightarrow}
Th(X)
\,.
\]
When the [[topological space]] $X$ has the [[mathematical structure|structure]] of a [[CW-complex]] then a [[cofibration]] which models the [[homotopy type]] of $p_{S(\mathcal{V}_X)}$ in the [[classical model structure on topological spaces]] is the inclusion of the unit [[sphere bundle]] into the unit disk bundle $D_X(\mathcal{V}_X)$ of $\mathcal{V}_X$ (with respect to any choice of fiberwise [[metric]]) since this is then a [[relative cell complex]]-inclusion. Therefore the homotopy cofiber (eq:ThomSpaceAsHomotopyCofiber) is then represented by the [[1-category|1-category theoretic]] [[cofiber]]
\[
\label{ThomSpaceAsActualCofiber}
S_X(\mathcal{V}_X)
\overset{ i_{S_X(\mathcal{V}_X)} }{\longrightarrow}
D_X(\mathcal{V}_X)
\overset{ cofib }{\longrightarrow}
Th(X)
\,.
\]
Finally, since the [[zero section]] of the unit disk bundle is manifestly the [[weak homotopy equivalence]] that exibits this [[cofibrant resolution]], we may call
\[
\label{ZeroSectionIntoTheThomSpace}
0_X
\;\colon\;
X
\underoverset
{\simeq}
{
0_{D_X(\mathcal{V}_X)}
}
{\longrightarrow}
D(\mathcal{V}_X)
\overset
{cofib\big( i_{S_X(\mathcal{V}_X)} \big)}
{\longrightarrow}
Th(X)
\]
the zero-section _into_ the Thom spaces.
## Statement
\begin{prop}\label{TheStatement}
The zero-section (eq:ZeroSectionIntoTheThomSpace) into the [[Thom space]] of the universal $\mathbb{K}$-line bundle (eq:UniversalKLineBundle)
$$
\array{
Th
\Big(
E (S(\mathbb{K}))
\underset{
S(\mathbb{K})
}{\times}
\mathbb{K}
\Big)
\\
{}^{{}_{\mathllap{
0_{
B \big(
S(\mathbb{K})
\big)
}
}}}
\big\uparrow
{}^{{}_{
\mathrlap{\simeq}}
}
\\
B \big(
S(\mathbb{K})
\big)
}
$$
is a [[weak homotopy equivalence]].
\end{prop}
\begin{proof}
The point is that for the universal _line_ bundle, the associated [[sphere bundle]] is [[homotopy equivalent]] to the [[universal principal bundle]] and hence [[contractible homotopy type|weakly contractible]].
One way to see it is to unwind the definition of the unit sphere bundle in the universal line bundle as follows:
\[
\label{SphereBundleOfUniversalLineBundleIsContractible}
S_{B \big(S(\mathbb{K})\big)}
\Big(
E\big(S(\mathbb{K})\big)
I \underset{S(\mathbb{K})}{\times}
\mathbb{K}
\Big)
\;=\;
\Big(
E\big(S(\mathbb{K})\big)
\underset{S(\mathbb{K})}{\times}
S(\mathbb{K})
\Big)
\;=\;
E\big(S(\mathbb{K})\big)
\;\simeq\;
\ast
\,.
\]
{#ViaContractibleInfiniteDimensionalSpheres} Another way to see the same is to observe that the sphere bundle associated to the universal line bundle is the [[sequential colimit]] over the [[tautological line bundle|tautological]] [[principal bundles]] over the finite-dimensional [[complex projective space]], which themselves are [[n-spheres]] (see [there](tautological+line+bundle#eq:TautologicalPrincipalBundleOverProjectiveSpace)). With this, the statement (eq:SphereBundleOfUniversalLineBundleIsContractible) follows from the fact that [[the infinite-dimensional sphere is weakly contractible]] (see [there](infinite-dimensional+sphere#AsAnInfiniteSphericalCellComplex)):
$$
S_{\mathbb{K}P^\infty}
\big(
\mathcal{L}^\ast_{\mathbb{K}P^\infty}
\big)
\;\simeq\;
\underset{ \underset{n}{\longrightarrow} }{\lim}
\;
\big(
S_{\mathbb{K}P^n}
\left(
\mathcal{L}^\ast_{\mathbb{K}P^n}
\right)
\big)
\;\simeq\;
\underset{ \underset{n}{\longrightarrow} }{\lim}
\;
\left(
S^{ (n+1) \cdot dim_{{}_{\mathbb{R}}}(\mathbb{K}) - 1 }
\right)
\;\simeq\;
S^\infty
\;\simeq\;
\ast
\,.
$$
In any case, this means that we have the following solid [[commuting diagram]], where the solid vertical morphisms are all [[weak homotopy equivalences]]:
\begin{xymatrix@R=25pt@C=50pt}
&& &&
B(S(\mathbb{K}))
\ar[d]
_-{
0
}
|-{
\hspace{3pt}
\scalebox{.7}{
\rotatebox{-90}{$
\;\simeq\;
$}
}
}
\\
\ast
\;
\ar@{^{(}->}[rr]_-{ \in \, \mathrm{Cofibrations} }
\ar[d]
|-{
\hspace{3pt}
\scalebox{.7}{
\rotatebox{-90}{$
\;\simeq\;
$}
}
}
&&
D_{B(S(\mathbb{K}))}
\Big(
E\big( S(\mathbb{K}) \big)
\underset{S(\mathbb{K})}{\times}
\mathbb{K}
\Big)
\ar[rr]
^-{ \mbox{\tiny cofiber} }
_-{ \mbox{\tiny $\Rightarrow$ homotopy cofiber} }
\ar@{=}[d]
&&
D_{B(S(\mathbb{K}))}
\Big(
E\big( S(\mathbb{K}) \big)
\underset{S(\mathbb{K})}{\times}
\mathbb{K}
\Big)
\ar@{-->}[d]
\\
S_{B(S(\mathbb{K}))}
\Big(
E\big( S(\mathbb{K}) \big)
\underset{S(\mathbb{K})}{\times}
\mathbb{K}
\Big)
\;
\ar@{^{(}->}[rr]
_-{ \in \, \mathrm{Cofibrations} }
&&
D_{B(S(\mathbb{K}))}
\Big(
E\big( S(\mathbb{K}) \big)
\underset{S(\mathbb{K})}{\times}
\mathbb{K}
\Big)
\ar[rr]
^-{ \mbox{\tiny cofiber} }
_-{ \mbox{\tiny $\Rightarrow$ homotopy cofiber} }
&&
\mathrm{Th}
\Big(
E\big( S(\mathbb{K}) \big)
\underset{S(\mathbb{K})}{\times}
\mathbb{K}
\Big)
\end{xymatrix}
(Here the left vertical map picks any point of the sphere bundle. There is then a unique horizontal map on the left to make the left square [[commuting diagram|commute]].)
Now, since the [[classifying space]] $B(S(\mathbb{K}))$ does have the structure of a [[CW-complex]] (given, for instance, by its realization as infinite [[real projective space|real]]/[[complex projective space|complex]]/[[quaternionic projective space|quaternionic]] [[projective space]] $\mathbb{K}P^\infty$, via the [[cell structure of projective space]]), the bottom [[cofiber]] here represents, as in (eq:ThomSpaceAsActualCofiber), the defining [[homotopy cofiber]].
Since [[homotopy cofibers]] are preserved, up to [[weak equivalence]], by weak equivalences of their diagrams (by [this Prop.](Introduction+to+Homotopy+Theory#FiberOfFibrationIsCompatibleWithWeakEquivalences)), it follows that the dashed vertical morphism is a weak equivalence in the [[classical model structure on topological spaces]], hence a [[weak homotopy equivalence]].
This is the statement that was to be shown. Or more explicitly: By [[two-out-of-three]] also the [[composition|composite]] vertical morphism on the right is a weak homotopy equivalence, which is the desired morphism in the form (eq:ZeroSectionIntoTheThomSpace).
\end{proof}
## Related statements
* [[cell structure of projective space]]
* [Conner-Floyd E-Chern classes are E-Thom classes](universal+complex+orientation+on+MU#ConnerFloydChernClassesAreThomClasses)
See also:
* [[complex oriented cohomology]], [[universal complex orientation on MU]]
* [[quaternionic oriented cohomology]]
## References
* {#Adams74} [[Frank Adams]], Part I, Example 2.1 in: _[[Stable homotopy and generalised homology]]_, 1974
* {#Kochmann96} [[Stanley Kochman]], Lemma 2.6.5 in: _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996
[[!redirects zero-section into Thom space of universal line bundle is weak homotopy equivalence]]
|
zero-set structure | https://ncatlab.org/nlab/source/zero-set+structure |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--
\tableofcontents
## Definition
Given a [[set]] $X$, a **zero-set structure** on $X$ is a [[sigma-topology|$\sigma$-topology]] $\mathcal{Z}$ such that
* For every pair of distinct points in $X$ there is an open set $Z \in \mathcal{Z}$ containing precisely one of these points.
* If $Z \in \mathcal{Z}$ then there are $Z_n \in \mathcal{Z}$ for all natural numbers $n$ such that $Z^c = \bigcup_{n \in \mathbb{N}} Z_n$.
* If $Z_1, Z_2 \in \mathcal{Z}$ and $Z_1 \cap Z_2 = \emptyset$, then there are $V_1, V_2 \in \mathcal{Z}$ with $Z_1 \subseteq V_1^c$, $Z_2 \subseteq V_2^c$, and $V_1^c \cap V_2^c = \emptyset$.
## See also
* [[zero set]]
* [[sigma-topology]]
## References
* Fedor Petrov, *"countable" topology*, MathOverflow ([web](https://mathoverflow.net/questions/173255/countable-topology))
* Hugh Gordon, *Rings of functions determined by zero-sets*. Hugh Gordon. Pacific J. Math. Volume 36, Number 1 (1971), 133-157. ([pdf](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-36/issue-1/Rings-of-functions-determined-by-zero-sets/pjm/1102971274.pdf))
[[!redirects zero-set structure]]
[[!redirects zero-set structures]] |
zeta function | https://ncatlab.org/nlab/source/zeta+function |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Theta functions
+--{: .hide}
[[!include theta functions - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The concept of _zeta function_ originates in [[number theory]], but to get an idea of what they "really are" it is helpful to proceed anachronistically:
$\zeta$-functions are [[meromorphic functions]] $s \mapsto \zeta(s)$ on the [[complex plane]], which behave like [[analytic continuations]] of [[traces]] of powers
$$
s \mapsto Tr \left(\frac{1}{H}\right)^s
$$
of suitable [[elliptic differential operators]] $H$ (in [[physics]] these are [[regularization (physics)|regularized]] [[traces]] of [[Feynman propagators]] leading to expressions for [[vacuum amplitudes]]), which means that for sufficiently nice such $H$ these are analytic continuations in $s$ of sums of the form
$$
s \mapsto \underset{\lambda}{\sum} \lambda^{-s}
\,,
$$
where the summation is over the [[eigenvalues]] $\lambda$ of $H$.
Indeed, such _[[zeta functions of elliptic differential operators]]_ constitutes one class of examples of zeta functions. Of particular interest is the case where $H$ is a [[Laplace operator]] of a [[hyperbolic manifold]] and in particular on a hyperbolic [[Riemann surface]], for that case one obtains the _[[zeta function of a Riemann surface]]_, in particular the _[[Selberg zeta function]]_.
In modern language one also speaks of _[[L-functions]]_. Where a zeta function of some space is like the [[Feynman propagator]] of _the_ canonical [[Laplace operator]] of that space, an L-function is defined from an extra "twisting" information such as that of a [[flat bundle]]/[[local system of coefficients]] on the space (and is hence like the [[Feynman propagator]] of the corresponding twisted/coupled [[Laplace operator]]). The major properties satisfied by anything that qualifies as a zeta function or [[L-functions]] are: these are [[meromorphic functions]] $s \mapsto L(s)$ on the [[complex plane]] such that
1. for $\Re(s) \gt 1$ they have a [[convergence|converging]] [[series]] expansion of the above form, and/or a [[infinite product|multiplicative series]] expression, the _[[Euler product]]_;
1. such that [[analytic continuation]] of the series expression exists to a meromorphic function $L(-)$ on the complex plane;
1. and such that the result satisfies a _[[functional equation]]_ which says that the product $\hat L$ of $L$ with some correcion functions satisfies $\hat L(1-s) = \hat L(s)$.
Proceeding from the above class of examples in [[complex analytic geometry]] one may wonder if there are [[analogy|analogs]] also in [[arithmetic geometry]]. Indeed, by the [[function field analogy]] there are. All the way down "on [[Spec(Z)]]" the analog of the [[Selberg zeta function]] is the [[Riemann zeta function]], which _historically_ is the first of all zeta functions, defined by [[analytic continuation]] of the [[series]]
$$
s \mapsto \underoverset{n = 1}{\infty}{\sum} n^{-s}
\,.
$$
The _[[Riemann hypothesis]]_ [[conjecture|conjectures]] a characterization of the [[roots]] of this zeta function and is regarded as one of the outstanding problems in [[mathematics]]. It has evident analogs for all other zeta functions (for some of which it has been proven).
More generally, over [[arithmetic curves]] which are [[spectrum of a commutative ring|spectra]] of [[rings of integers]] of more general [[number fields]], the Riemann zeta function has generalization to the _[[Artin L-functions]]_ defined intrinsically in terms of [[characteristic polynomials]] of [[Galois representations]]. When the Galois representation is 1-dimensional, then the Artin L-function may be expressed (by "[[Artin reciprocity]]") in terms of "more arithmetic" data by [[Dirichlet L-functions]] and [[Hecke L-functions]]. When the Galois representation is higher dimensional, then the [[Langlands correspondence]] [[conjecture]] asserts that the Artin L-function may be expressed "arithmetically" as the [[automorphic L-function]] of an [[automorphic form]].
Similarly on [[arithmetic curves]] given by [[function fields]] there is the [[Goss zeta function]] and in [[higher dimensional arithmetic geometry]] the [[Weil zeta function]], famous from the _[[Weil conjectures]]_. The
When interpreting the [[Frobenius morphisms]] that appear in the [[Artin L-functions]] geometrically as flows (as discussed at _[Borger's arithmetic geometry -- Motivation](Borger's+absolute+geometry#Motivation)_) then this induces an evident analog of [[zeta function of a dynamical system]]. This in turn has strong analogies with [[Alexander polynomials]] in [[knot theory]] (see at _[[arithmetic topology]]_).
[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]
## Properties
### Function field analogy
{#FunctionFieldAnalogy}
One way to understand the plethora of different zeta functions is to see them as the incarnation of the same general concept in different flavors of [[geometry]]. This is expressed at least in parts by the
[[!include function field analogy -- table]]
## Related concepts
* [[multiple zeta values]], [[motivic multiple zeta values]], [[motivic integration]], [[motive]]
* [[Weil conjecture]]
* [[Riemann hypothesis]]
* there are attempts to understand the Riemann zeta function as the spectrum of a [[Hamiltonian]] of a [[quantum mechanical system]]. See at _[[Riemann hypothesis and physics]]_.
* [[Beilinson regulator]]
* [[zeta function of a Riemann surface]]
## References
### General
A useful survey of the zoo of zeta functions is in
* [[Jeffrey Lagarias]], _Number theory zeta functions and Dynamical zeta functions_ ([pdf](http://www.math.lsa.umich.edu/~lagarias/doc/numberthzeta.pdf))
Further general review includes
* {#Kowalski} E. Kowalski, first part of _Automorphic forms, L-functions and number theory (March 12–16) Three Introductory lectures_ ([pdf](http://www.math.ethz.ch/~kowalski/lectures.pdf))
* [[Alain Connes]], [[Matilde Marcolli]], chapter II of _[[Noncommutative Geometry, Quantum Fields and Motives]]_
Discussion in the more general context of [[higher dimensional arithmetic geometry]] is in
* {#Fesenko08} [[Ivan Fesenko]], _Adelic approch to the zeta function of arithmetic schemes in dimension two_, Moscow Math. J. 8 (2008), 273–317 ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/ada.pdf))
### In algebraic geometry
* [[Yuri Manin]], _Lectures on zeta functions and motives (according to Deninger and Kurokawa)_, Astérisque __228__:4 (1995) 121--163, and preprint MPIM1992-50 [pdf](http://www.mpim-bonn.mpg.de/preblob/4793)
* Nobushige Kurokawa, _Zeta functions over $F_1$_, Proc. Japan Acad. Ser. A Math. Sci. __81__:10 (2005) 180-184 [euclid](http://projecteuclid.org/euclid.pja/1135791771)
* [[Bruno Kahn]], _Fonctions zêta et $L$ de variétés et de motifs_, [arXiv:1512.09250](http://arxiv.org/abs/1512.09250v1).
### Categorical approaches
* [[Michael Larsen|M. Larsen]], [[Valery Lunts|V. A. Lunts]], _Motivic measures and stable birational geometry_, Mosc. Math. J. __3__, 1 (2003) 85--95; _Rationality criteria for motivic zeta functions_, Compos. Math. __140__:6 (2004) 1537–1560
* [[Vladimir Guletskii]], _Zeta functions in triangulated categories_, Mathematical Notes __87__, 3 (2010) 369--381, [math/0605040](http://arxiv.org/abs/math/0605040)
* [[Maxim Kontsevich|M. Kontsevich]], _Notes on motives in finite characteristics_, [math.AG/0702206](http://arxiv.org/abs/math.AG/0702206)
* [[John Baez]], _[[johnbaez:Zeta functions]]_
* [[Sergey Galkin]], [[Evgeny Shinder]], _On a zeta-function of a dg-category_, [arXiv:1506.05831](http://arxiv.org/abs/1506.05831).
[[!redirects zeta functions]] |
zeta function of a dynamical system | https://ncatlab.org/nlab/source/zeta+function+of+a+dynamical+system |
#Contents#
* table of contents
{:toc}
## Idea
When interpreting the [[Frobenius morphisms]] that appear in the [[Artin L-functions]] geometrically as flows (as discussed at _[Borger's arithmetic geometry -- Motivation](Borger's+absolute+geometry#Motivation)_) then this induces an evident analog of _[[zeta function]] of a [[dynamical system]]_.
The archetypal example is the _[[Ruelle zeta function]]_.
## Definition
## References
Lecture notes include
* [[Mark Pollicott]], _Dynamical zeta functions_ ([pdf](https://homepages.warwick.ac.uk/~masdbl/grenoble-16july.pdf))
* _Dynamical zeta functions_ ([pdf](http://homepages.warwick.ac.uk/~masdbl/leiden-lectures.pdf))
* [[Jeffrey Lagarias]], _Number theory zeta functions and Dynamical zeta functions_ ([pdf](http://www.math.lsa.umich.edu/~lagarias/doc/numberthzeta.pdf))
[[!redirects zeta functions of dynamical systems]]
[[!redirects dynamical zeta function]]
[[!redirects dynamical zeta functions]]
|
zeta function of a Riemann surface | https://ncatlab.org/nlab/source/zeta+function+of+a+Riemann+surface |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
#### Functional analysis
+--{: .hide}
[[!include functional analysis - contents]]
=--
#### Operator algebra
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The [[zeta function]] naturally associated to a [[Riemann surface]]/[[complex curve]], hence the [[zeta function of an elliptic differential operator]] for the [[Laplace operator]] on the Riemann surface (and hence hence essentially the [[Feynman propagator]] for the [[scalar fields]] on that surface) is directly analogous to the zeta functions associated with [[arithmetic curves]], notably the [[Artin L-functions]].
([Minakshisundaram-Pleijel 49](#MinakshisundaramPleijel49)) considered the [[zeta function of an elliptic differential operator]] for the [[Laplace operator]] on a [[Riemann surface]].
Motivated by the resemblance of the [[Selberg trace formula]] to Weil's formula for the sum of zeros of the [[Riemann zeta function]], ([Selberg 56](#Selberg56)) defined for any compact hyperbolic [[Riemann surface]] a [[zeta function]]-like expression, the _[[Selberg zeta function]] of a Riemann surface_. (e.g. [Bump, below theorem 19](#Bump)).
Much of this is more generally defined/considered on higher dimensional [[hyperbolic manifolds]].
That the Selberg zeta function is indeed proportional to the [[zeta function of an elliptic differential operator|zeta function]] of a [[Laplace operator]] is due to ([D'Hoker-Phong 86](#DHokerPhong86), [Sarnak 87](#Sarnak87)), and that it is similarly related to the [[eta function of a self-adjoint operator|eta function]] of a [[Dirac operator]] on the given Riemann surface/hyperbolic manifold goes back to ([Milson 78](#Milson78)), with further development including ([Park 01](#Park01)). For review of the literature on this relation see also the beginning of ([Friedman 06](#Friedman06)).
## Examples
### For a complex torus / complex elliptic curve
For $\mathbb{C}/(\mathbb{Z}\oplus \tau \mathbb{Z})$
a [[complex torus]] (complex [[elliptic curve]]) equipped with its
standard flat [[Riemannian metric]], then the [[zeta function of an elliptic differential operator|zeta function]] of the corresponding [[Laplace operator]] $\Delta$ is
$$
\zeta_{\Delta} = (2\pi)^{-2 s} E(s)
\coloneqq
(2\pi)^{-2 s} \underset{(k,l)\in \mathbb{Z}\times\mathbb{Z}-(0,0)}{\sum}
\frac{1}{{\vert k +\tau l\vert}^{2s}}
\,.
$$
The corresponding [[functional determinant]] is
$$
\exp(
E^\prime_{\Delta}(0)
)
= (Im \tau)^2 {\vert \eta(\tau)\vert}^4
\,,
$$
where $\eta$ is the [[Dedekind eta function]].
(recalled e.g. in [Todorov 03, page 3](#Todorov03))
### Of Dirac operators twisted by a flat connection
{#OfDiracOperatorTwistedByFlatConnection}
For $A$ a [[flat connection]] on a [[Riemannian manifold]], write $D_A$ for the [[Dirac operator]] twisted by this connection.
On a suitable [[hyperbolic manifold]], the [[partition function]]/[[theta function]] for $D_A$ appears in ([Bunke-Olbrich 94, prop. 6.3](#BunkeOlbrich94)) (and [Bunke-Olbrich 94a, def. 3.1](#BunkeOlbrich94a)) for the odd dimensional case). The corresponding Selberg zeta formula is ([Bunke-Olbrich 94a, def. 4.1](#BunkeOlbrich94a)). This has a form analogous to that of [[Artin L-functions]] with the flat connection replaced by a [[Galois representation]].
## Properties
### Analogy with Artin L-function
{#AnalogyWithArtinLFunction}
That the Selberg/Ruelle zeta function is equivalently an [[Euler product]] of [[characteristic polynomials]] is due to ([Gangolli 77, (2.72)](#Gangolli77) [Fried 86, prop. 5](#Fried86)).
That it is in particular the Euler product of characteristic polynomials of the [[monodromies]]/[[holonomies]] of the [[flat connection]] corresponding to the given [[group representation]] is ([Bunke-Olbrich 94, prop. 6.3](#BunkeOlbrich94)) for the even-dimensional case and ([Bunke-Olbrich 94a](#BunkeOlbrich94a)) for the odd-dimensional case.
Notice that this is analogous to the standard definition of an [[Artin L-function]] if one interprets a [[Frobenius map]] $Frob_p$ (as discussed there) as an element of the arithmetic fundamental group of an [[arithmetic curve]] and a [[Galois representation]] as a [[flat connection]].
### Function field analogy
[[!include function field analogy -- table]]
## Related concepts
[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]
## References
Original articles include
* {#MinakshisundaramPleijel49} S. Minakshisundaram, ; Å Pleijel, _Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds_ (1949), Canadian Journal of Mathematics 1: 242–256, doi:10.4153/CJM-1949-021-5, ISSN 0008-414X, MR 0031145 ([web](http://cms.math.ca/10.4153/CJM-1949-021-5))
* {#Selberg56} [[Atle Selberg]], _Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series_, Journal of the Indian Mathematical Society 20 (1956) 47-87.
* {#Milson78} [[John Milson]], _Closed geodesic and the $\eta$-invariant_, Ann. of Math., 108, (1978) 1-39 ([](http://www.jstor.org/stable/1970928))
Review includes
* Wikipedia, _[Selberg zeta function](http://en.wikipedia.org/wiki/Selberg_zeta_function)_
* Wikipedia, _[Minakshisundaram–Pleijel zeta function](http://en.wikipedia.org/wiki/Minakshisundaram–Pleijel_zeta_function)_
* {#Watkins} [[Matthew Watkins]], citation collection on _[Selberg trace formula and zeta functions](http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/physics4.htm)_
* {#Bump} Bump, below theorem 19 in _Spectral theory of $\Gamma \backslash SL(2,\mathbb{R})$_ ([[BumpSpectralTheory.pdf:file]])
Expression of the Selberg/Ruelle zeta function as an [[Euler product]] of [[characteristic polynomials]] is due to
* {#Gangolli77} Ramesh Gangolli, _Zeta functions of Selberg's type for compact space forms of symmetric spaces of rank one_, Illinois J. Math. Volume 21, Issue 1 (1977), 1-41. ([Euclid](http://projecteuclid.org/euclid.ijm/1256049498))
* {#Fried86} [[David Fried]], _The zeta functions of Ruelle and Selberg. I_, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 19 no. 4 (1986), p. 491-517 ([Numdam](http://www.numdam.org/item?id=ASENS_1986_4_19_4_491_0))
Discussion of the relation between, on the one hand, [[zeta function of an elliptic differential operator|zeta function]] of [[Laplace operators]]/[[eta function of a self-adjoint operator|eta funcstions]] of [[Dirac operators]] and, on the other hand, Selberg zeta functions includes
* {#DHokerPhong86} [[Eric D'Hoker]] [[Duong Phong]], _Communications in Mathematical Physics_, Volume 104, Number 4 (1986), 537-545 ([Euclid](http://projecteuclid.org/euclid.cmp/1104115166))
* {#Sarnak87} [[Peter Sarnak]], _Determinants of Laplacians_, Communications in Mathematical Physics, Volume 110, Number 1 (1987), 113-120. ([Euclid](http://projecteuclid.org/euclid.cmp/1104159171))
* [[Ulrich Bunke]], [[Martin Olbrich]], Andreas Juhl, _The wave kernel for the Laplacian on the classical locally symmetric spaces of rank one, theta functions, trace formulas and the Selberg zeta function_, Annals of Global Analysis and Geometry February 1994, Volume 12, Issue 1, pp 357-405
* {#BunkeOlbrich94} [[Ulrich Bunke]], Martin Olbrich, _Theta and zeta functions for locally symmetric spaces of rank one_ ([arXiv:dg-ga/9407013](http://arxiv.org/abs/dg-ga/9407013))
and for odd-dimensional spaces also in
* {#BunkeOlbrich94a} [[Ulrich Bunke]], [[Martin Olbrich]], _Theta and zeta functions for odd-dimensional locally symmetric spaces of rank one_ ([arXiv:dg-ga/9407012](http://arxiv.org/abs/dg-ga/9407012))
* {#BunkeOlbrich94b} [[Ulrich Bunke]], [[Martin Olbrich]] _$\Gamma$-Cohomology and the Selbeg zeta function_ ([arXiv:dg-ga/9411004](http://arxiv.org/abs/dg-ga/9411004))
* [[Ulrich Bunke]], [[Martin Olbrich]], _Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian group_ ([arXiv:dg-ga/9603003](http://arxiv.org/abs/dg-ga/9603003))
* {#BunkeOlbrich95} [[Ulrich Bunke]], [[Martin Olbrich]], _Selberg zeta and theta functions: a differential operator approach_, Akademie Verlag 1995
* {#Park01} Jinsung Park, _Eta invariants and regularized determinants for odd dimensional hyperbolic manifolds with cusps_ ([arXiv:0111175](http://arxiv.org/abs/math/0111175))
* {#Friedman04} [[Joshua Friedman]], _The Selberg trace formula and Selberg zeta-function for cofinite Kleinian groups with finite-dimensional unitary representations_ ([arXiv:math/0410067](http://arxiv.org/abs/math/0410067))
* {#Friedman06} [[Joshua Friedman]], _Regularized determinants of the Laplacian for cofinite Kleinian groups with finite-dimensional unitary representations_, Communications in Mathematical Physics ([arXiv:math/0605288](http://arxiv.org/abs/math/0605288))
See also
* {#Todorov03} [[Andrey Todorov]], _The analogue of the Dedekind eta function for CY threefolds_, 2003 [pdf](http://www.ma.huji.ac.il/conf/crelle.pdf)
[[!redirects Selberg zeta function of a Riemann surface]] |
zeta function of an elliptic differential operator | https://ncatlab.org/nlab/source/zeta+function+of+an+elliptic+differential+operator |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
For a suitable [[linear operator]] $H$ (say on [[section]] of a [[line bundle]] over a [[Riemann surface]]), its _zeta function_ is the [[analytic continuation]] of the [[trace]]
$$
\zeta_H(s) \coloneqq Tr(H^{-s})
$$
of the $-s$ power of $H$, which, if $H$ is suitably self-adjoint, is
the [[sum]] of the $-s$-powers of all its [[eigenvalues]], as a function of $s$. This is analogous to the _[[Riemann zeta function]]_ and the [[Dedekind zeta function]] (or would be if there were something like a [[Laplace operator]] on [[Spec(Z)]] or more generally on an [[arithmetic curve]], see at _[[function field analogy]]_).
The exponential of the derivative of the zeta function at $n = 0$ also encodes the [[functional determinant]] of $H$, a [[regularization (physics)|regularized]] version ("[[zeta function regularization]]") of the naive and generally ill-defined product of all eigenvalues. As such, zeta functions play a central role in [[quantum field theory]].
Generally, the values of $\zeta_H(s)$ of interest in [[physics]] (when regarding $H$ as a [[Hamilton operator]]) are those for (low) integral $s$. These are just the _[[special values of L-functions]]_.
## Definition
### The zeta function
Given an [[elliptic differential operator]] with positive lower bound $c$, write $H$ for its [[self-adjoint extension]] and write
$$
0 \lt \lambda_1 \leq \lambda_2 \leq \cdots
$$
for its [[eigenvalues]].
+-- {: .num_defn #ZetaBySeries}
###### Definition
The _zeta function_ of $H$ is the [[holomorphic function]] defined by the [[series]]
$$
\begin{aligned}
\zeta_H(s)
& \coloneqq
Tr( H^{-s} )
\\
& \coloneqq
\underoverset{n = 1}{\infty}{\sum} \frac{1}{(\lambda_n)^s}
\end{aligned}
\,.
$$
where this [[convergence|converges]] and then extended by [[analytic continuation]].
=--
(e.g. ([Duistermaat-Guillemin 75 (2.13)](#DuistermaatGuillemin75), [Berline-Getzler-Vergne 04, section 9.6](#BerlineGetzlerVergne04) ) ).
### Functional determinant and zeta-function regularization
{#FunctionalDeterminant}
Notice that the first [[derivative]] $\zeta^\prime_H$ of this zeta function is, where the original series converges, given by
$$
\zeta_H^\prime(s)
=
\sum_{n = 1}^\infty \frac{- \ln \lambda_n}{ (\lambda_n)^s}
\,.
$$
Therefore one says ([Ray-Singer 71](#RaySinger71)) that the _[[functional determinant]]_ of $H$ is the exponential of the derivative of zeta function of $H$ at 0:
$$
det_{reg} H
\coloneqq
\exp(- \zeta_H^\prime(0))
\,.
$$
Via the [[analytic continuation]] involved in defining $\zeta_H(0)$ in the first place, this may be thought of as a _[[regularization (physics)|regularization]]_ of the ill-defined naive definition "$\prod_n \lambda_n$" of the [[determinant]] of $H$. As such functional determinants often appear in [[quantum field theory]] as what is called _[[zeta function regularization]]_.
Conversely, the [[logarithm]]
$$
Z \coloneqq - \frac{1}{2}\zeta_H^\prime(0) = \tfrac{1}{2} log\,det_{reg} H
$$
is what is called the _[[vacuum energy]]_ in [[quantum field theory]] (for $H^{-1}$ the [[Feynman propagator]]).
If $H = D^2$ has a square root $D$ (a [[Dirac operator]]-type square root as in [[supersymmetric quantum mechanics]]) then under some conditions on the growth of the eigenvalues, then the functional determinant may also be expressed in terms of the [[eta function]] of $D$ as
$$
det H
=
det (D^2)
=
\exp( \frac{\partial}{\partial s}\frac{\partial}{\partial c} \eta_{D}(0))
\,.
$$
See at _[eta invariant -- Relation to zeta function](eta+invariant#RelationToTheZetaFunction)_ for more on this.
### Relation to partition functions and number-theoretic zeta/theta functions
{#AnalogyWithNumberTheoreticZetaFunctions}
By basic [[integration]] identities we have that
+-- {: .num_prop #IntegralKernelExpression}
###### Proposition
The series expression in def. \ref{ZetaBySeries} is equal to the [[Mellin transform]] of the [[partition function]]
$$
\zeta_H(s)
=
\int_{(0,\infty)}
t^{s-1}
\left(
\underset{\lambda_k \neq 0}{\sum}
\exp(-t \lambda_k)
\right)
d t
\,.
$$
=--
(see e.g. [Quine-Heydari-Song 93 (8)](#QuineHeydariSong93), [Richardson, pages 8-9](#Richardson), [BCEMZ 03, section A.2](#BCEMZ03), [Connes-Marcolli 06, theorem 13.11](#ConnesMarcolli06)).
+-- {: .num_remark}
###### Remark
If one thinks of the operation $H$ as a [[Hamiltonian]] of a [[quantum mechanical system]], then the term
$$
Tr(\exp(-\beta H))
\coloneqq
\underset{\lambda_k \neq 0}{\sum}
\exp(-\beta \lambda_k)
+ 1
$$
is the _[[partition function]]_ of this system. Accordingly, prop. \ref{IntegralKernelExpression} says that the zeta function of $H$ is obtained from its partition function by
$$
\zeta_H(s)
=
\int_{(0,\infty)}
\beta^{s-1}
\;
\left(Tr(\exp(-\beta H)) - 1 \right)
\;
d \beta
\,.
$$
=--
Further, by a change of integration variable $t\coloneqq x^2$
in the expression in prop. \ref{IntegralKernelExpression}
one obtains
+-- {: .num_prop #IntegralKernelExpressionInSquares}
###### Proposition
The series expression in def. \ref{ZetaBySeries} is equal to
$$
\begin{aligned}
\zeta_H(s)
& =
2
\int_{(0,\infty)}
x^{2s-1}
\left(
\underset{\lambda_k \neq 0}{\sum}
\exp(- x^2 \lambda_k)
\right)
d x
\end{aligned}
\,.
$$
In particular if $H = D^2$ is the square of a [[Dirac operator]]/[[supersymmetric quantum mechanics]]-type square root operator $D$ with [[eigenvalues]] $\pm \alpha_k$,then $\lambda_k = \alpha_k^2$ and hence in this case the series is
$$
\begin{aligned}
\zeta_H(s)
& =
2
\int_{(0,\infty)}
x^{2s-1}
\left(
\underset{\alpha_k \neq 0}{\sum}
\exp(- (x \alpha_k)^2)
\right)
d x
\end{aligned}
\,.
$$
=--
By comparison one observes:
+-- {: .num_remark}
###### Remark
The integral expression in prop. \ref{IntegralKernelExpressionInSquares}
is analogous to the expression of [[zeta functions]] in [[number theory]]/[[arithmetic geometry]] as integrals of a [[theta function]] (for instance discussed [here](Riemann%20zeta%20function#RelationToThetaFunctions) for the [[Riemann zeta function]])
$$
\hat\zeta_f(2 s)
=
\int_{(0,\infty)} (\theta(x^2) - 1) x^{2s-1} d x
\,.
$$
Under this analogy the [[theta function]] in the case of the differential operator $H$ is
$$
\theta_H(x)
\coloneqq
\underset{\lambda_k \neq 0}{\sum}
\exp(- x \lambda_l)
\,.
$$
This is formally the same definition as that of adelic theta functions (e.g.[Garrett 11, section 1.8](Iwasawa-Tate%20theory#Garrett11))
=--
+-- {: .num_remark}
###### Remark
The [[determinant line bundle]] of the [[functional determinant]] of the [[Dirac operator]] on a [[complex torus]] is a complex-analytic [[theta function]] as above, quotiented by the [[Dedekind eta function]].
Early references explaining this include [Alvarez-Gaumé & Moore & Vafa 86](#AlvaresGaumeMooreVafa86), [Alvarez-Gaumé & Bost & Moore & Nelson & Vafa 87](#AlvaresGaumeBostMooreNelsonVafa87). In a bigger perspective, this relation plays a central role in the general discussion of [[self-dual higher gauge theory]] ([Witten 96](self-dual%20higher%20gauge%20theory#Witten96)).
=--
## Examples
### Of Laplace operator on complex torus and Dedekind eta function
For $\mathbb{C}/(\mathbb{Z}\oplus \tau \mathbb{Z})$
a [[complex torus]] (complex [[elliptic curve]]) equipped with its
standard flat [[Riemannian metric]], then the [[zeta function of an elliptic differential operator|zeta function]] of the corresponding [[Laplace operator]] $\Delta$ is
$$
\zeta_{\Delta} = (2\pi)^{-2 s} E(s)
\coloneqq
(2\pi)^{-2 s} \underset{(k,l)\in \mathbb{Z}\times\mathbb{Z}-(0,0)}{\sum}
\frac{1}{{\vert k +\tau l\vert}^{2s}}
\,.
$$
The corresponding [[functional determinant]] is
$$
\exp(
E^\prime_{\Delta}(0)
)
= (Im \tau)^2 {\vert \eta(\tau)\vert}^4
\,,
$$
where $\eta$ is the [[Dedekind eta function]].
(recalled e.g. in [Todorov 03, page 3](#Todorov03))
For more see also at _[[zeta function of a Riemann surface]]_.
### Analytic torsion
The functional determinant of a [[Laplace operator]] of a [[Riemannian manifold]] acting on [[differential n-forms]] is up to a sign in the exponent a factor in what is called the _[[analytic torsion]]_ of the manifold.
## Related concepts
* [[partition function]]
[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]
## References
### General
An early reference is
* {#DuistermaatGuillemin75} [[Hans Duistermaat]], [[Victor Guillemin]], _The Spectrum of Positive Elliptic Operators and Periodic Bicharacteristics_,Inventiones mathematicae (1975) Volume: 29, page 39-80 ([EuDML](https://eudml.org/doc/142329))
(see also at _[[Duistermaat-Guillemin trace formula]]_)
Textbook accounts include
* {#BerlineGetzlerVergne04} [[Nicole Berline]], [[Ezra Getzler]], [[Michèle Vergne]], section 9.6 of _Heat Kernels and Dirac Operators_
Review includes
* {#Richardson} [[Ken Richardson]], section 3 of _Introduction to the Eta invariant_ ([pdf](http://faculty.tcu.edu/richardson/Seminars/etaInvariant.pdf))
* {#QuineHeydariSong93} J. R. Quine, S. H. Heydari, R. Y. Song, _Zeta regularized products_, Transactions of the AMS volume 338, number 1, 1993 ([[QuineZetaRegularization.pdf:file]])
* Wikipedia, _[Functional determinant -- Zeta function version](http://en.wikipedia.org/wiki/Functional_determinant#Zeta_function_version)_
* Wikipedia, _[Zeta function regularization](http://en.wikipedia.org/wiki/Zeta_function_regularization)_
* {#ConnesMarcolli06} [[Alain Connes]], [[Matilde Marcolli]], _A walk in the noncommutative garden_ ([arXiv:0601054](http://arxiv.org/abs/math/0601054))
### Zeta function regularization
* {#Speer71} [[Eugene Speer]], _On the structure of Analytic Renormalization_, Comm. math. Phys. 23, 23-36 (1971) ([Euclid](http://projecteuclid.org/euclid.cmp/1103857549))
* {#BCEMZ03} A. Bytsenko, G. Cognola, [[Emilio Elizalde]], [[Valter Moretti]], S. Zerbini, section 2 of _Analytic Aspects of Quantum Fields_, World Scientific Publishing, 2003, ISBN 981-238-364-6
### Functional determinant
The definition of a _[[functional determinant]]_ via the exponential of the derivative of the zeta function at 0 originates in
* {#RaySinger71} D. Ray, [[Isadore Singer]], _R-torsion and the Laplacian on Riemannian manifolds_, Advances in Math. 7: 145–210, (1971) doi:10.1016/0001-8708(71)90045-4, MR 0295381
Discussion in the special case of [[2d CFT]] ([[worldsheet]] [[string theory]]) is in
* {#AlvaresGaumeMooreVafa86} [[Luis Alvarez-Gaumé]], [[Gregory Moore]], [[Cumrun Vafa]], _Theta functions, modular invariance, and strings_, Communications in Mathematical Physics Volume 106, Number 1 (1986), 1-4 ([Euclid](http://projecteuclid.org/euclid.cmp/1104115581))
* {#AlvaresGaumeBostMooreNelsonVafa87} [[Luis Alvarez-Gaumé]], [[Jean-Benoit Bost]], [[Gregory Moore]], Philip Nelson, [[Cumrun Vafa]], _Bosonization on higher genus Riemann surfaces_, Communications in Mathematical Physics, Volume 112, Number 3 (1987), 503-552 ([Euclid](http://projecteuclid.org/euclid.cmp/1104159982))
* {#Todorov03} [[Andrey Todorov]], _The analogue of the Dedekind eta function for CY threefolds_, 2003 [pdf](http://www.ma.huji.ac.il/conf/crelle.pdf)
[[!redirects zeta functions of elliptic differential operators]]
|
zeta function regularization | https://ncatlab.org/nlab/source/zeta+function+regularization |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Theta functions
+--{: .hide}
[[!include theta functions - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In the context of [[regularization (physics)|regularization in physics]], _zeta function regularization_ is a method/prescription for extracing finite values for [[traces]] of powers of [[Laplace operators]]/[[Dirac operators]] by
1. considering $s$-powers for all values of $s$ in the [[complex plane]] where the naive trace does make sense and then
1. using [[analytic continuation]] to obtain the desired [[special values of L-functions|special value]] at $s = 1$ -- as for [[zeta functions]].
### Analytic regularization of propagators
One speaks of _analytic regularization_ ([Speer 71](#Speer71)) or _zeta function regularization_ (e.g. [M 99](#M99), [BCEMZ 03, section 2](#BCEMZ03)) if a [[Feynman propagator]]/[[Green's function]] for a [[boson|bosonic]] [[field (physics)|field]], which is naively given by the expression
"$Tr\left(\frac{1}{H}\right)$" (for $H$ the given [[wave operator]]/[[Laplace operator]]) is made well defined by interpreting it as the [[principal value]] of the [[special values of L-functions|special value]] at $s= 1$
$$
Tr_{reg}
\left(\frac{1}{H}\right)
\coloneqq
pv\, \zeta_H(1)
$$
of the [[zeta function of an elliptic differential operator|zeta function]] which is given by the expression
$$
\zeta_H(s)
\coloneqq
Tr\left(
\frac{1}{H}
\right)^s
$$
for all values of $s \in \mathbb{C}$ for which the right hand side exists, and is defined by [[analytic continuation]] elsewhere.
Analogously the zeta function regularization of the [[Dirac propagator]] for a [[fermion]] [[field (physics)|field]] with [[Dirac operator]] $D$ is defined by
$$
Tr_{reg}
\left(\frac{D}{D^2} \right)
\coloneqq
pv\, \eta_D(1)
$$
where $\eta$ is the [[eta function]] of $D$.
### Functional determinants
Notice that the first [[derivative]] $\zeta^\prime_H$ of this [[zeta function of an elliptic differential operator|zeta function]] is, where the original series converges, given by
$$
\zeta_H^\prime(s)
=
\sum_{n = 1}^\infty \frac{- \ln \lambda_n}{ (\lambda_n)^s}
\,.
$$
Therefore the _[[functional determinant]]_ of $H$ ([Ray-Singer 71](#RaySinger71)) is the exponential of the zeta function of $H$ at 0:
$$
Det_{reg} H
\coloneqq
\exp(- \zeta_H^\prime(0))
\,.
$$
(see also [BCEMZ 03, section 2.3](#BCEMZ03))
Via the [[analytic continuation]] involved in defining $\zeta_H(0)$ in the first place, this may be thought of as a _[[regularization (physics)|regularization]]_ of the ill-defined naive definition "$\prod_n \lambda_n$" of the [[determinant]] of $H$. As such functional determinants often appear in [[quantum field theory]] as what is called _[[zeta function regularization]]_.
### Higher amplitudes
Accordingly, more general [[scattering amplitudes]] are controled by [[multiple zeta functions]] (...).
## Examples
### Of Laplace operator on complex torus and Dedekind eta function
For $\mathbb{C}/(\mathbb{Z}\oplus \tau \mathbb{Z})$
a [[complex torus]] (complex [[elliptic curve]]) equipped with its
standard flat [[Riemannian metric]], then the [[zeta function of an elliptic differential operator|zeta function]] of the corresponding [[Laplace operator]] $\Delta$ is
$$
\zeta_{\Delta} = (2\pi)^{-2 s} E(s)
\coloneqq
(2\pi)^{-2 s} \underset{(k,l)\in \mathbb{Z}\times\mathbb{Z}-(0,0)}{\sum}
\frac{1}{{\vert k +\tau l\vert}^{2s}}
\,.
$$
The corresponding [[functional determinant]] is
$$
\exp(
E^\prime_{\Delta}(0)
)
= (Im \tau)^2 {\vert \eta(\tau)\vert}^4
\,,
$$
where $\eta$ is the [[Dedekind eta function]].
(recalled e.g. in [Todorov 03, page 3](#Todorov03))
### Zeta regularization for divergent integrals ###
the zeta regularizatio method can be extended to include also a regularization for the divergent integrals $ \int_{a}^{\infty}x^{m}dx $ which appears in QFT, this is made by means of the identity
$$\begin{array}{l}
\int_{a}^{\infty }x^{m-s} dx =\frac{m-s}{2} \int_{a}^{\infty }x^{m-1-s} dx +\zeta (s-m)-\sum_{i=1}^{a}i^{m-s} +a^{m-s} \\
-\sum_{r=1}^{\infty }\frac{B_{2r} \Gamma (m-s+1)}{(2r)!\Gamma (m-2r+2-s)} (m-2r+1-s)\int_{a}^{\infty }x^{m-2r-s} dx \end{array} $$
for the case of $ m=-1$ although the harmonic series has a pole we can regularize by the 2 possibilities
$ \sum_{n=0}^{\infty} \frac{1}{n+a} = -\Psi (a) $ or $ \sum_{n=0}^{\infty} \frac{1}{n+a} = -\Psi (a)+log(a) $ in particular
$ \sum_{n=1}^{\infty} \frac{1}{n} = \gamma $ Euler-Mascheroni constant, and $ \Psi(a)= -\frac{\Gamma '(a)}{\Gamma (a)} $
So within this reuglarization there wouldn't be any UV ultraviolet divergence
### Analytic torsion
The functional determinant of a [[Laplace operator]] of a [[Riemannian manifold]] acting on [[differential n-forms]] is up to a sign in the exponent a factor in what is called the _[[analytic torsion]]_ of the manifold.
## Related concepts
[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]
## References
Original articles include
* {#Speer71} [[Eugene Speer]], _On the structure of Analytic Renormalization_, Comm. Math. Phys. 23, 23-36 (1971) ([Euclid](http://projecteuclid.org/euclid.cmp/1103857549))
* {#RaySinger71} D. Ray, [[Isadore Singer]], _R-torsion and the Laplacian on Riemannian manifolds_, Advances in Math. 7: 145–210, (1971) doi:10.1016/0001-8708(71)90045-4, MR 0295381
Modern accounts and reviews include
* {#Freed87} [[Daniel Freed]], page 8 of _On determinant line bundles_, Math. aspects of [[string theory]], ed. S. T. Yau, World Sci. Publ. 1987, (revised [pdf](http://www.math.utexas.edu/~dafr/Index/determinants.pdf), [dg-ga/9505002](http://arxiv.org/abs/dg-ga/9505002))
* {#Elizalde95} [[Emilio Elizalde]], _Ten Physical Applications of Spectral Zeta Functions_ (1995)
* {#M99} [[Valter Moretti]], _Local z-function techniques vs point-splitting procedures: a few rigorous results_
Commun. Math. Phys. 201, 327 (1999).
* {#BCEMZ03} A. Bytsenko, G. Cognola, [[Emilio Elizalde]], [[Valter Moretti]], S. Zerbini, section 2 of _Analytic Aspects of Quantum Fields_, World Scientific Publishing, 2003, ISBN 981-238-364-6
* {#Robles09} [[Nicolas Robles]], _Zeta function regularization_, 2009 ([[RoblesZetaRegularization.pdf:file]])
See also
* {#Todorov03} [[Andrey Todorov]], _The analogue of the Dedekind eta function for CY threefolds_, 2003 [pdf](http://www.ma.huji.ac.il/conf/crelle.pdf)
[[!redirects zeta function regularizations]]
[[!redirects functional determinant]]
[[!redirects functional determinants]]
[[!redirects zeta-function regularization]]
[[!redirects zeta-function regularizations]]
|
zeta polynomial | https://ncatlab.org/nlab/source/zeta+polynomial | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Combinatorics
+-- {: .hide}
[[!include combinatorics - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The **zeta polynomial** $Z_P(n)$ of a finite [[partially ordered set]] $P$ counts the number of multichains (also known as "weakly increasing sequences") of length $n$ in $P$.
## Definition
By a _multichain_ of length $n$ in $P$, we mean a sequence of elements $x_0 \le x_1 \le \cdots \le x_n$, which can be identified with an [[order-preserving function]] from the [[linear order]] $[n] = \{ 0 \lt 1 \lt \cdots \lt n \}$ into $P$.
To see that
$$Z_P(n) = |Hom([n],P)|$$
defines a [[polynomial]] in $n$,[^Note] first observe that any function $[n] \to P$ factors as a [[surjection]] from $[n]$ onto some $[k] = \{ 0 \lt 1 \lt \cdots \lt k \}$ (where $k \le n$), followed by an [[injection]] from $[k]$ to $P$. The total number of order-preserving functions from $[n]$ to $P$ can therefore be calculated explicitly as
$$Z_P(n) = \sum_{k=0}^{d} b_k \binom{n}{k}$$
where $b_k$ is the number of _chains_ $x_0 \lt x_1 \lt \cdots \lt x_k$ in $P$ (i.e., injective order-preserving functions from $[k]$ to $P$), and where $d$ is the length of the longest chain. Hence $Z_P(n)$ is a polynomial of degree equal to the length of the longest chain in $P$.
[^Note]: Note that the definition we use here for $Z_P(n)$ has an index shift from the definition that seems to be more standard in combinatorics. For example, the definition in [(Stanley, 3.12)](#StanleyEC1) counts multichains of length $n-2$ rather than of length $n$. Accordingly, one should apply a substitution to get some of the properties stated here to match equivalent results in the literature.
## Examples
The zeta polynomial of $[2] = \{ 0 \lt 1 \lt 2 \}$ is
$$3 + 3n + \binom{n}{2} = \frac{n^2 + 5n + 6}{2}$$
For example, evaluating the polynomial at $n=0$ and $n=1$ confirms that $[2]$ contains 3 points and 6 [[intervals]], while evaluating it at $n=2$ confirms that there are 10 order-preserving functions from $[2]$ to itself.
The zeta polynomial of the 5-element poset
$$P = \array{&&v&& \\ &&\uparrow&& \\ &&u&& \\ &\nearrow& &\nwarrow& \\ y &&&& z \\ &\nwarrow& &\nearrow& \\ &&x&&}$$
is $5 + 9n + 7\binom{n}{2} + 2\binom{n}{3} = \frac{2n^3 + 15n^2 + 37n + 30}{6}$.
Evaluating at $n=1$, we compute that $P$ contains 14 distinct intervals.
## Properties
### Relation to order polynomial
The [[order polynomial]] is related to the zeta polynomial by the equation
$$
\Omega_P(n+2) = Z_{P^\downarrow}(n)
$$
where $P^\downarrow \cong (2)^P$ is the lattice of [[lower sets]] in $P$.
This can be seen as a consequence of the [[currying]] isomorphisms
$$Hom([n], P^\downarrow) \cong Hom([n] \times P, (2)) \cong Hom(P, [n]^\downarrow)$$
together with the isomorphisms $[n]^\downarrow \cong [n+1] \cong (n+2)$.
### Relation to zeta function
Using the formalism of [[incidence algebras]], the zeta polynomial has a simple expression in terms of the _zeta function_ of $P$ (defined by $\zeta_P(x,y) = 1$ if $x\le y$ and $\zeta_P(x,y) = 0$ otherwise):
$$Z_P(n) = \sum_{x,y\in P} \zeta_P^n(x,y)$$
where $\zeta_P^n$ is the $n$-fold convolution product of $\zeta_P$. (In other words, if we view the zeta function as a square matrix, then the zeta polynomial is the sum of the entries in its $n$-fold matrix product.) This follows immediately from the definition of the convolution product,
$$(f\cdot g)(x,y) = \sum_{x \le z \le y} f(x,z) \cdot g(z,y)$$
since $\zeta_P^n(x,y)$ computes the number of multichains of length $n$ in $P$ from $x$ to $y$.
As a special case, if $P$ has both a [[bottom]] element 0 and a [[top]] element 1, then
$$Z_P(n) = \zeta_P^{n+2}(0,1)$$
since an arbitrary multichain $x_0 \le x_1 \le \cdots \le x_n$ of length $n$ can be extended to a multichain $0 \le x_0 \le x_1 \le \cdots \le x_n \le 1$ of length $n+2$ between 0 and 1.
## Related concepts
* [[order polynomial]]
* [[incidence algebra]]
* [[Möbius inversion]]
* [[nerve]]
## References
* Paul H. Edeleman. Zeta Polynomials and the Möbius Function. European Journal of Combinatorics 1(4), 1980.
{#Edelman80}
* Richard P. Stanley, _Enumerative combinatorics_, vol.1 ([pdf](http://www-math.mit.edu/~rstan/ec/ec1.pdf))
{#StanleyEC1}
* Joseph P. S. Kung, [[Gian-Carlo Rota]], Catherine H. Yan. Combinatorics: The Rota Way. Cambridge, 2009.
{#KungRotaYan}
category: combinatorics |
zeta-functions and eta-functions and theta-functions and L-functions -- table | https://ncatlab.org/nlab/source/zeta-functions+and+eta-functions+and+theta-functions+and+L-functions+--+table |
| context/[[function field analogy]] | [[theta function]] $\theta$ | [[zeta function]] $\zeta$ (= [[Mellin transform]] of $\theta(0,-)$) | [[L-function]] $L_{\mathbf{z}}$ (= [[Mellin transform]] of $\theta(\mathbf{z},-)$) | [[eta function]] $\eta$ | [[special values of L-functions]] |
|---|-----|-----------|---|-----|----|
| [[physics]]/[[2d CFT]] | [[partition function]] $\theta(\mathbf{z},\mathbf{\tau}) = Tr(\exp(-\mathbf{\tau} \cdot (D_\mathbf{z})^2))$ as function of [[complex structure]] $\mathbf{\tau}$ of [[worldsheet]] $\Sigma$ (hence [[polarization]] of [[phase space]]) and [[background field|background]] [[gauge field]]/[[source]] $\mathbf{z}$ | analytically continued [[trace]] of [[Feynman propagator]] $\zeta(s) = Tr_{reg}\left(\frac{1}{(D_{0})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tau$ | analytically continued [[trace]] of [[Feynman propagator]] in [[background field|background]] [[gauge field]] $\mathbf{z}$: $L_{\mathbf{z}}(s) \coloneqq Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(\mathbf{z},\tau)\, d\tau$ | analytically continued [[trace]] of [[Dirac propagator]] in [[background field|background]] [[gauge field]] $\mathbf{z}$ $\eta_{\mathbf{z}}(s) = Tr_{reg} \left(\frac{sgn(D_{\mathbf{z}})}{ { \vert D_{\mathbf{z}} } \vert }\right)^s $ | [[zeta function regularization|regularized]] [[1-loop vacuum amplitude]] $pv\, L_{\mathbf{z}}(1) = Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)$ / [[zeta function regularization|regularized]] fermionic [[1-loop vacuum amplitude]] $pv\, \eta_{\mathbf{z}}(1)= Tr_{reg} \left( \frac{D_{\mathbf{z}}}{(D_{\mathbf{z}})^2} \right)$ / [[vacuum energy]] $-\frac{1}{2}L_{\mathbf{z}}^\prime(0) = Z_H = \frac{1}{2}\ln\;det_{reg}(D_{\mathbf{z}}^2)$ |
| [[Riemannian geometry]] ([[analysis]]) | | [[zeta function of an elliptic differential operator]] | [[zeta function of an elliptic differential operator]] | [[eta function of a self-adjoint operator]] | [[functional determinant]], [[analytic torsion]] |
| [[complex analytic geometry]] | [[section]] $\theta(\mathbf{z},\mathbf{\tau})$ of [[line bundle]] over [[Jacobian variety]] $J(\Sigma_{\mathbf{\tau}})$ in terms of covering coordinates $\mathbf{z}$ on $\mathbb{C}^g \to J(\Sigma_{\mathbf{\tau}})$ | [[zeta function of a Riemann surface]] | [[Selberg zeta function]] | | [[Dedekind eta function]] |
| [[arithmetic geometry]] for a [[function field]] | | [[Goss zeta function]] (for [[arithmetic curves]]) and [[Weil zeta function]] (in [[higher dimensional arithmetic geometry]]) | | | |
| [[arithmetic geometry]] for a [[number field]] | [[Hecke theta function]], [[automorphic form]] | [[Dedekind zeta function]] ([being](Artin+L-function#RelationToDedekindZeta) the [[Artin L-function]] $L_{\mathbf{z}}$ for $\mathbf{z} = 0$ the [[trivial representation|trivial]] [[Galois representation]]) | [[Artin L-function]] $L_{\mathbf{z}}$ of a [[Galois representation]] $\mathbf{z}$, expressible "in coordinates" (by [[Artin reciprocity]]) as a finite-order [[Hecke L-function]] (for 1-dimensional representations) and generally (via [[Langlands correspondence]]) by an [[automorphic L-function]] (for higher dimensional reps) | | [[class number]] $\cdot$ [[regulator]] |
| [[arithmetic geometry]] for $\mathbb{Q}$ | [[Jacobi theta function]] ($\mathbf{z} = 0$)/ [[Dirichlet theta function]] ($\mathbf{z} = \chi$ a [[Dirichlet character]]) | [[Riemann zeta function]] (being the [[Dirichlet L-function]] $L_{\mathbf{z}}$ for [[Dirichlet character]] $\mathbf{z} = 0$) | [[Artin L-function]] of a [[Galois representation]] $\mathbf{z}$ , expressible "in coordinates" (via [[Artin reciprocity]]) as a [[Dirichlet L-function]] (for 1-dimensional Galois representations) and generally (via [[Langlands correspondence]]) as an [[automorphic L-function]] | | |
[[!redirects zeta-functions and eta-functions and L-functions -- table]]
[[!redirects zeta functions table]] |
ZFA | https://ncatlab.org/nlab/source/ZFA |
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### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--
# Zermelo--Fraenkel set theory with atoms
* table of contents
{: toc}
## Idea
**ZFA** is a variant of the [[material set theory]] [[ZF]] which allows for objects, called _atoms_ or _[[urelements]]_ (hence the alternative name ZFU), which may be members of [[sets]], but are not made up of other elements. ZFA featured in early independence proofs, notably [[Fraenkel-Mostowski permutation models]], for example showing AC is independent of the rest of the axioms of ZFA.
[[Zermelo|Zermelo's]] original 1908 axiomatisation of set theory included atoms, but they were soon discarded as a foundational approach as they could be modeled inside of atomless set theory.
## Definition
There are two possible approaches to formulating ZFA. In both cases, we can further require that the [[axiom of choice]] is satisfied, and obtain ZFCA.
### Empty atoms
In this approach, atoms are empty. We start by adding an additional unary predicate $A$, where we interpret $A(x)$ as saying "$x$ is an atom". We write $(\forall set x)$ to mean $\forall x, \neg A(x) \Rightarrow$, and similarly write $(\forall atom x)$ to mean $\forall x, A(x) \Rightarrow$.
Then the [[axiom of extensionality]] says
$$
(\forall set x) (\forall set y) (\forall z, z \in x \Leftrightarrow z \in y) \Rightarrow x = y,
$$
and the [[axiom of empty set]] says
$$
(\exists set x) (\forall y) (y \notin x).
$$
We also add the axiom that says atoms are empty:
$$
(\forall atom a) (\forall x) x \notin a.
$$
Sometimes it is also convenient to assume that we have a set of atoms:
$$
(\exists X) (\forall x) (A(x) \Leftrightarrow x \in X),
$$
but in some cases, we might also like to consider models with a proper class of atoms.
### Reflexive/Quine atoms
We can give up on the [[axiom of foundation]], and introduce the urelements as [[reflexive sets]], ie. sets $x$ such that $x = \{x\}$. In place of the axiom of foundation, we can have an axiom of _weak_ foundation, where we require the existence of a set A such that every element of A is reflexive, and the [[cumulative hierarchy]] built up from $A$ is the whole universe. In other words, if we define
* $R(0) = A$,
* $R(\alpha + 1) = P(R(\alpha))$ for any ordinal $\alpha$,
* $R(\lambda) = \bigcup_{\gamma \lt \lambda} R(\gamma)$ for $\lambda$ a limit ordinal,
then $V = \bigcup_\alpha R(\alpha)$.
## Models
### Fraenkel--Mostowski models
By allowing atoms in our models, we lend ourselves to the method of [[Fraenkel-Mostowski models]], where we can obtain models in which the [[axiom of choice]] fails by imposing some symmetry on the atoms (so that we cannot uniformly pick an atom out of many). Such models are closely related to [[categories of G-sets]].
## Related concepts
* [[ZFC]]
* [[Fraenkel-Mostowski models]]
* [[permutation model]]
* [[urelements]]
[[!redirects ZFA]]
[[!redirects ZFU]]
[[!redirects ZFCA]]
[[!redirects ZFAC]]
|
ZFC | https://ncatlab.org/nlab/source/ZFC |
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### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--
# Contents
* table of contents
{:toc}
## Idea
The most commonly accepted standard [[foundation of mathematics]] today is a [[material set theory]] commonly known as _Zermelo--Fraenkel set theory with the [[axiom of choice]]_ or $ZFC$ for short. There are many variations on that theory (including [[constructive mathematics|constructive]] and [[class]]-based versions, which are also discussed here).
Accompanying ZFC, especially taking into account the [[axiom of foundation]], is a picture (or 'ontology') of material sets forming a [[cumulative hierarchy]] organized by an [[ordinal number|ordinal]]-valued rank function. This picture (sometimes referred to as the 'iterative conception') considers sets as generated by starting at bottom with the [[empty set]] and building to higher ranks by applying a [[power set]] operation to get to a next successor ordinal rank, and taking [[unions]] to get to limit ordinal ranks. This iterative conception finds alternative expression in [[algebraic set theory]].
## History
The first version was developed by [[Ernst Zermelo]] in 1908; in 1922, [[Abraham Fraenkel]] and [[Thoralf Skolem]] independently proposed a precise [[first order logic|first-order]] version with the [[axiom of replacement]]; von Neumann added the [[axiom of foundation]] in 1925. All of these versions included the [[axiom of choice]], but this was considered controversial for some time; one has merely $ZF$ if it is taken out.
$ZFC$ is similar to the [[class]] theories [[NBG]] (due to [[John von Neumann]], [[Paul Bernays]], and [[Kurt Gödel]]) and [[Morse–Kelley set theory|MK]] (due to [[Anthony Morse]] and [[John Kelley]]). The former is a conservative, finitely axiomatisable extension of $ZFC$, while the latter is stronger and cannot be finitely axiomatised (although a conservative extension involving meta-classes could be).
Contemporary set theorists often accept additional [[large cardinal]] [[axioms]], which can greatly increase the strength of $ZFC$, far beyond even $MK$. Other additional axioms which are sometimes added are the [[axiom of determinacy]] (or various weaker versions of it) or the [[axiom of constructibility]]. There are also weaker variants of $ZFC$, especially for [[constructive mathematics|constructive]] and [[predicative mathematics|predicative]] mathematics. Then there are alternatives on a different basis, notably [[New Foundations|NFU]] (a very [[predicative mathematics|impredicative]] [[material set theory]] with a set of all sets) and [[ETCS]] (a [[structural set theory]]).
(The source for this history, especially the dates, is mostly [the English Wikipedia](http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory).)
## Axioms
$ZFC$ is a [[simple type theory|simply typed]] [[first-order theory|first-order]] [[material set theory]], with a single type $V$ called the __[[cumulative hierarchy]]__ and a binary __membership__ [[predicate]] $\in$ on $V$. Every term of $V$ in standard $ZFC$ is a [[pure set]], which we will call simply a set.
A set $a$ is said to belong to, be in, be a __member__ of, or be an __element__ of a set $b$ if $a \in b$, and set $b$ is said to have the member $a$.
There are also variations of ZFC with non-set terms such as [[urelements]] and [[classes]]. Urelements may be distinguished from sets and classes since they have no elements (although the [[empty set]] also has no elements); sets are usually those classes that are themselves elements (members) of sets. Urelements are also called *atoms*, and $ZF$ with atoms included is sometimes called [[ZFA]] or $ZFU$.
1. [[axiom of extensionality|Extensionality]]: If two sets have the same members, then they are __[[equality|equal]]__ and themselves members of the same sets. See [[axiom of extensionality]] for variations, such as whether this is taken as a definition or an axiomatisation of equality of sets, and how the condition might be strengthened if (10) is left out.
2. [[axiom of the null set|Null Set]]: There is an __[[empty set]]__: a set $\empty$ with no elements. By (1), it follows that this set is unique; by even the weakest version of (5), it is enough to state the existence of some set. (Analogous remarks apply to axioms (3), (4), (6), (7), and (8), except when (5) is omitted and (4) and (6) are needed to derive it.)
3. [[axiom of pairing|Pairing]]: If $a$ and $b$ are sets, then there is a set $\{a,b\}$, the __[[unordered pairing]]__ of $a$ and $b$, whose elements are precisely those sets equal to $a$ or $b$. Between them, (2) and (3) form a nullary/binary pair; the unary version follows from (3), since $\{a\} = \{a,a\}$ by (1). The higher finitary versions rely also on (4) (with $\{a,b,c\} = \{a\} \cup \{b,c\}$, etc), so they should probably be stated explicitly as an [[axiom scheme]] if (4) is omitted (but nobody seems to do that).
4. [[axiom of union|Union]]: If $\mathcal{C}$ is a set, then there is a set $\bigcup \mathcal{C}$, the __[[union]]__ of $\mathcal{C}$, whose elements are precisely the elements of the elements of $\mathcal{C}$. It is normal to write $A \cup B$ for $\bigcup \{A,B\}$, etc, $A \cup B \cup C$ for $\bigcup \{A,B,C\}$, etc. Note that $\bigcup \{A\} = A$ and $\bigcup \empty = \empty$ (using (1) to prove these results), so no special notation is needed for these.
5. [[axiom of separation|Separation/Specification/Comprehension]]: Given any [[predicate]] $\phi[x]$ in the language of set theory with the chosen [[free variable]] $x$ of type $V$, if $U$ is a set, then there is a set $\{x \in U \;|\; \phi[x]\}$, the __[[subset]]__ of $U$ given by $\phi$, whose elements are precisely those elements $x$ of $U$ such that $\phi[x]$ holds. There are many variations, from Bounded Separation to Full Comprehension, which we should probably describe at [[axiom of separation]]; the version listed here is Full Separation. This is an [[axiom scheme]], but it can be made a single axiom in $NBG$ (but not completely in $MK$). Note that (5) follows from (4) and (6) using [[classical logic]], so it is often left out, but it must be included in intuitionistic variations, variations where (6) is omitted or weakened (or where (4) is omitted or weakened, in principle, although that never happens in practice), and variations where (5) itself is strengthened.
6. [[axiom of replacement|Replacement/Collection]]: Given a [[predicate]] $\psi[x,Y]$ with the chosen [[free variables]] $x$ and $Y$ of type $V$, if $U$ is a set and if for every $x$ in $U$ there is a unique $Y$ such that $\psi[x,Y]$ holds, then there is set $\{\iota\,Y.\;\psi[x,Y] \;|\; x \in U\}$, the __[[image]]__ of $U$ under $\psi$, whose elements are precisely those sets $Y$ such that there is an element $x$ of $U$ such that $\psi[x,Y]$ holds; if $\Psi[x]$ is a defined term, then we write $\{\Psi[x] \;|\; x \in U\}$ for $\{\iota\,Y.\; Y = \Psi[x] \;|\; x \in U\}$. Again there are many variations, from Weak Replacement to Strong Collection, which we should probably describe at [[axiom of replacement]]; the one described here is Replacement. This is also an [[axiom scheme]], but it can be made into a single axiom in both $NBG$ and $MK$. One could combine this with (5) to produce $\{\iota\,Y.\;\psi[x,Y] \;|\; x \in U \;|\; \phi[x]\}$, but nobody seems to do this.
7. [[axiom of power sets|Power Sets]]: If $U$ is a set, then there is a set $\mathcal{P}U$, the __[[power set]]__ of $U$, whose elements are precisely the [[subsets]] of $U$, that is the sets $A$ whose elements are all elements of $U$. When using [[intuitionistic logic]], it is possible to accept only a weak version of this, such as [[subset collection|Subset Collection]] or (even weaker) [[function sets|Exponentiation]]. But in classical logic, Power Sets follows from Exponentiation and the weakest form of (5).
8. [[axiom of infinity|Infinity]]: There is a set $\omega$ of __[[ordinal number|finite ordinals]]__ as pure sets. Normally one states that $\empty \in \omega$ and $a \cup \{a\} \in \omega$ whenever $a \in \omega$, although variations are possible. Using any but the weakest version of (6), it is enough to state that there is a set satisfying [[Peano arithmetic|Peano's axioms]] of [[natural numbers]], or even any [[Dedekind-infinite set]]. It seems to be uncommon to incorporate (2) into (8), but in principle (8) implies (2).
9. [[axiom of choice|Choice]]: If $\mathcal{C}$ is a set of pairwise disjoint sets, each of which has an element, then there is a set with exactly one element from each element of $\mathcal{C}$. Note that this set is *not* unique, nor can we construct a canonical version which is, so we do not give it any name or notation. This version is the simplest to state in the language of $ZFC$; see [[axiom of choice]] for further discussion and weak versions. It is possible to incorporate (9) into (5) or (6), but this seems to be rare.
10. [[axiom of foundation|Foundation/Regularity/Induction]]: Given a formula $\phi$ with a chosen [[free variable]] $X$ of type $V$, if $\phi$ holds whenever $\phi[a/X]$ holds for every $a \in X$, then $\phi$ holds absolutely. For variations (including the axiom of anti-foundation), see [[axiom of foundation]]. This [[axiom scheme]] can be made into a single axiom even in $ZFC$ itself (although not in versions with intuitionistic logic; in that case it can be made a single axiom only in a class theory).
## Variations
Zermelo\'s original version consists of axioms (1--5) and (7--9), in a somewhat imprecise form (which affects the interpretation of 5) of [[higher-order logic|higher-order]] [[classical logic]].
The modern $ZF$ consists of (1--8) and (10), using [[first-order logic|first-order]] [[classical logic]], the strongest form of (6) (that is, Strong Collection, although the standard Replacement is sufficient with [[classical logic]]), and the strongest form of (5) possible using only sets and not classes (Full Separation). Since Full Separation follows from Replacement with [[classical logic]], it is often omitted from the list of axioms.
$ZFC$ adds (9) and is thus the strongest version without classes or additional axioms. The version originally formulated by Fraenkel and Skolem did not include (10), although the three founders all eventually accepted it.
It is common to take __Zermelo set theory__ ($\mathrm{Z}$) to be $ZF$ without (6), although Zermelo never accepted the first-order formulation, and his axioms did not include the Axiom of Foundation (10). Note that the weakest form (Weak Replacement) of (6) follows from (7) and (5), so it holds even in $\mathrm{Z}$.
Another variant is __bounded Zermelo set theory__ ($BZ$), which is like $\mathrm{Z}$ but with only Bounded Separation; this is of interest to category theorists because $BZC$ is equivalent to [[ETCS]].
### Constructive versions
See also [[constructive set theory]].
The most well-known foundations for [[constructive mathematics]] through material set theory are [[Peter Aczel]]\'s __constructive Zermelo--Fraenkel set theory__ ($CZF$) and [[John Myhill]]\'s __intuitionistic Zermelo--Fraenkel set theory__ ($IZF$).
$CZF$ uses axioms (1--8) and (10), usually weak forms, in [[intuitionistic logic]]; specifically, it uses Bounded Separation for (5), Strong Collection for (6), and an intermediate (Subset Collection) form of (7). $IZF$ is simliar, but it uses Full Separation for (5) and the full strength of (7); Myhill\'s original version uses only Replacement for (6), but Collection (equivalent to Strong Collection using Full Separation) is standard now.
Note that adding (9) to $IZF$ implies [[excluded middle]] and so makes $ZFC$. However, some authors like to include a weak form of (9), such as [[dependent choice]] or [[COSHEP]].
[[Mike Shulman]]\'s survey of material and structural set theories ([Shulman 2018](#Shulman2018)) takes $CPZ^{\circlearrowleft-}$ as the most basic form; it consists of (1--4) and the weakest versions (Bounded Separation and Weak Replacement) of (5&6) in intuitionistic logic. Adding (10) gives $CPZ^{-}$, adding (8) gives $CPZ^{\circlearrowleft}$, and adding both gives $CPZ$, __constructive pre-Zermelo set theory__. Shulman gives systematic notation for other versions, which includes those (constructive and classical) listed above.
Myhill has another version, __constructive set theory__ ($CST$); this consists of (1--4), Bounded Separation for (5), Replacement for (6), the weakest (Exponentiation) form of (7), (8), and a weak version (Dependent Choice) of (9). It also uses a variation of the language, with [[urelements]] for natural numbers; note that the existence of $\omega$ still follows using (6). This classifies $CST$ as $\mathrm{C}{\Pi}ZF^{\circlearrowleft} + DC$ in Shulman\'s system if one ignores the use of urelements and strengthens Replacement to Strong Collection.
### Class theories
__[[Morse--Kelley class theory]]__ ($MK$) features both [[sets]] and [[proper classes]]. This allows it to strengthen (5) to Full Comprehension, since $\phi$ can include [[quantification]] over classes; the same holds in (6) and (10), although this does not add any additional strength.
__Von Neumann--Bernays--Gödel class theory__ ($NBG$) uses the same language as $MK$, but it still uses only Full Separation for (5). This makes it conservative over $ZFC$ and also allows for a finite axiomatisation; we replace the formulas in (5) and (6) with classes, and add some special cases of (5) for subclasses, one for each logical connective. (It is provable that plain $ZF$, if consistent, cannot be finitely axiomatized in its own first-order language; $NBG$ escapes this conclusion by extending the language with the notion of classes.)
One can also rework all of the weak versions of set theory above into a class theory like $NBG$, which is conservative over the original set theory. One can also use a class theory like $MK$, although this destroys any attempt to use a weak version of (5).
### Large cardinals
One often adds axioms for [[large cardinals]] to $ZFC$. Even (8) can be seen as a large cardinal axiom, stating that $\aleph_0$ exists. These additional axioms are most commonly studied in the context of a material set theory, but they work just as well in a structural set theory.
Note that adding an [[inaccessible cardinal]] (commonly considered the smallest sort of large cardinal) to $ZFC$ is already stronger than $MK$: given an inaccessible cardinal $\kappa$, one can interpret the sets and classes in $MK$ as the sets in $V_\kappa$ and $V_{\kappa+1}$, respectively. Of course, one can add a large cardinal to $MK$ to get something even stronger.
It is often convenient to assume that one always has more large cardinals when necessary. You cannot say this in an absolute sense, but you can adopt the axiom that every set belongs to some [[Grothendieck universe]]. Adding this axiom to $ZFC$ makes __[[Tarski-Grothendieck set theory]]__ ($TG$). This is not the last word, however; you can make it stronger by adding classes in the style of $MK$, or even adding a cardinal which is inaccessible from $TG$. In fact, we have barely begun the large cardinals known to modern set theory!
### Miscellaneous variations
The [[axiom of constructibility]], usually notated "$V = L$", is a very strong axiom that can be added to $ZF$; it asserts that all sets belong to the [[constructible universe]] $L$, which can be "constructed" in a definable way through a transfinite procedure. This notion of "constructible" should not be confused with [[constructive mathematics]]; for instance, $V = L$ implies the [[axiom of choice]] and thus also [[excluded middle]] even with intuitionistic logic. $V = L$ also implies the [[generalized continuum hypothesis]] ($GCH$), which is how Gödel originally proved that $GCH$ was consistent with $ZFC$. However, it is incompatible with the sufficiently large cardinals: the existence of a [[measurable cardinal]] implies that $V \neq L$. Most contemporary set theorists do not regard $V = L$ as potentially "true."
The [[axiom of determinacy]] ($AD$) is another axiom that can be added to $ZF$; it asserts that a certain class of infinite [[game]]s is determined (one player or the other has a winning strategy). $AD$ is inconsistent with the full axiom of choice, although it is consistent with dependent choice. A weaker form of $AD$ called "projective determinacy" is consistent with $AC$ and is equiconsistent with certain large cardinal assertions.
The $GCH$ itself, or its negation, could also be regarded as an additional axiom that can be added to $ZF$. Many set theorists would prefer to find a more "natural" axiom, such as a large cardinal axiom, which implies either $GCH$ or its negation. The equiconsistency of projective determinacy with a large cardinal assertion can be regarded as a step in this direction.
### Relation to structural set theories
The structural set theory **[[ETCS]]** is equivalent to $BZC$ in that the [[category of sets]] in that theory satisfies $ETCS$ while the well-founded [[pure sets]] in $ETCS$ satisfy $BZC$. This uses (1--4), Bounded Separation for (5), and (7--10), with Weak Replacement following from (5) and (7).
[[Mike Shulman]]\'s **[[SEAR|SEARC]]** is equivalent to $ZFC$ in the same way. $SEAR$, which lacks the axiom of choice, is equivalent to $ZF^{\circlearrowleft}$, which is $ZF$ without (10), in a weaker sense of equivalence.
[Shulman 2018](#Shulman2018) is an extensive survey of variations of $ZFC$ and variations of $ETCS$ (mostly weak ones), showing how these correspond.
## Related entries
* [[material set theory]]
* [[algebraic set theory]]
* [[cumulative hierarchy]]
* [[structural ZFC]]
## References
Zermelo's axiomatisation grew out of his reflections on his proofs of the well-ordering theorem (1904/08) and was published in
* [[Ernst Zermelo]], _Untersuchungen über die Grundlagen der Mengenlehre I_ , Math. Ann. **65** (1908) pp.261-81. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002262002))
English versions of the early key texts on set theory by Zermelo, Fraenkel, Skolem, von Neumann et al. can be found in
* J. van Heijenoort, _From Frege to Gödel - A Sourcebook in Mathematical Logic 1879-1931_ , Harvard UP 1967.
There are many texts which discuss ZFC and the [[cumulative hierarchy]] from a traditional (material) set-theoretic perspective. A good example is
* Kenneth Kunen, _Set Theory: An Introduction to Independence Proofs_, Studies in Logic and the Foundations of Mathematics Vol. 102 (2006), Elsevier.
A classification of axioms of variants of $ZFC$, with an eye towards corresponding [[structural set theories]], is
* [[Michael Shulman]] (2018). Comparing material and structural set theories. [arXiv:1808.05204](https://arxiv.org/abs/1808.05204).
{#Shulman2018}
[[!redirects ZF]]
[[!redirects ZFC]]
[[!redirects Zermelo-Fraenkel set theory]]
[[!redirects Zermelo–Fraenkel set theory]]
[[!redirects Zermelo--Fraenkel set theory]]
[[!redirects Zermelo-Fraenkel]]
[[!redirects Zermelo–Fraenkel]]
[[!redirects Zermelo--Fraenkel]]
[[!redirects Zermelo-Fraenkel axioms]]
[[!redirects Zermelo–Fraenkel axioms]]
[[!redirects Zermelo--Fraenkel axioms]]
[[!redirects Zermelo set theory]]
[[!redirects CPZ]]
|
Zhaohui Luo | https://ncatlab.org/nlab/source/Zhaohui+Luo |
* [webpage](http://www.cs.rhul.ac.uk/~zhaohui/)
## Selected writings
On [[type theory]] in [[computer science]] (via the [[calculus of constructions]]):
* [[Zhaohui Luo]], *Computation and Reasoning -- A Type Theory for Computer Science*, Clarendon Press (1994) $[$[ISBN:9780198538356](https://global.oup.com/academic/product/computation-and-reasoning-9780198538356), [pdf](https://www.researchgate.net/profile/Zhaohui-Luo-4/publication/243770570_Computation_and_Reasoning_A_Type_Theory_for_Computer_Science/links/54d0dc6b0cf29ca81103f26a/Computation-and-Reasoning-A-Type-Theory-for-Computer-Science.pdf)$]$
## Related entries
* [[type universe]]
* [[coercion]]
* [[dependent type theoretic methods in natural language semantics]]
category: people
|
Zhen Huan | https://ncatlab.org/nlab/source/Zhen+Huan |
* [webpage](https://huanzhen84.github.io/zhenhuan/)
* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=233130)
## Selected writings
On [[equivariant elliptic cohomology]] in the form of [[orbifold K-theory]] on [[Huan's inertia orbifolds]] (a form of [[twisted ad-equivariant Tate K-theory]]):
* [[Zhen Huan]], _Quasi-elliptic cohomology_, 2017 ([hdl](http://hdl.handle.net/2142/97268))
* [[Zhen Huan]], *Quasi-Elliptic Cohomology and its Power Operations*, J. Homotopy and Related Structures **13** (2018) 715–767 [[arXiv:1612.00930](https://arxiv.org/abs/1612.00930), [doi:10.1007/s40062-018-0201-y](https://doi.org/10.1007/s40062-018-0201-y)]
* [[Zhen Huan]], *Quasi-elliptic cohomology and its Spectrum* ([arXiv:1703.06562](https://arxiv.org/abs/1703.06562))
* {#Huan18} [[Zhen Huan]], _Quasi-Elliptic Cohomology I_, Advances in Mathematics, Volume 337, 15 October 2018, Pages 107-138 ([arXiv:1805.06305](https://arxiv.org/abs/1805.06305), [doi:10.1016/j.aim.2018.08.007](https://doi.org/10.1016/j.aim.2018.08.007))
* [[Zhen Huan]], *Quasi-theories* [[arXiv:1809.06651](https://arxiv.org/abs/1809.06651)]
* [[Zhen Huan]], *Quasi-theories and their equivariant orthogonal spectra* [[arXiv:1809.07622](https://arxiv.org/abs/1809.07622)]
> (via [[orthogonal spectra]])
* [[Zhen Huan]], [[Matthew Spong]], *Twisted Quasi-elliptic cohomology and twisted equivariant elliptic cohomology* ([arXiv:2006.00554](https://arxiv.org/abs/2006.00554))
* [[Zhen Huan]], [[Nathaniel Stapleton]], *Level structures on p-divisible groups from the Morava E-theory of abelian groups* ([arXiv:2001.10075](https://arxiv.org/abs/2001.10075))
Review:
* [[Zhen Huan]], *Quasi-elliptic cohomology theory and the twisted, twisted Real theories*, talk at Perimeter Institute (2020) [video: [doi:10.48660/20050059](https://doi.org/10.48660/20050059)]
Generalization to [[twisted equivariant K-theory|twisted equivariant]] [[KR-theory]]:
* [[Zhen Huan]], [[Matthew B. Young]], *Twisted Real quasi-elliptic cohomology* [[arXiv:2210.07511](https://arxiv.org/abs/2210.07511)]
On [[2-representations]]:
* [[Zhen Huan]], _2-Representations of Lie 2-groups and 2-Vector Bundles_ ([arXiv:2208.10042](https://arxiv.org/abs/2208.10042))
## Related entries
* [[Tate K-theory]]
* [[orbifold K-theory]]
* [[equivariant elliptic cohomology]]
* [[Huan's inertia orbifold]], [[cyclic loop space]]
category: people
|
Zhen Lin > history | https://ncatlab.org/nlab/source/Zhen+Lin+%3E+history |
see *[[Zhen Lin Low]]* |
Zhen Lin Low | https://ncatlab.org/nlab/source/Zhen+Lin+Low |
* [webpage](https://zll22.user.srcf.net/dpmms/)
* [blog](http://semmibol.wordpress.com/)
## Selected writings
On [[Grothendieck universes]] for [[category theory]]:
* [[Zhen Lin Low]], _Universes for category theory_ [[arxiv/1304.5227](http://arxiv.org/abs/1304.5227)]
On [[geometric homotopy theory|geometric]] [[homotopy theory]] via [[simplicial objects]] [[internalization|internal]] to a [[Grothendieck topos]] ([[simplicial sheaves]]):
* {#Low14} [[Zhen Lin Low]], _Internal and local homotopy theory_ (2014) $[$[arXiv:1404.7788](https://arxiv.org/abs/1404.7788)$]$
On [[simplicial homotopy theory]], [[(∞,1)-category theory]] and [[(∞,1)-topos theory]]:
* {#Low15} [[Zhen Lin Low]], *[[Notes on homotopical algebra]]*, 2015 ([pdf](http://zll22.user.srcf.net/writing/homotopical-algebra/2015-11-10-Main.pdf))
On generalization of the notion of [[schemes]] (to something like [[Structured Spaces]]):
* {#Low16} [[Zhen Lin Low]], *Categories of spaces built from local models*, 2016 ([pdf](https://www.repository.cam.ac.uk/bitstream/handle/1810/256998/Low-2016-PhD.pdf;jsessionid=76AD1098F39D92EDDFE705324F14BBA5?sequence=1), [[LowCategoriesOfSpaces.pdf:file]])
category: people
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