Allan J. Silberger

* [personal page](https://academic.csuohio.edu/silberger/)

## Related concepts

On [[L-parameters]] in the [[local Langlands correspondence]]

* {#SilbergerZink14} [[Allan J. Silberger]], [[Ernst-Wilhelm Zink]], Section 3 of: *Langlands Classification for L-Parameters*, Journal of Algebra Volume 511, 1 October 2018, Pages 299-357 ([arXiv:1407.6494](https://arxiv.org/abs/1407.6494), [doi:10.1016/j.jalgebra.2018.06.012](https://doi.org/10.1016/j.jalgebra.2018.06.012))

Talk notes:

* [[Ernst-Wilhelm Zink]], with [[Allan J. Silberger]], *Langlands classification for L-parameters* ([pdf](https://www.math.hu-berlin.de/~zyska/zink/Lvortrag1.pdf))

category: people

[[!redirects Allan Silberger]]


Allan L. Edelson

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=1087)

## Selected writings

On [[Real vector bundles]] and their [[KR-theory]]:

* [[Allan L. Edelson]], *Real Vector Bundles and Spaces with Free Involutions*, Transactions of the American Mathematical Society **157** (1971) 179-188 [[doi:10.2307/1995841](https://doi.org/10.2307/1995841), [jstor:1995841](https://www.jstor.org/stable/1995841)]

category: people


Allan Merino

* [personal page](http://allanmerino.com/)

## Selected writings

On [[reductive dual pair|reductive dual pairs]]/[[Howe duality]] for [[super Lie algebras]]:

* [[Allan Merino]], Hadi Salmasian, *Classification and double commutant property for dual pairs in an orthosymplectic Lie supergroup* [[arXiv:2208.09746](https://arxiv.org/abs/2208.09746)]

category: people


allegorical set theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

There are two general approaches to [[structural set theory]]; those that attempt to axiomatise the [[category]] [[Set]] of [[sets]] and [[functions]], and those that attempt to axiomatise the [[allegory]] [[Rel]] of [[sets]] and [[relations]]. The latter is called **allegorical set theory**.

## Examples

* [[SEAR]] is an example of a dependently sorted allegorical set theory with primitives of [[sets]], [[elements]], and [[relations]]

## See also

* [[structural set theory]]

* [[categorical set theory]], [[allegorical set theory]]

[[!redirects allegorical set theory]]
[[!redirects allegorical set theories]]


allegory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Relations
+-- {: .hide}
[[!include relations - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea ##

In [[category theory]], an _allegory_ is a [[category]] with properties meant to reflect properties that hold in a category [[Rel]] of [[relations]]. The notion was first introduced (as far as we know) and certainly first made famous in the book _[[Categories, Allegories]]_ ([Freyd-Scedrov](#FreydScedrov)).

Freyd and Scedrov argue that a categorical calculus of relations is an alternative and often more amenable framework for developing concepts traditionally couched in "functional" language (i.e., concepts which apply to [[sets]] and [[functions]]); for instance, a principal _raison d'etre_ for [[regular categories]] is precisely that one can do [[relational calculus]] in them (as had been long known, e.g., [[Saunders MacLane]] showed how to calculate with relations to do [[diagram chasing|diagram chases]] in [[abelian categories]]). Allegories, and correlative notions such as [[bicategory of relations|bicategories of relations]], also offer a smooth approach to [[regular and exact completion|regular and exact completions]], as used for example in the construction of [[realizability topos|realizability toposes]].

A signal feature of allegories is emphasis on the _modular law_ (see def. \ref{Allegory} below), which generalizes the [[modular lattice|modular law]] in lattices to more general relations, and which generalizes also so-called [[Frobenius reciprocity]] in [[categorical logic]].

## Definition

+-- {: .num_defn #Allegory}
###### Definition

An **allegory** is a [[locally posetal 2-category]] $A$ equipped with an [[involution]] $(-)^o \colon A^{op} \to A$ which is the [[identity-on-objects]], such that

1. the involution is order preserving and distributes over composition, i.e. $ (\psi\phi)^o = \phi^o\psi^o $,
1. each [[hom-object|hom-poset]] $A(x,y)$ has binary [[meet|meets]], and
1. the *[[modular lattice|modular law]]* holds: for $\phi\colon x\to y$, $\psi\colon y\to z$, and $\chi\colon x\to z$, we have $\psi \phi \cap \chi \le \psi (\phi \cap \psi^o \chi)$.

=--

From these properties we immediately get that
\[
(\phi \cap \psi)^o = \phi^o \cap \psi^o
\]
and
\[
  (\phi \cap \psi) \chi \leq \phi\chi \cap \psi\chi
\quad\text{and}\quad
  \chi (\phi \cap \psi) \leq \chi\phi \cap \chi\psi.
\]
\begin{proof}
The first claim follows from the observation that
$$
\begin{aligned}
    \chi \leq (\phi \cap \psi)^o
  &\iff
    \chi^o \leq \phi \cap \psi
&\quad&(.)^o\;\text{monotone and involutive}
\\
  &\iff
    \chi^o \leq \phi \; \text{ and } \; \chi^o \leq \psi
&&\text{meet properties}
\\
  &\iff
    \chi \leq \phi^o \; \text{ and } \; \chi \leq \psi^o
&&(.)^o\;\text{ monotone and involutive}
\\
  &\iff
    \chi \leq \phi^o \cap \psi^o
&&\text{meet properties.}
\end{aligned}
$$

For the claim $(\phi \cap \psi) \chi \leq \phi\chi \cap \psi\chi$ we use the [[horizontal composition]] in a [[locally posetal 2-category]], i.e. the fact that composition is monoton. Due to $\phi \cap \psi \leq \phi$ we get $(\phi \cap \psi)\chi \leq \phi\chi$. Analogously, we get that $(\phi \cap \psi)\chi \leq \psi\chi$. Hence $(\phi \cap \psi) \chi \leq \phi\chi \cap \psi\chi$. The claim $\chi (\phi \cap \psi) \leq \chi\phi \cap \chi\psi$ follows the the same arguments or by applying involution and the first claim.
\end{proof}

## Examples

* If $C$ is a [[regular category]] and $Rel(C)$ is the [[locally posetal 2-category|locally posetal]]  [[bicategory]] of [[internal relation|internal relations]], then $Rel(C)$ is an allegory.

* Any [[first-order hyperdoctrine with equality]] similarly gives rise to an allegory, as does any abstract [[bicategory of relations]] in the sense of Carboni-Walters.

* Any [[modular lattice]] can be regarded as a one-object allegory if we take composition to be union and the involution to be the identity.

## Maps, tabulations, and units

A __map__ $ r\colon x \to y $ in an allegory is a morphism that has a [[right adjoint]]. If $r \dashv s$, then $s = r^o$ (hint: use the modular law to show $r \dashv s \cap r^o$ and $r \cap s^o \dashv s$).
The [[unit of an adjunction|unit of the adjunction]] $ id_x \leq r^o r $ entails that the morphism is __[[entire relation|entire]]__ (sometimes also called _total_) while the [[unit of an adjunction|counit of the adjunction]] $ r r^o \leq id_y $ states the fact hat the morphism is __[[functional relation|functional]]__ (sometimes also called _univalent_).

Any 2-category has a [[bicategory of maps]].  In an allegory, the ordering between maps is discrete, meaning that if $f \leq g$ then $f = g$.  Consequently, the bicategory of maps of an allegory is a [[category]].

A __tabulation__ of a morphism $\phi$ is a pair of maps $f,g$ such that $\phi = g f^o$ and $f^o f \cap g^o g = 1$.  An allegory is **tabular** if every morphism has a tabulation, and **pretabular** if every morphism is contained in one that has a tabulation.

Every [[regular category]], and indeed every [[locally regular category]], has a tabular allegory of internal binary relations.  Conversely, by restricting to the morphisms with left adjoints ("maps") in a tabular allegory, we obtain a locally regular category.  These constructions are inverse, so tabular allegories are equivalent to locally regular categories.

A locally regular category has finite products if and only if its tabular allegory of relations has top elements in its hom-posets.

Finally, a **unit** in an allegory is an object $U$ such that $1_U$ is the greatest morphism $U\to U$, and every object $X$ admits a morphism $\phi\colon X\to U$ such that $1_X\le \phi^o\phi$.  A locally regular category has a terminal object (hence is regular) if and only if its tabular allegory of relations has a unit.

Thus, regular categories are equivalent to unital (or unitary) tabular allegories.  For more details, see [[Categories, Allegories]] ([Freyd-Scedrov](#FreydScedrov)), the [[Elephant]] ([Johnstone](#Johnstone)), or [[toddtrimble:Theory of units and tabulations in allegories]].

## Division allegories

A **union allegory** is an allegory whose hom-posets have finite joins that are preserved by composition.  Thus a union allegory is locally a [[lattice]].  If additionally it is locally a [[distributive lattice]], it is called a **distributive allegory**.  The category of maps in a unitary tabular union allegory is a [[coherent category]] (a "pre-logos"), and conversely the bicategory of relations in a coherent category is a (unitary tabular) distributive allegory.  In particular, every unitary tabular union allegory is distributive, but in the non-tabular case this can fail: for instance, any [[modular lattice]] can be regarded as a one-object union allegory and need not be distributive.

A **division allegory** is a distributive allegory in which composition on one (and therefore the other) side has a right adjoint (left or right division). That is: given $r: A \to B$ and $s \colon A \to C$, there exists $s/r: B \to C$ such that

$$t \leq s/r \in \hom(B, C) \Leftrightarrow t \circ r \leq s \in \hom(A, C)$$

(so that $- \circ r$ has right adjoint $-/r$: an example of a right [[Kan extension]]). Composition on the other side, $r \circ -$, has a right adjoint (an example of a right Kan lift) given by

$$r\backslash u \coloneqq (u^o/r^o)^o.$$

In the bicategory of sets and relations, with notation as above, we have

$$(s/r)(b, c) \dashv \vdash \forall_{a \colon A} r(a, b) \Rightarrow s(a, c)$$

where $r(a, b)$ is shorthand for "$(a, b)$ belongs to $r$".

The category of maps (functional relations) of a unitary/unital tabular division allegory is a [[Heyting category]] (a "[[logos]]"), and conversely the bicategory of relations in a Heyting category is a unitary tabular division allegory. ([Freyd-Scedrov](#FreydScedrov), 2.32, p. 227.)

## Power allegories

A **power allegory** is, *more or less*, an allegory $\mathcal{A}$ such that the inclusion functor $i: Map(\mathcal{A}) \to \mathcal{A}$ has a right adjoint $P$. The idea is that $P$ assigns to an object $A$ a power object $P(A)$, as in topos theory; if we summarize the notion of topos as a regular category $\mathbf{E}$ for which the inclusion $i: \mathbf{E} \to Rel(\mathbf{E})$ has a right adjoint $P$, then it becomes apparent that the notion of power allegory is similar except that it takes the "relation side" as primary and derives the "function side" as $Map(\mathcal{A})$, whereas in topos theory it's just the other way around. But in either case the adjunction $i \dashv P$ is fundamental.

Since the inclusion $i$ is the identity on objects, the counit of the adjunction $i \dashv P$ at an object $B$ may be written

$$\ni_B: P(B) \to B$$

and we have a kind of [[comprehension scheme]] that to each $r: A \to B$ there is a unique map $\chi_r: A \to P(B)$, the *characteristic map* of $r$, such that $r = \ni_B \circ \chi_r$. (If $A$ and $B$ are related by a property $r$, then for each $a$ there is a subobject $\chi_r(a)$ of $B$ consisting of elements $b$ so related to $a$.) In the case $r = 1_A: A \to A$, the characteristic map $\chi_{1_A}: A \to P(A)$ is called the *singleton map* of $A$, and more general $\chi_r$ may be defined as $P(r) \sigma_A$.

### Freyd-Scedrov definition

The exact definition of power allegory is a matter for consideration. One can get a certain distance just by adopting the naive definition suggested above, that a power allegory is nothing more than an allegory for which the inclusion $Map(\mathcal{A}) \to \mathcal{A}$ has a right adjoint $P$. (Below we introduce a similar notion that we call a $P$-allegory.) But it seems hard to develop a theory from the naive notion that rises to a level comparable to topos theory.

Freyd and Scedrov start with a structure of division allegory (thus packing in a good amount of internal logic from the start) and introduce the fundamental adjunction $i \dashv P$ in terms of that structure. For them, a *power allegory* is defined to be a division allegory which associates to each object $B$ a morphism $\ni_B \colon P(B) \to B$ such that for all $r \colon A \to B$

* $1_A \leq (r \backslash \ni_B) \circ (\ni_B \backslash r)$

which expresses the truth of the formula $\forall_{a \colon A} \exists_{S: P(B)} \forall_{b \colon B} S \ni_B b \Leftrightarrow r(a, b)$, and

* $(\ni_B \backslash \ni_B) \wedge (\ni_B \backslash \ni_B)^o \leq 1_{P(B)}$

which internalizes an axiom of extensionality, which reads $\forall_{b \colon B} (S \ni_B b) \Leftrightarrow (T \ni_B b) \vdash S = T$. Given those axioms, and given $r: A \to B$, one may define

$$\chi_r \coloneqq (\ni_B \backslash r) \wedge (r\backslash \ni_B)^o,$$

which internalizes the formula-definition $\chi_r(a, S) \coloneqq \forall_b S \ni_B b \Leftrightarrow r(a, b)$, and then show $\chi_r$ is a map. ([Freyd-Scedrov](#FreydScedrov), pp. 235-236.)

The bicategory of relations in a [[topos]] is a power allegory; conversely, the category of maps in a unitary tabular power allegory is a topos.

### Variant notion

Nevertheless, the spare elegance of the naive definition gives one something to shoot for. It appears that quite a decent theory can be developed just by adding the assumption of coproducts in an allegory: a reasonable and fairly mild assumption. (Most allegories don't admit many colimits; for example having coequalizers is pretty rare. But the standard examples do have finite coproducts, coinciding with coproducts on the maps/functional side.)

+-- {: .num_defn}
###### Definition
A $P$-allegory is an allegory $\mathcal{A}$ with finite coproducts[^1] for which the inclusion $i: Map(\mathcal{A}) \to \mathcal{A}$ has a right adjoint $P$.
=--

[^1]: We mean coproducts as certain conical colimits *qua* [[locally posetal 2-category]].

As before, the counit is denoted $\ni: i P \to 1_{\mathcal{A}}$. It is perhaps surprising that the notion of $P$-allegory is at least as strong as power allegory in the Freyd-Scedrov sense:

+-- {: .num_theorem}
###### Theorem
Any $P$-allegory is a division allegory in which the Freyd-Scedrov conditions on $\ni$ are satisfied.
=--

+-- {: .proof}
###### Proof (sketch)
For now we content ourselves with a description of the division structure.
Let $r: A \to C$ and $s: B \to C$ be morphisms; we construct a right Kan lift $s \backslash r$ of $r$ through $s$. This means that for all $t: A \to B$ we have $s \circ t \leq r$ if and only if $t \leq s \backslash r$. We begin by constructing the right Kan lift for the case $r = s = \ni_C: P C \to C$.

Define the internal union $\bigcup_C$ by $\bigcup_C = P(\ni_C): P P C \to P C$, and define an internal order relation $[\Rightarrow]_C: P C \to P C$ by $P C \stackrel{\in_{P C}}{\to} P P C \stackrel{\bigcup_C}{\to} P C$. (Here $\in \coloneqq \ni^o$.) Define $[\Leftarrow]_C \coloneqq [\Rightarrow]_C^o$.

We now show $[\Leftarrow]_C = \ni_C \backslash \ni_C$. That is, for $R: P C \to P C$, we show $\ni_C R \leq \ni_C$ is equivalent to $R \leq [\Leftarrow]$. The backward implication is easy; for the forward implication, one may prove it first for *reflexive* $R$ where we have an actual equation $\ni_C R = \ni_C$. It follows that $P(\ni_C) P(R) = P(\ni_C)$, whence $P(R) \leq \bigcup_C^o \bigcup_C$. A short calculation then yields $R = \ni_C P(R) \sigma_{P C} = \ni_C \bigcup_C^o \bigcup_C \sigma_{P C} = \ni_C \bigcup_C^o = [\Leftarrow]$. The case for general $R$ reduces to the reflexive case: form the reflexive completion $1 \vee R$ as the composite

$$C \stackrel{\langle 1_{P C}, R \rangle}{\to} C + C \stackrel{\nabla}{\to} C$$

which is where coproducts come in; here $\nabla$ is the codiagonal and we use the fact that $C + C$ in an allegory is a biproduct to form the pairing for the first arrow. It is easy to show that $\ni_C(1 \vee R) = \ni_C \vee \ni_C R = \ni_C$, and then we derive $R \leq 1 \vee R \leq [\Leftarrow]$ from before. Thus we have shown $[\Leftarrow]_C = \ni_C \backslash \ni_C$.

For general $r: A \to C, s: B \to C$, we claim the right Kan lift $s \backslash r: A \to B$ is given by $\chi_s^o (\ni_C \backslash \ni_C) \chi_r$. For, we have $r = \ni_C \chi_r$, whence

$$
\begin{aligned}
    s t &\leq r &
  \;&\iff & \ni_C \chi_s t &\leq \ni_C \chi_r
\\
 &&& \iff & \ni_C \chi_s t \chi_r^o &\leq \ni_C
\\
 &&& \iff & \chi_s t \chi_r^o &\leq \ni_C \backslash \ni_C
\\
 &&& \iff & t &\leq \chi_s^o (\ni_C \backslash \ni_C) \chi_r
\end{aligned}
$$

thus proving the claim.

Further details may be found [here](https://ncatlab.org/toddtrimble/published/Note+on+power+allegories).
=--

Thus the notion of $P$-category is just as strong as the Freyd-Scedrov notion of power allegory, and one can then piggy-back on their further developments.

## Syntactic allegories

Let $T$ be a [[regular theory]].  There is then an allegory $\mathcal{A}_T$ given as follows:

* the objects are finite strings of [[sorts]] of $T$;

* a morphism $\vec X \to \vec Y$ is a predicate $P(\vec x, \vec y)$ of sort $\vec X, \vec Y$ (or rather a provable-equivalence class of such predicates);

* the identity $\vec X \to \vec X$ is (named by) $x_1 = x_1 \wedge x_2 = x_2 \wedge \cdots \wedge x_n = x_n$;

* the composite of $R \colon \vec X \to \vec Y$ and $S \colon Y \to \vec X$ is named by $\exists \vec y. R(\vec x, \vec y) \wedge S(\vec y, \vec z)$.

That $\mathcal{A}_T$ is an allegory is B.311 in [Freyd--Scedrov](#FreydScedrov); that it is in fact unitary and pre-tabular is B.312.

Further structure on $T$ gives rise to further structure on $\mathcal{A}_T$ (B.313): if $T$ is a [[coherent theory|coherent]], [[first-order theory|first-order]] or [[higher-order logic|higher-order]] theory, then $\mathcal{A}_T$ will be a distributive, division or power allegory respectively.

Every pre-tabular allegory has a tabular completion, given by splitting its coreflexive morphisms (i.e. those endomorphisms $R$ such that $R \subset id$).  The category of maps in the coreflexive splitting of $\mathcal{A}_T$ is precisely the [[syntactic category]] of $T$.

### The existential quantifier

There are two possible ways to interpret a regular formula of the form $\exists y. R(x,y) \wedge S(y,z)$ in a unitary pre-tabular allegory, if $R$ and $S$ are interpreted as $r \colon X \to Y$ and $s \colon Y \to Z$ respectively:  as the composite $s \cdot r$, or more 'literally' by:

* pulling $r$ and $s^o$ back to the same hom set and taking their intersection: $(p_1^o r p_1) \cap (p_2^o s^o p_2) \colon X \times Z \to Y \times Y$;

* then forcing the two $Y$s to be equal by post-composing with $\Delta_Y^o \colon Y \times Y \to Y$, applying the existential quantifier by post-composing with the unique map $!_Y$ to the unit, and post-composing with $!_Z^o$ to get a morphism into $Z$;

* then forcing the $Z$ in the domain to be equal to the $Z$ in the codomain by taking the meet with $p_2 \colon X \times Z \to Z$;

* and finally pulling back along (precomposing with the right adjoint of) $p_1 \colon X \times Z \to X$ to get a morphism $X \to Z$.

We would like to know that these morphisms are equal, so that an existential formula will have a unique interpretation:

+-- {: .num_prop}
###### Proposition
$$ s r = (p_2 \cap !_Z^o !_Y \Delta_Y^o ((p_1^o r p_1) \cap (p_2^o s^o p_2))) p_1^o  $$
=--

+-- {: .proof}
###### Proof
We show inclusion in each direction.

Firstly, $s r = s r \cap p_2 p_1^o$, because the product projections tabulate the top morphism.  Notice also that the RHS above is equal to
$$ (\top_{Y Z} (r p_1 \cap s^o p_2) \cap p_2) p_1^o  $$
where $\top_{Y Z}$ is the top morphism $Y \to Z$.

Now we can calculate:
$$
\begin{aligned}
  s r \cap p_2 p_1^o
  & = (s r p_1 \cap p_2) p_1^o &\quad& \text{modular law} \\
  & = (s r p_1 \cap p_2 \cap p_2) p_1^o \\
  & \leq (s(r p_1 \cap s^o p_2) \cap p_2) p_1^o && \text{modular law} \\
  & \leq (\top_{Y Z} (r p_1 \cap s^o p_2) \cap p_2) p_1^o
\end{aligned}
$$

In the other direction, we have
$$
\begin{aligned}
  (\top_{Y Z} (r p_1 \cap s^o p_2) \cap p_2) p_1^o
  & \leq (\top_{Y Z} s^o (s r p_1 \cap p_2) \cap p_2) p_1^o
&\quad& \text{modular law} \\
  & \leq (\top_{Y Y} (s r p_1 \cap p_2) \cap p_2) p_1^o
&& \top_{Y Z} s^0 \leq \top_{Y Y}\\
  & = (p_2 (p_1^o s r p_1 \cap p_1^o p_2) \cap p_2) p_1^o
&& \top{Y Y} = p_2 p_1^o \\
  & = p_2 (p_1^o s r p_1 \cap p_1^o p_2 \cap p_2^o p_2) p_1^o
&& \text{modular law} \\
  & = p_2 (p_1^o s r p_1 \cap (p_1^o \cap p_2^o) p_2) p_1^o
&& \text{maps distribute} \\
  & = p_2 (p_1^o s r p_1 \cap \Delta p_2) p_1^o
&& \text{see below} \\
  & = p_2 \Delta (\Delta^o p_1^o s r p_1 \cap p_2) p_1^o
&& \text{modular law} \\
  & = (s r p_1 \cap p_2) p_1^o
&& p_1 \Delta = p_2 \Delta = id \\
  & = s r \cap p_2 p_1^o
&& \text{modular law}
\end{aligned}
$$
and we are back to where we started.  In the fourth-last step we used the fact that if $p_1, p_2 \colon Z \times Z \to Z$ are the projections, then $p_1 \cap p_2 = \Delta^o$.  But $\Delta = \langle id, id \rangle$, and from lemma 1 [[finnlawler:allegory|here]] we have that $\langle id, id \rangle = p_1^o \cap p_2^o$.
=--

## Related entries

* [[dagger 2-poset]]

Other attempted axiomatizations of the same idea "something that acts like the category of relations in a regular category" include:

* [[bicategory of relations]] (a special sort of [[cartesian bicategory]])

* [[1-category equipped with relations]]

Discussion of the relation between pretabular unitary allegories and bicategories of relations, and also between tabular unitary allegories and regular categories is in

* [[toddtrimble:Theory of units and tabulations in allegories]]

## References

The standard monograph is

* {#FreydScedrov}[[Peter Freyd]] and Andre Scedrov, _[[Categories, Allegories]]_, Mathematical Library Vol 39, North-Holland (1990). ISBN 978-0-444-70368-2.

The notion is discussed also in chapter A3 of

* {#Johnstone}[[Peter Johnstone]], _[[Sketches of an Elephant]]_

In

* [[Bob Walters]], _Categorical algebras of relations_ ([blog post](http://rfcwalters.blogspot.com/2009/10/categorical-algebras-of-relations.html) )

it is shown that any [[bicategory of relations]] is an allegory.

See also

* Wikipedia, _[Allegory](http://en.wikipedia.org/wiki/Allegory_%28category_theory%29)_

An introduction headed towards applications in computer science and, in particular, fuzzy controllers can be found in

* Michael Winter, _Goguen Categories_ (2007)

[[!redirects tabular allegory]]
[[!redirects union allegory]]
[[!redirects division allegory]]
[[!redirects power allegory]]

[[!redirects allegories]]
[[!redirects tabular allegories]]
[[!redirects union allegories]]
[[!redirects division allegories]]
[[!redirects power allegories]]


Allen Hatcher

* [website](https://math.cornell.edu/allen-hatcher)

## Selected writings

On [[algebraic topology]]:

* {#Hatcher02} [[Allen Hatcher]], *Algebraic Topology*, Cambridge University Press (2002) [[ISBN:9780521795401](https://www.cambridge.org/gb/academic/subjects/mathematics/geometry-and-topology/algebraic-topology-1?format=PB&isbn=9780521795401), [webpage](https://pi.math.cornell.edu/~hatcher/AT/ATpage.html)]

## Related entries

* [[topology]], [[algebraic topology]]

* [[vector bundles]], [[topological K-theory]]

* [[Whitehead tower]]

category: people

[[!redirects Alan Hatcher]]


Allen Knutson

* [webpage](http://www.math.cornell.edu/m/People/Faculty/knutson)

category: people


almost connected topological group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

+-- {: .num_defn #AlmostConnected}
###### Definition

A [[locally compact topological space|locally compact]] [[topological group]] $G$ is called **[[almost connected topological group|almost connected]]** if the underlying [[topological space]] of the [[quotient]] topological group $G/G_0$ (of $G$ by the [[connected component]] of the [[neutral element]], also called the [[identity component]]) is [[compact topological space|compact]].

=--

See for instance ([Hofmann-Morris, def. 4.24](#HofmannMorris)).

\begin{remark}
Since the connected component $G_0$ is [[closed subspace|closed]], the [[neutral element]] in $G/G_0$ is a [[closed point]]. It follows that $G/G_0$ is [[T1-space|$T_1$]] and therefore (because it is a [[uniform space]]) also $T_{3 \frac1{2}}$ (a [[Tychonoff space]]; see at *[[uniform space]]* for details). In particular, $G/G_0$ is a [[compact Hausdorff space]].
\end{remark}

+-- {: .num_defn }
###### Example

Every [[compact topological space|compact]] and every [[connected topological space|connected]] [[topological group]] is almost connected.

Also every [[quotient]] of an almost connected group is almost connected.

=--

## Related concepts

* [[connected topological space]]

* [[maximal compact subgroup]]

## References

Textbooks with relevant material:

* {#Stroppel} M. Stroppel, _Locally compact groups_, European Math. Soc., (2006)

* {#HofmannMorris} [[Karl Hofmann]] Sidney Morris, _The Lie theory of connected pro-Lie groups_, Tracts in Mathematics 2, European Mathematical Society,  (2000)

Original articles:

* Chabert, Echterhoff, Nest, _The Connes-Kasparov conjecture for almost connected groups and for linear $p$-adic groups_ ([pdf](http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_2003__97_/PMIHES_2003__97__239_0/PMIHES_2003__97__239_0.pdf))

[[!redirects almost connected group]]

[[!redirects almost connected topological groups]]
[[!redirects almost connected groups]]


almost equality

In [[measure theory]], two [[functions]] $f, g\colon X \to Y$ are __almost-everywhere equal__, or __almost equal__, if their [[equaliser]]
$$ \{ a\colon X \;|\; f(a) = g(a) \} $$
is a [[full set]].  We are usually only interested when the functions $f$ and $g$ are [[measurable function|measurable]], although this is not technically necessary.  We also may consider when $f$ and $g$ are [[almost functions]] ([[partial functions]] whose domains are full).

As we need to know what a full set is, there is no notion of almost equality on an arbitrary classical [[measurable space]].  However, if the measurable space is equipped with such a notion (as is always the case with a [[Cheng measurable space]] or a [[localisable measurable space]]), then we have almost functions.  Of course, a [[measure space]] also has plenty of structure for this.  The morphisms between [[measurable locales]] are also inherently considered only up to almost equality.

Besides measure theory, the concept applies whenever we have a notion of something being true (in this case, that two functions are equal) [[almost everywhere]].

[[!redirects almost equal]]
[[!redirects almost equality]]
[[!redirects almost-everywhere equal]]
[[!redirects almost-everywhere equality]]


almost function

In [[measure theory]], an __almost-everywhere-defined function__, or __almost function__, is a [[partial function]] whose domain is a [[full set]].  We are usually only interested in *[[measurable function|measurable]]* almost functions.  Typically, we consider almost functions up to the [[equivalence relation]] of almost equality, whereby two almost functions are __[[almost equality|almost equal]]__ if their [[equaliser]] is also a full set, that is if they are equal almost everywhere.

As we need to know what a full set is, there is no notion of almost function on an arbitrary classical [[measurable space]].  However, if the measurable space is equipped with such a notion (as is always the case with a [[Cheng measurable space]] or a [[localisable measurable space]]), then we have almost functions.  Of course, a [[measure space]] also has plenty of structure for this.  The morphisms between [[measurable locales]] also inherently correspond to measurable almost functions (up to almost equality).

It is a commonplace that one really only cares about measurable functions up to almost equality.  That one only needs measurable functions to be defined almost everywhere is the same idea.  However, in [[classical mathematics]], every almost function may be extended (using [[excluded middle]]) to an actual (everywhere-defined) [[function]], and this extension is unique up to almost equality.  Accordingly, the notion of almost function is only necessary in [[constructive mathematics]].  However, even classically, using them from the start can avoid annoying but trivial technicalities.  (For example, one does not need the notion of [[essentially bounded function]]; [[bounded function|bounded]] almost functions will do.)

Besides measure theory, the concept applies whenever we have a notion of something being true (in this case, that a partial function is defined) [[almost everywhere]].

[[!redirects almost function]]
[[!redirects almost functions]]

[[!redirects almost-everywhere-defined function]]
[[!redirects almost-everywhere-defined functions]]
[[!redirects almost-everywhere defined function]]
[[!redirects almost-everywhere defined functions]]
[[!redirects almost everywhere-defined function]]
[[!redirects almost everywhere-defined functions]]
[[!redirects almost everywhere defined function]]
[[!redirects almost everywhere defined functions]]

[[!redirects almost-everywhere-defined]]
[[!redirects almost-everywhere defined]]
[[!redirects almost everywhere-defined]]
[[!redirects almost everywhere defined]]


almost Hermitian structure

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

An _almost Hermitian structure_ a [[reduction of the structure group]] along the inclusion $U(n) \hookrightarrow GL(n,\mathbb{C})$ of the [[unitary group]] into the [[complex numbers|complex]] [[general linear group]].

Under further embedding $U(n) \hookrightarrow GL(n,\mathbb{C}) \hookrightarrow GL(2n,\mathbb{R})$ an almost hermitian structure on the [[frame bundle]] of a [[smooth manifold]], hence a [[G-structure]] for $G = U(n)$, is first of all the choice of an [[almost complex structure]] and then an [[almost Hermitian manifold]] structure.

An [[integrability of G-structure|first-order intgrable]] $U(n)$-structure (almost Hermitian manifold) structure is _[[Kähler manifold]]_ structure.

By the fact (see at _[unitary group -- relation to orthogonal, symplectic and general linear group](unitary+group#RelationToOrthogonalSymplecticAndGeneralLinearGroup)_) that $U(n) \simeq O(2n) \underset{GL(2n,\mathbb{R})}{\times} Sp(2n,\mathbb{R}) \underset{GL(2n,\mathbb{R})}{\times} GL(n,\mathbb{C})$ this means that an almost Hermitian structure is precisely a joint [[orthogonal structure]], [[almost symplectic structure]] and [[almost complex manifold]].

## Properties

### Relation to almost complex structure

Since the inclusion $U(n) \hookrightarrow GL(2n,\mathbb{R})$ factors through the [[symplectic group]] via the [[maximal compact subgroup]] inclusion

$$
  U(n) \hookrightarrow Sp(2n,\mathbb{R}) \hookrightarrow GL(2n,\mathbb{R})
$$

an almost Hermitian manifold structure is in particular an [[almost complex structure]]. Conversely, since the [[maximal compact subgroup]] inclusion is a [[homotopy equivalence]], there is no [[obstruction]] to lifting an almost complex structure to an almost Hermitian structure.

### Relation to Kähler manifolds

An [[integrable G-structure|first-order integrable]] almost Hermitian structure is a _[[Kähler manifold]] structure_.

## Related concepts

* [[Kähler manifold]]

* [[Hermitean Yang-Mills connection]]

[[!redirects almost Hermitian structures]]

[[!redirects Hermitean structure]]
[[!redirects Hermitean structures]]

[[!redirects almost Hermitean structure]]
[[!redirects almost Hermitean structures]]


almost Kähler geometric quantization

#Contents#
* table of contents
{:toc}

## Idea

A [[Kähler structure]] on a [[symplectic manifold]] induces a [[polarization]], and [[geometric quantization]] with respect to such [[Kähler polarizations]] works particularly well and has numerous examples (e.g. the [[orbit method]] and [[quantization of Chern-Simons theory]]).

An _[[almost Kähler structure]]_ does not induce a [[polarization]] in the usual sense, unless it is actually a Kähler structure. Nevertheless, via [[geometric quantization by push-forward]] there is still a concept of [[geometric quantization]] of almost Kähler structures (in the case of finite-dimensional manifolds), which is "as good" as Kähler quantization ([Borthwick-Uribe 96](#BorthwickUribe96)).

Examples of almost-Kähler structures on [[phase spaces]] include the phase space of the [[scalar field]] with reasonable [[interaction]] terms, such as the [[free field]] or the [[phi^4 theory|phi^4 interaction]] ([Collini 16, section 3.2.2](#Collini16)).

## Related concepts

* [[pseudo-holomorphic vector bundle]]

* [[Fedosov deformation quantization]] naturally applies to almost-Kähler structures, too

## References

A discussion of a kind of [[geometric quantization via push-forward]] on finite-dimensional almost Kähler manifolds is in

* {#BorthwickUribe96} [[David Borthwick]], [[Alejandro Uribe]], _Almost complex structures and geometric quantization_ ([arXiv:dg-ga/9608006](https://arxiv.org/abs/dg-ga/9608006))

Discussion of the almost-Kähler structure on the [[phase space]] of the [[scalar field]] is in

* {#Collini16} [[Giovanni Collini]], chapter 3 of _Fedosov Quantization and Perturbative Quantum Field Theory_ ([arXiv:1603.09626](https://arxiv.org/abs/1603.09626))

On [[symplectic reduction]] in the almost-Kähler case:

* Martin Otto, _A reduction scheme for phase spaces with almost Kähler symmetry: Regularity results for momentum level sets_, Journal of Geometry and Physics
Volume 4, Issue 2, 1987, Pages 101-118

[[!redirects almost Kähler quantization]]


almost module

#Contents#
* table of contents
{:toc}

_Under construction: Extracted from a series of tweets by Syzygay_

## Idea

Let $R$ be a (commutative unital) [[ring]]. Suppose $I$ is a flat idempotent $R$-ideal. To be a flat ideal means $-\otimes I$ is an [[exact functor]]. The tensor product is always right exact, so in particular, $-\otimes I$ preserves injections. To be idempotent, $I^2 = I$.

Fixing a pair $(R,I)$ like this, we construct the category of "$I$-almost $R$-modules", $alMod R$.

We take the full subcategory of $R$-modules spanned by $M$ so that $I\otimes M \equiv M$. (This is not the actual definition, but this is an equivalent category. The correct definition is the quotient of the category of $R$-modules by the category of $I$-almost $0$ modules. See below.)

the category structure (it's the essential image of the functor ModR→ModR via M↦I⊗M, it's an Abelian category, and it's a localization of ModR).

There is an inclusion of categories j_! : alModR → ModR.

There is an exact right adjoint of j_!, denoted j* : ModR → alModR which sends M to I⊗M.

And j* has a right adjoint j_* : alModR → ModR via
j_*(M) = Hom(I, j_!(M)).

j* does legitimately map into alModR. To prove this, we must show that I⊗j*M≅j*M.

I⊗j*M = I⊗(I⊗M).
Since I is a flat ideal, ≅ I²⊗M.
Since I is idempotent, ≅ I⊗M = j*M.

we can think of j* as sending a module to the nearest almost module.

In some sense, the ideal I acts like an identity (under the tensor product) for almost modules. So the idea here is that I doesn't really do anything to the (I-)almost R modules.

An R-module M is (I-)almost 0 if IM = 0.

Since I is idempotent, if IM = 0, then j*M = I⊗M = 0 as well.

So the nearest "almost module" to an "almost 0" module is 0 itself. The only defect to being 0 is killed by the identity-like ideal.

Let f : M → N be an R-linear map.

f is almost injective if ker f is almost 0.
f is almost surjective if coker f is almost 0.
f is an almost isomorphism if ker f and coker f are almost 0.

An "almost" property of a map gets sent to the actual property by j*.

M is almost flat if j*M is flat in alModR; equivalently, if Torᵢ(M,N) is almost 0 for all i > 0 and for all N.

The problem with alModR is that the projective objects behave strangely, so instead, we define "almost Hom", alHom.

For two almost modules M and N,
alHom(M,N) = j*(Hom(M,N)).

Note: alHom(j*M,j*N) ≅ j*(Hom(M,N)).

Note: For L, M, N in alModR,
Hom(L⊗M,N) ≅ Hom(L, alHom(M,N)) where Hom's are taken in the category alModR.
So we do also get a form of "almost" hom-tensor adjunction.

A module M is almost projective if alHom(j*M,-) is exact in alModR, or equivalently, if Extⁿ(M,N) is almost 0 for all N and for all n > 0.

What are examples of rings R with flat idempotent ideals I?

It turns out that if R is a perfectoid ring and I is generated by a pseudo-uniformizer, then (R,I) satisfies these conditions!

One application is proving the tilting equivalence of perfectoid algebras. Given a perfectoid field K whose pseudo-uniformizer ϖ satisfies |p|≤|ϖ|≤1 for some prime p, there is a perfectoid field K♭ of char p (the tilt of K).

If K is a perfectoid field, then the category of perfectoid K-algebras is equivalent to the category of perfectoid K♭-algebras. Hence, we can reduce some problems in mixed char to char p.

The category of I-almost 0 modules is a Serre subcategory of Mod R. The "correct" definition of the category of I-almost R-modules is the quotient of the category of R-modules by the category of I-almost 0 modules.

## Related entries

* [[almost scheme]]

* [[p-adic Hodge theory]]

* [[perfectoid space]]

## References

* [[Bhargav Bhatt]],  _Almost Ring Theory I_, 2014 ([[BhattAlmostRing.pdf:file]])

* [[Ofer Gabber]], [[Lorenzo Ramero]], _Almost ring theory_, Springer 2003 ([arXiv:math/0002064](https://arxiv.org/abs/math/0002064), [doi:10.1007/b10047](https://link.springer.com/book/10.1007/b10047))

  [[Ofer Gabber]], [[Lorenzo Ramero]], *Foundations for almost ring theory* ([arXiv:math/0409584](https://arxiv.org/abs/math/0409584))

* [[Gerd Faltings]], _Almost étale extensions_,  Cohomologies p-adiques et applications arithmétiques (II), Astérisque no. 279  (2002), p. 185-270 ([numdam:AST_2002__279__185_0](http://www.numdam.org/item/AST_2002__279__185_0))

See also

* Wikipedia, _[Almost ring](https://en.wikipedia.org/wiki/Almost_ring)_

[[!redirects almost modules]]

[[!redirects almost ring]]
[[!redirects almost rings]]

[[!redirects almost ring theory]]

[[!redirects almost mathematics]]


almost open subspace

A [[subspace]] $A$ of a [[space]] $X$ is __almost open__ if it is [[open subspace|open]] modulo the $\sigma$-[[sigma-ideal|ideal]] of [[meagre subspace|meagre subspaces]].  We also say that $A$ has the __Baire property__.

Explicitly, $A$ is almost open if there exist an open subspace $G$ and an [[infinite sequence]] $N_1, N_2, \ldots$ of [[nowhere dense subspace|nowhere dense subspaces]] (meaning that their [[topological closure|closures]] have [[empty subspace|empty]] [[topological interior|interiors]]) such that
$$ A \cup \bigcup_i N_i = G \cup \bigcup_i N_i .$$

That every subspace of the [[real line]] is almost open follows from the [[axiom of determinacy]] but contradicts the [[axiom of choice]].  In the absence of choice, it is a convenient assumption to make and is one of the axioms of [[dream mathematics]].

category: foundational axiom

[[!redirects almost open set]]
[[!redirects almost open sets]]
[[!redirects almost open subset]]
[[!redirects almost open subsets]]
[[!redirects almost open subspace]]
[[!redirects almost open subspaces]]

[[!redirects baire property]]
[[!redirects Baire property]]


almost scheme

#Contents#
* table of contents
{:toc}

## Motivation

Gabber and Ramero have developed a chapter in abstract algebra and [[algebraic geometry]] which should, in the style of Grothendieck's theory of [[schemes]], capture some of the ideas of

* [[Gerd Faltings]], _Almost étale extensions_,  Cohomologies p-adiques et applications arithmétiques (II), Astérisque no. 279  (2002), p. 185-270 ([numdam:AST_2002__279__185_0](http://www.numdam.org/item/AST_2002__279__185_0))

To this aim they define a notion of almost ring/algebra, almost module, and finally, __almost scheme__. The framework is akin to many ideas in [[noncommutative algebraic geometry]] and in particular uses localization theory (compare also the generalized algebraic geometry of [[Nikolai Durov]]).

## Definitions

One starts with a "basic setup" consisting of a fixed _commutative_ unital ("[[base ring|base]]") ring $V$ containing an ideal $I$ such that $I^2 = I$. Usually one also assumes that $\tilde{I} := I \otimes_V I V$ is a flat $V$-module. A $V$-module $M$ is __almost zero__ if $IM=0$. A morphism $f:M\to N$ of $V$-modules is __almost isomorphism__ if $Ker f$ and $Coker f$ are almost zero, or equivalently if $\tilde{I}\otimes_V f : \tilde{I}\otimes_V M\to \tilde{I}\otimes_V N$ is an isomorphism. Almost isomorphisms form a category of fractions in $V-Mod$. The full subcategory $\Sigma$ of $V-Mod$ of $V$-modules which are almost isomorphic to $0$ is a Serre subcategory of $V-Mod$, hence one can construct the quotient category $V^a-Mod := V-Mod/\Sigma$ with the localization functor $Q^* : V-Mod\to V-Mod/\Sigma$; the image of $V$ in $V-Mod/\Sigma$ is denoted by $V^a$. The usual tensor product on $V-Mod$ induces a tensor product on $V^a-Mod$ making it into an abelian symmetric monoidal category. An __almost algebra__ is a commutative monoid in this tensor category; almost algebras form a category $V^a-Alg$; in the usual way one defines the (categories of) left and right (unital or not) modules over any given almost algebra.

An __affine almost scheme__ is just an object of an opposite category to $V^a-Alg$. A number of Grothendieck topologies are available on the category of affine almost schemes $V^a-Aff$; leading to the notions of locally affine spaces in these topologies; or better in relative version $R^a-Aff = (R^a-Alg)^{op}$ where $R$ is a $V$-algebra. By convention, the default topology is an appropriate version of flat topology, leading to the site $(R^a-Aff)_{fpqc}$. An __almost $R$-scheme__ is by the definition a sheaf of sets on $(R^a-Aff)_{fpqc}$.

More general "basic setups" in terms of topoi are also considered in the book.

##Literature##

* [[Ofer Gabber]], Lorenzo Ramero, _Almost Ring Theory_, Lec. Notes in Math. 1800, Springer 2003.

Abstract of the first arXiv release (51 pages [math.0002064](http://arxiv.org/abs/math/0002064))

>The categories of almost modules and almost algebras are introduced as a convenient setting for the development of Faltings' method of almost etale extensions. After some preliminaries of general "almost homological algebra" we construct the almost version of the cotangent complex and we use it to generalise some results of Faltings on the lifting of almost etale morphisms and almost etale algebras over nilpotent extensions. We also study the "almost trace" of an almost flat and almost finitely presented morphism, in particular we show that the almost trace is (almost) perfect if and only if the morphism is almost etale. Finally we study some cases of non-flat descent for almost rings, and establish the invariance of almost etale morphisms under Frobenius.

Abstract of 6th public release at [arXiv:math/0201175] (http://arxiv.org/abs/math/0201175) (about 230 pages)

>We develop almost ring theory, which is a domain of mathematics somewhere halfway between ring theory and category theory (whence the difficulty of finding appropriate MSC-class numbers). We apply this theory to valuation theory and to p-adic analytic geometry. You should really have a look at the introductions (each chapter has one).

In the introduction (6th public release), the authors heuristically place their work in the general field of _abstract algebraic geometry_:

>The purpose of the game is to reconstruct in this new framework as much as possible (and useful) of classical linear and commutative algebra. Essentially, this is the same as the ideology informing Deligne's paper [[Catégories Tannakiennes]], which sets out to develop algebraic geometry in the context of abstract tannakian categories. We could also claim an even earlier ancestry, in that some of the leading motifs resonating throughout our text, can be traced as far back as Gabriel's memoir [[Des catégories abéliennes]].

>In evoking Deligne's and Gabriel's works, we have unveiled another source of motivation whose influence has steadily grown throughout the long gestation of our paper. Namely, we have come to view almost ring theory as a contribution to that expanding body of research of still uncertain range and shifting boundaries, that we could call "abstract algebraic geometry". We would like to encompass under this label several heterogeneous developments: notably, it should include various versions of non-commutative geometry that have been proposed in the last twenty years, but also the relative schemes of M.HAKIM, _Topos annelés et schémas relatifs_, Springer Ergebnisse Math. Grenz. 64 (1972), as well as Deligne's ideas for algebraic geometry over symmetric monoidal categories.

> The common thread loosely unifying these works is the realization that "geometric spaces" do not necessarily consist of set-theoretical points, and - perhaps more importantly -functions on such "spaces" do not necessarily form (sheaves of) commutative rings. Much effort has been devoted to extending the reach of geometric intuition to non-commutative algebras; alternatively, one can retain commutativity, but allow "structure sheaves" which take values in tensor categories other than the category of rings. As a case in point, to any given almost ring $A$ one can attach its spectrum $Spec A$, which is just $A$ viewed as an object of the opposite of the category $V^a-Alg$. $Spec A$ has even a natural flat topology, which allows to define more general almost schemes by gluing (i.e. taking colimits of) diagrams of affine spectra; all this is explained in section 5.7, where we also introduce quasi-projective almost schemes and investigate some basic properties of the smooth locus of a quasi-projective almost scheme. By way of illustration, these generalities are applied in section 5.8 in order to solve a deformation problem for torsors over affine almost group schemes; let us stress that the problem in question is stated purely in terms of affine objects (i.e. almost rings and "almost Hopf algebras"), but the solution requires the introduction of certain auxiliary almost schemes that are not affine.

## Related concepts

* [[almost module]]

[[!redirects almost schemes]]


almost surely

+-- {: .rightHandSide}
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### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

In [[probability theory]] an [[event (probability theory)|event]] of [[probability]] 1 is said to happen *almost surely*.

In [[measure theory]], such subsets are also known as [[full subsets]].
Their [[complements]] are known as [[null subsets]] or [[negligible subsets]].

## Related concepts

* [[null subset]]

* [[full subset]]

* [[deterministic random variable]]

## References

See also:

* Wikipedia, *[Almost surely](https://en.wikipedia.org/wiki/Almost_surely)*


Alonso Botero

* [Institute page](https://fisica.uniandes.edu.co/en/professors/alonso-botero-mejia)

* [GoogleScholar page](https://scholar.google.com/citations?user=e06A7mUAAAAJ&hl=en)

## Selected writings

Relating [[quantum information theory]] to the [[representation theory of the symmetric group]]:

* [[Alonso Botero]], *Quantum Information and the Representation Theory of the Symmetric Group*, Rev. colomb. mat. vol.50 no. 2 Bogotá July/Dec. 2016 ([doi:10.15446/recolma.v50n2.62210](https://doi.org/10.15446/recolma.v50n2.62210), [pdf](http://www.scielo.org.co/pdf/rcm/v50n2/v50n2a05.pdf))

category: people


Alonso Perez-Lona

doctoral student at Virginia Tech in the group of [[Eric Sharpe]].

* [MO page](https://mathoverflow.net/users/172910/alonso-perez-lona)

## Selected writings

On [[orbifolds]] by [[2-groups]] in view of [[sigma-models]] inspired from [[string theory]]:

* [[Alonso Perez-Lona]], [[Eric Sharpe]], *Three-dimensional orbifolds by 2-groups* [[arXiv:2303.16220](https://arxiv.org/abs/2303.16220)]

category: people

[[!redirects Alonso Perez Lona]]
[[!redirects perezl.alonso]]


Alonzo Church

* [Wikipedia entry](http://en.wikipedia.org/wiki/Alonzo_Church)

## Selected writings

On [[simple type theory]]:

* {#Church40} [[Alonzo Church]], §5 of: *A Formulation of the Simple Theory of Types*, The Journal of Symbolic Logic **5** 2 (1940) 56-68  [[doi:10.2307/2266170](https://doi.org/10.2307/2266170)]

## Related entries

* [[judgement]]

category:people


alpha particle

[[!redirects alpha particles]]

## Idea

The [[atomic nucleus]] of [[Helium]] is also called the _alpha-particle.

## References

* Wilipedia, _[Alpha particle](https://en.m.wikipedia.org/wiki/Alpha_particle)


alpha-equivalence

+-- {: .rightHandSide}
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### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Equality and Equivalence
+--{: .hide}
[[!include equality and equivalence - contents]]
=--
=--
=--

\tableofcontents

## Idea

In [[logic]] and [[type theory]], $\alpha$-equivalence is the principle that two syntactic expressions ([[types]], [[terms]], [[propositions]], [[contexts]], whatever) are equivalent for all purposes if their only difference is the renaming of bound [[variables]].

Depending on the technicalities of how variables are managed, $\alpha$-equivalence may be a necessary axiom, a provable theorem, or entirely trivial.  In any case, it is often (usually? always?) seen as a technicality devoid of conceptual interest.  However, it can be a technically nontrivial task to implement $\alpha$-conversion (which is necessary to avoid capture of free variables upon [[substitution]]) when programming logic or type theory into a computer.

## Definition

We work in a [[dependent type theory]] which acts as the metatheory for our object type theory.

The type $\mathbb{A}$ is a [[countable set]] (and thus a [[type]] with [[decidable equality]]) whose terms are called *names* or *atoms*. The [[type]] $\mathrm{Exp}$ of *syntactic expressions* is an [[h-set]] which is inductively defined by binary functions $P:\mathrm{Exp} \times \mathrm{Exp} \to \mathrm{Exp}$ and $B([(-)](-)):\mathbb{A} \times \mathrm{Exp} \to \mathrm{Exp}$. There are dependent types $E:\mathrm{Exp} \vdash \mathrm{occ} \; \mathrm{type}$ representing the occuring atoms, $E:\mathrm{Exp} \vdash \mathrm{free} \; \mathrm{type}$ representing the free atoms, and $E:\mathrm{Exp} \vdash \mathrm{bnd} \; \mathrm{type}$ representing the bound atoms, such that for each expression $E:\mathrm{Exp}$, there are [[embeddings]]

$$i_\mathrm{occ}(E):\mathrm{occ}(E) \hookrightarrow \mathbb{A}$$
$$i_\mathrm{free}(E):\mathrm{free}(E) \hookrightarrow \mathbb{A}$$
$$i_\mathrm{bnd}(E):\mathrm{bnd}(E) \hookrightarrow \mathbb{A}$$

which may be defined by recursion on $\mathrm{Exp}$.

We define the following dependent types $a:\mathbb{A}, E:\mathrm{Exp} \vdash a \lhd E \; \mathrm{type}$ as
$$a \lhd E \coloneqq \sum_{x:\mathrm{occ}(E)} i_\mathrm{occ}(E)(x) =_\mathbb{A} a$$
representing that $a$ occurs in $E$ at least once,
$$a \lhd_\mathrm{free} E \coloneqq \sum_{x:\mathrm{free}(E)} i_\mathrm{free}(E)(x) =_\mathbb{A} a$$
representing that $a$ freely occurs in $E$ at least once, and
$$a \# E \coloneqq (a \lhd_\mathrm{free} E) \to \emptyset$$
representing that $a$ is fresh (does not freely occurs in $E$ at least once)

...

## See also

* [[equality]], [[syntactic equality]], [[definitional equality]]
* [[substitution]]
* [[rewriting]]

## References

* [[Roy L. Crole]], *Alpha equivalence equalities*, Theoretical Computer Science, Volume 433, 18 May 2012, Pages 1-19, ([doi:10.1016/j.tcs.2012.01.030](https://doi.org/10.1016/j.tcs.2012.01.030))

The distinction between $\alpha$-equivalence and syntactic equality of expressions is briefly discussed in:

* [[Benno van den Berg]], [[Martijn den Besten]], *Quadratic type checking for objective type theory* ([arXiv:2102.00905](https://arxiv.org/abs/2102.00905))

[[!redirects alpha equivalent]]
[[!redirects alpha-equivalent]]
[[!redirects α-equivalent]]
[[!redirects alpha equivalence]]
[[!redirects alpha-equivalence]]
[[!redirects α-equivalence]]
[[!redirects alpha conversion]]
[[!redirects alpha-conversion]]
[[!redirects α-conversion]]


alternating group

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###Context###
#### Group Theory
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[[!include group theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

An [[alternating group]] $A_n$ is a [[subgroup]] of a [[symmetric group]] $S_n$ consisting of the _even_ [[permutations]].

The alternating group $A_n$ is to the [[symmetric group]] $S_n$ as the [[special orthogonal group]] $SO(n)$ is to the [[orthogonal group]] $O(n)$. See also at _[symmetric group -- Whitehead tower](https://ncatlab.org/nlab/show/permutation#WhietheadTowerAndSupersymmetry)_

## Examples

* The alternating group $A_4$ on four elements is [[isomorphism|isomorphic]] to the orientation-preserving [[tetrahedral group]].

* The alternating group $A_5$ on five elements, of order $60$, is the smallest [[nonabelian group|nonabelian]] [[simple group]]. Geometrically, it may be realized as [[finite subgroup of SO(3)]] which carries a regular [[icosahedron]] into itself: the [[icosahedral group]].

* For all $n \geq 5$, the alternating group $A_n$ is simple. This is true even if $n$ is infinite: define $Alt(X)$ for any set $X$ to consist of all permutations of $X$ each of which fixes all but finitely elements, and which is an even permutation on that finite subset.

## Related concepts

* [[finite group]]

* [[alternating representation]]

## References

* Wikipedia, _[Alternating group](https://en.wikipedia.org/wiki/Alternating_group)_

[[!redirects alternating groups]]


alternating multifunction

# Alternating functions
* table of contents
{: toc}

## Idea

In [[linear algebra]], alternating [[multilinear functions]] are well known, and are in many cases (over the [[real numbers]], for example) are equivalent to [[antisymmetric multifunction|antisymmetric functions]].  In cases where they differ (such as in [[characteristic]] $2$), it is often the alternating functions that behave better.

Actually, being alternating is not, in itself, really about linearity, and we can abstract away to a nonlinear concept of _alternating function_.  (That said, there is one bit of very mild linear structure that is needed: a [[basepoint]] in the [[target]] set.)

The property of being alternating is called _alternation_ (rather than alternatingess), although in principle one could also use _alternating_ as a noun.

## Definitions

Let $X$ be a [[set]], and let $(Y,0)$ be a [[pointed set]] (so $Y$ is a set and $0$ is one of its elements).  Let $n$ be a [[natural number]] (or indeed any [[cardinal number]]).  Recall that a [[multifunction]] of [[arity]] $n$ to $Y$ from $X$ is the same thing as a [[function]] to $Y$ from the $n$-fold [[cartesian power]] $X^n$.

An __alternating multifunction__ (or simply _alternating function_) of arity $n$ from $X$ to $(Y,0)$ is a multifunction of arity $n$ from $X$ to $Y$ such that, whenever two of the function\'s arguments are equal, the value of the function is $0$.  In arity $0$ or $1$, every multifunction is trivially alternating; in arity $2$, we can write this as the [[equational law]] $f(a,a) = 0$; in arity $3$, we have the equational laws $f(a,a,b) = 0$, $f(a,b,a) = 0$, and $f(a,b,b) = 0$; etc.

## Properties

There are many nice properties of alternating *[[multilinear function|multilinear]]* functions.  So suppose that $X$ and $Y$ are [[modules]] over a [[base rig]] $K$ and that $f$ is a multilinear function from $X$ to $Y$; use the usual [[zero]] element of the module $Y$ as the basepoint $0$.  In the case where $Y$ is $K$ itself, we speak of an __alternating form__ (a phrase which is usually taken to include multilinearity).

We will sometimes want to assume that scalar multiplication by $2$ is cancellable in $Y$ (which for example is always the case when $2$ is invertible in $K$, in particular when $K$ is a [[field]] of [[characteristic]] other than $2$), but only when stated.  Since alternation requires looking at two arguments of $f$, we will often, when this leads to no loss of generality, assume that these are the first two arguments, writing $\vec{z}$ to represent all of the other arguments.

+-- {: .num_prop #alterationImpliesAntisymmetry}
###### Proposition

An alternating [[multilinear function]] is [[antisymmetric function|antisymmetric]].
=--

+-- {: .proof}
By multilinearity,
$$ f(x+y,x+y,\vec{z}) = f(x,x,\vec{z}) + f(x,y,\vec{z}) + f(y,x,\vec{z}) + f(y,y,\vec{z}) .$$
Applying alternation, most of these terms vanish:
$$ 0 = 0 + f(x,y,\vec{z}) + f(y,x,\vec{z}) + 0 .$$
Therefore,
$$ f(x,y,\vec{z}) + f(y,x,\vec{z}) = 0 ,$$
which is antisymmetry.
=--

+-- {: .num_prop #antisymmetryImpliesAlternation}
###### Proposition

If multiplication by $2$ is cancellable in $Y$, then an [[antisymmetric function|antisymmetric]] function to $Y$ is alternating.
=--

+-- {: .proof}
###### Proof

By antisymmetry,
$$ f(x,x,\vec{z}) + f(x,x,\vec{z}) = 0 ,$$
or equivalently
$$ 2 f(x,x,\vec{z}) = 2 \cdot 0 .$$
Cancelling $2$,
$$ f(x,x,\vec{z}) = 0 ,$$
which is alternation.
=--

+-- {: .num_remark #antisymmetryWarning}
###### Remark

It is false in both directions to state in general that alternating functions and antisymmetric functions are the same, but for different reasons.  An alternating function must be antisymmetric *if* it is multilinear, regardless of the behaviour of $2$, but not when it is nonlinear; an antisymmetric function must be alternating *if* multiplication by $2$ is cancellable in the target, regardless of linearity, but not when $2$ is noncancellable.

The simplest strongest-possible counterexamples are
$$ (x,y \mapsto |y-x|)\colon \mathbb{R}^2 \to \mathbb{R} ,$$
which is alternating but not antisymmetric, and
$$ (x,y \mapsto x y)\colon \mathbb{F}_2^2 \to \mathbb{F}_2 ,$$
which is antisymmetric but not alternating.

Of course, alternating and antisymmetric functions *are* the same in the context of multilinear functions to a module in which $2$ is cancellable, in particular for multilinear functions between [[vector fields]] over the [[real numbers]].
=--

+-- {: .num_remark}
###### Remark

The [[alternating groups]] are really about antisymmetric functions rather than alternating functions as such.  (Whereas a [[symmetric function]] is preserved by the application of any element of the [[symmetric group]], an antisymmetric function is preserved by and only by the elements of the alternating group.)  Nevertheless, this precise distinction between 'alternating' and 'antisymmetric' is well established in the theory of vector spaces over a field of [[characteristic]] $2$ (in which multiplication by $2$ is as uncancellable as possible).
=--

## Constructive aspects

In [[constructive mathematics]], we usually assume that the arity $n$ of $f$ has [[decidable equality]], which is true if $n$ is a [[natural number]] (which is most common) or even a (possibly infinite) [[extended natural number]].  However, as long as the arity is equipped with an [[inequality]], then we can state the definition: whenever equal arguments have inequal indices, the value of the multifunction $f$ there is zero.  If $X$ and $Y$ are also equipped with inequalities, then $f$ is __strongly alternating__ if, whenever its value is inequal to $0$ in $Y$, then arguments with inequal indices must be inequal in $X$.  (In arity $2$, for example, if $f(a,b) \ne 0$, then $a \ne b$.)  If the inequality on $Y$ is [[tight relation|tight]] (so that its [[negation]] is [[equality]] in $Y$), then every strongly alternating function is alternating, but the reverse requires [[excluded middle]] in general.

[[!redirects alternating multifunction]]
[[!redirects alternating multifunctions]]
[[!redirects alternating multimap]]
[[!redirects alternating multimaps]]

[[!redirects alternating function]]
[[!redirects alternating functions]]
[[!redirects alternating map]]
[[!redirects alternating maps]]

[[!redirects alternating multilinear function]]
[[!redirects alternating multilinear functions]]
[[!redirects alternating multilinear map]]
[[!redirects alternating multilinear maps]]

[[!redirects alternating form]]
[[!redirects alternating forms]]


alternating representation

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### Context
#### Representation theory
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[[!include representation theory - contents]]
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#Contents#
* table of contents
{:toc}

## Definition

For $S_n$ a [[symmetric group]], its _alternating representation_ is the 1-[[dimension|dimensional]] [[linear representation]]

$$
  alt \;\; S_n \longrightarrow GL(1)
$$

which sends even [[permutations]] to $+1$ and odd permutations to $-1$.

More generally, for $G$ any [[finite group]] and $H \subset G$ a [[subgroup]] of [[index of a subgroup|index]] 2, then the corresponding alternating representation of $G$ sends elements of $H$ to $+1$ and all other elements to -1.

This reduces to the previous special case by setting $G \coloneqq S_n$ and $H \coloneqq A_n \subset S_n$ the [[alternating group]].

## Related concepts

* [[trivial representation]]

* [[regular representation]]

* [[permutation representation]]

## References

For instance

* João Pedro Martins dos Santos, Def. 3 in _Representation Theory of Symmetric Groups_, 2012 ([pdf](https://www.math.tecnico.ulisboa.pt/~ggranja/joaopedro.pdf))

[[!redirects alternating representations]]

[[!redirects sign representation]]
[[!redirects sign representations]]


alternative algebra

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### Context
#### Algebra
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# Contents
* table of contents
{: toc}

## Definitions

+-- {: .num_defn #AlternativeAlgebra}
###### Definition

Consider the following equational laws of a [[binary operation]] (written multiplicatively):

* __left alternativity__: $(x x) y = x (x y)$;
* __flexibility__: $(x y) x = x (y x)$;
* __right alternativity__: $(y x) x = y (x x)$.

An operation satisfying one of these conditions is __left-alternative__, __flexible__, or __right-alternative__, respectively, and it is __alternative__ if it is both left-alternative and right-alternative.  A [[magma]] is so if its binary operation is so, and a [[nonassociative algebra]] (or [[nonassociative ring]]) is so if its multiplication operation is so.
=--

## Properties

A [[commutative operation|commutative]] operation/magma/algebra must be flexible, and it is left-alternative iff it is right-alternative (and so simply alternative).  An [[associative operation|associative]] operation/magma/algebra is both alternative and flexible.

Among algebras (but not magmas), any algebra with two of these three properties must have the third.  In particular, an alternative algebra must be flexible.  This follows from the characterization in terms of the associator below.

### In terms of the associator

+-- {: .num_prop #InTermsOfAssociator}
###### Proposition

For a [[nonassociative algebra]] $A$, alternativity according to def. \ref{AlternativeAlgebra} is equivalent to the condition that the [[associator]], i.e. the [[multilinear map|tri-linear map]]

$$
  [-,-,-] \;\colon\; A \otimes A \otimes A \longrightarrow A
$$

given by

$$
  [a,b,c] \coloneqq (a b) c - a (b c)
$$

is [[alternating map|alternating]], in that whenever two of the three arguments are equal, the result is zero.
=--

+-- {: .num_cor #skewsymmetry}
###### Corollary

Alternativity implies that the associator is [[skew-symmetric function|skew-symmetric]], in that for any [[permutation]] $\sigma$ of three elements then $[a_{\sigma_1}, a_{\sigma_2}, a_{\sigma_3}] = (-1)^{\vert \sigma\vert} [a_1, a_2, a_3]$ for ${\vert\sigma\vert}$ the [[signature of a permutation|signature of the permutation]].  Over a [[field]] whose [[characteristic]] is different from $2$, or more generally over any [[commutative ring]] in which $2$ is invertible or even cancellable, alternativity is equivalent to skew-symmetry of the associator.
=--

+-- {: .proof}
###### Proof

In one direction, left alternativity says directly that the associator is alternating in the first two arguments:

$$
  [x,x,y] = (x x) y - x (x y) = (x x) y - (x x) y = 0 ,
$$

and right alternativity says the same thing in the last two arguments:

$$
  [x,y,y] = (x y) y - x (y y) = (x y) y - (x y) y = 0 .
$$

To be fully alternating, we then argue using multi-linearity of the associator:

$$
  [x,y,x] = [x,x,x] + [x,y,x] + [y,x,x] + [y,y,x] = [(x+y),(x+y),x] = 0 .
$$

Multilinearity also proves that the associator is skew-symmetric, in adjacent arguments:

$$
  0 = [(x+y),(x+y),z] = [x,y,z] + [y,x,z]
$$

$$
  0 = [z,(x+y),(x+y)] = [z,x,y] + [z,y,x]
$$

and hence in all arguments.

In the other direction, skew-symmetry of the associator implies alternativity by

$$
  \begin{aligned}
    & [x,x,y] = -[x,x,y]
    \\
    \Leftrightarrow & 2 [x,x,y] = 0
    \\
    \Leftrightarrow & [x,x,y] = 0
  \end{aligned}
$$

using the assumption that $2$ is cancellable in $A$, and similarly for $[y,x,x] = 0$.
=--

+-- {: .num_prop #ArtinTheorem}
###### Proposition

A [[nonassociative algebra]] is alternative, def. \ref{AlternativeAlgebra}, prop. \ref{InTermsOfAssociator}, precisely if the [[subalgebra]] generated by any two elements is an [[associative algebra]].
=--

This is due to [[Emil Artin]], see for instance ([Schafer 95, p. 18](#Schafer95)).

+-- {: .num_prop #ZornTheorem}
###### Proposition

The only alternative [[division algebras]] over the [[real numbers]] are the [[real numbers]] themselves, the [[complex numbers]], the [[quaternions]] and the [[octonions]].
=--

This is due to ([Zorn 30](#Zorn30)).

## Examples

Every [[associative algebra]] is alternative and flexible.  Every [[Lie algebra]] or [[Jordan algebra]] is flexible.

Every [[Cayley–Dickson algebra]] over a [[commutative ring]] $R$ is flexible.  The first three (corresponding, if we start with the [[real numbers]], to the real numbers, [[complex numbers]], and [[quaternions]]) are associative and hence alternative.  The next one (corresponding to the [[octonions]]) is still alternative despite not being associative (unless $R$ has characteristic $2$).  After that (corresponding to the [[sedenions]] and above), they are not even alternative (unless $R$ has characteristic $2$).

## Related concepts

* [[Cayley-Dickson construction]]

* [[composition algebra]]

## References

* {#Schafer95} R. D. Schafer, chapter III of _Introduction to Non-Associative Algebras_ Dover, New York, 1995. ([web](https://archive.org/details/anintroductionto25156gut))

* {#Zorn30} [[Max Zorn]], _Theorie der alternativen Ringe_, Abhandlungen des Mathematischen Seminars der Universität Hamburg 8 (1930), 123-147

* Wikipedia, _[Alternative algebra](https://en.wikipedia.org/wiki/Alternative_algebra)_, _[Flexible algebra](https://en.wikipedia.org/wiki/Flexible_algebra)_

[[!redirects alternativity]]
[[!redirects alternative binary operation]]
[[!redirects alternative binary operations]]
[[!redirects alternative operation]]
[[!redirects alternative operations]]
[[!redirects alternative magma]]
[[!redirects alternative magmas]]
[[!redirects alternative algebra]]
[[!redirects alternative algebras]]

[[!redirects left alternativity]]
[[!redirects left-alternativity]]
[[!redirects left-alternative binary operation]]
[[!redirects left-alternative binary operations]]
[[!redirects left alternative binary operation]]
[[!redirects left alternative binary operations]]
[[!redirects left-alternative operation]]
[[!redirects left-alternative operations]]
[[!redirects left alternative operation]]
[[!redirects left alternative operations]]
[[!redirects left-alternative magma]]
[[!redirects left-alternative magmas]]
[[!redirects left alternative magma]]
[[!redirects left alternative magmas]]
[[!redirects left-alternative algebra]]
[[!redirects left-alternative algebras]]
[[!redirects left alternative algebra]]
[[!redirects left alternative algebras]]

[[!redirects right alternativity]]
[[!redirects right-alternativity]]
[[!redirects right-alternative binary operation]]
[[!redirects right-alternative binary operations]]
[[!redirects right alternative binary operation]]
[[!redirects right alternative binary operations]]
[[!redirects right-alternative operation]]
[[!redirects right-alternative operations]]
[[!redirects right alternative operation]]
[[!redirects right alternative operations]]
[[!redirects right-alternative magma]]
[[!redirects right-alternative magmas]]
[[!redirects right alternative magma]]
[[!redirects right alternative magmas]]
[[!redirects right-alternative algebra]]
[[!redirects right-alternative algebras]]
[[!redirects right alternative algebra]]
[[!redirects right alternative algebras]]

[[!redirects flexibility]]
[[!redirects flexible binary operation]]
[[!redirects flexible binary operations]]
[[!redirects flexible operation]]
[[!redirects flexible operations]]
[[!redirects flexible magma]]
[[!redirects flexible magmas]]
[[!redirects flexible algebra]]
[[!redirects flexible algebras]]


alternative experimental definition of commutative diagram

##Discussion

[[Eric]]: Here is a degenerate loop:
$$\array{
X & \stackrel{e}{\to} & X \\
{} & \mathllap{\scriptsize{e}}{\nwarrow} & \darr\scriptsize{e} \\
{} & {} & X
}$$

To say this loop commutes, is to say $e^3 = 1_X$. Or is it?

>[[David Roberts]]: That is correct.

[[Eric]]: Since this loop is degenerate, we may or may not have (depending on convention?), parallel identity arrows running between each vertex.

If there are hidden identity morphisms, then to say the triangle commutes is to say $e = 1_X$.

>[[David Roberts]]: No: the above version $e^3 = id$ is correct.

[[Eric]]: Here is a degenerate triangle:

$$\array{
X & \stackrel{e}{\to} & X \\
{} & \mathllap{\scriptsize{e}}{\searrow} & \darr\scriptsize{e} \\
{} & {} & X
}$$

To say this triangle commutes is to say $e^2 = e$. Or is it?

>[[David Roberts]]: That is correct.

[[Eric]]: Since this triangle is degenerate, we may or may not have (depending on convention?), parallel identity arrows running between each vertex.

If there are hidden identity morphisms, then to say the triangle commutes is to say $e = 1_X$.

[[David Roberts]]: No: the above version $e^2 = e$ is correct. For both of these wrong statements, you are assuming that arrows somehow cancel. Especially the first one, think of it in terms of dimensional analysis. If you have a quantity in physics with dimensions of something else cubed, it cannot have the dimensions of that original thing (unless both are dimensionless). But this is only a rough analogy, please don't read anything deep into it. In my comment in reply to Harry's $e^2 = e$ being a loop, then I said I would draw this as a commuting triangle. I wouldn't draw a loop (i.e. an endomorphism) (resp. a degenerate loop) as a triangle (resp. commuting triangle) in any circumstances, because we are not given that the loop factors into a composite of other arrows.

[[Eric]]: Thanks David. I think I've managed to boil it down to the basic disagreement. It is about "shape dependence". I reject the notion of shape dependence. The morphism $f:X\to X$ is a loop and this loop is the same as $f:X\righttoleftarrow$. I [explain this](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=332&Focus=7368#Comment_7368) on the n-Forum. The point is, you can accept shape dependence, in which case I agree with everything you say. Or you reject shape dependence and work with semidiagrams instead.

+-- {: .query}
I don\'t think that there\'s any doubt that $e^2 = e$ is the intended interpretation of the second diagram by everybody who uses commutative diagrams.  The fact that the same object $X$ shows up twice is irrelevant; the whole point of putting $X$ in there twice is because you *don\'t* want parallel identity arrows.  Of course, you can invent an alternative interpretation if you want, but I don\'t think that you\'ll find any usage to match it in the wild.  ---Toby
=--

##Main Article

A stub for now.

The basic goal of this page is to turn Domenico's and Urs' statements about connections into statements about commutative diagrams.

The rough is idea is:

$$\text{Higher Connection Flat}\quad\quad\simeq\quad\quad\text{Higher Diagram Commutes}$$

+--{.standout}
[Domenico](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=723&Focus=5015#Comment_5015): I guess this is well known, but let me try writing it here and see what happens.

If a connection on a principal G-bundle is locally represented by the 1-form $\omega$ with values in $\mathfrak{g}$, then the connection is flat if and only if the curvature 2-form $F=d\omega+\frac{1}{2}[\omega,\omega]$ vanishes, that is, if $\omega$ is a solution of the Maurer-Cartan equation.

Now, what is interesting is that one can see curvature only on 2-dimensional paths (it is a 2-form): if one restricts $\omega$ to a 1-dimensional submanifold, then there $\omega$ is clearly a solution to the Maurer-Cartan equation. so, if I think of an infinitesimal 1-simplex and look at my connection there, I could say that my connection is 1-flat. then, moving to an infinitesimal 2-simplex I see that the connection is (generally) not 2-flat: holonomy along two sides of the 2-simplex is not the same thing as holonomy along the third side. not the same, but in a very precise way: the curvature $FF$ exactly measures the gap to go from a horn of the 2-simplex to the third edge. this is very 2-categorical, and suggests I could cure the lack of flatness of my original connection by adding a copy of $\mathfrak{g}$ in degree -1 and cooking up a 2-Lie algebra $\mathfrak{l}$. Maurer-Cartan equation for this 2-Lie algebra would coincide with the original one one the 1-simplex (since only degree 1 elements are 1-forms with values in $\mathfrak{l}_0$. but on the 2-simplex we would have, in addition to these elements, also 2-forms with coefficients in $\mathfrak{l}_{-1}$, and the Maurer-Cartan equation on teh simplex would look like $d\omega+\frac{1}{2}[\omega,\omega]+\delta F=0$ where $\delta\colon \mathfrak{l}_{-1}\to \mathfrak{l}_{0}$ is the differential of the $L_\infty-algebra \mathfrak{l}$ (and it should be induced by the identity of $\mathfrak{g}$, thought as a degree 1 map from $\mathfrak{g}[-1]$ to $\mathfrak{g}$). So the original equation telling that $\omega$ had curvature $F$ is now equivalent to say that $\omega+F$ is flat.

in other words, what seemed a non-flat connection was so since I was not seeing the 2-bundle, but only a 1-bundle approximation. and on a 1-bundle I can only clearly see up to 1-simplices, wher my connection was actually flat.

once curvature has come in, we can repeat the argument: now we have a 2-flat connection and test it on the 3-simplex. if it has 3-curvature, that will presumibly be because we are not seeing the 3-bundle, yet. so I find it natural to wonder (to conjecture) whether any connection on a principal bundle (and more generally any $n$-flat connection on an n-bundle) can be seen as a flat connection on an $\infty$-bundle.
=--

+--{.standout}
[Domenico](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=723&Focus=5050#Comment_5050): I think I could take the task of working a bit on oo-Lie algebroid valued forms in the direction sketched above. yet, I'd like to still discuss it a bit here, sicne I'm now thinking of some more radical change. what I'm thinking is that the one-step definition as it is now is too strict. I mean, we are now defining flat $\mathfrak{a}$-connections as a morphism $\prod^{inf}(X)\to\mathfrak{a}$, and then general connections as something which becomes flat in a single step, i.e., going from $\mathfrak{a}$ to $cone(\mathfrak{a}$). why should we restrict to this? it would be natural to consider thigs that become flat in 2 sterps, 3, steps, and so on. eventually we could have things which "become flat after infinity steps" (and by the way this vague notion seems to me to better fit with classical $\mathfrak{g}$-connections).

let us see things the other way round: when we say flat, we mean flat in an $\infty$-categorical sense, so let us stress this by saying that a morphism $\prod^{inf}(X)\to\mathfrak{a}$ is an $\infty$-flat connection with values in $\mathfrak{a}$. then, what we are currently calling non-necessarily-flat $\mathfrak{a}$-connections would be the $(\infty-1)$-flat connections with values in $\mathfrak{a}$. but starting from $\infty$ is not too practical.., let us start from 0 instead.

the one-step definition of non-flat connections is reflected in the one step going from the 0-groupoid $X$ to the $\infty$-groupoid $\prod^{inf}(X)$. but we have a whole tower of higher and higher groupoids in between: the $n$-skeleta of $\prod^{inf}(X)$. so we have

$$X=\prod^{inf}(X)_{(0)}\hookrightarrow \prod^{inf}(X)_{(1)}\hookrightarrow \prod^{inf}(X)_{(2)}\hookrightarrow \cdots \hookrightarrow \prod^{inf}(X)$$

and we can talk of an $n$-flat connection with values in $\mathfrak{a}$ as a morphism $\prod^{inf}(X)_{(n)}\to \mathfrak{a}$. this will not lift to a morphism $\prod^{inf}(X)_{(n+1)}\to \mathfrak{a}$ unless the connection is $(n+1)$-flat, but will lift to a morphism $\prod^{inf}(X)_{(n+1)}\to cone(\mathfrak{a})$. the limit of this procedure will produce a "true connection" with values in $cone(cone(\cdots(cone(\mathfrak{a}))\cdots)$.

I have a vague feeling that starting with a 0-falt $\mathfrak{o}(n)$-connection one will this way meet the series $\mathfrak{so}(n), \mathfrak{spin}(n), \mathfrak{string}(n), \mathfrak{fivebrane}(n), \mathfrak{uniray}(n)$,..., but I have to think more carefully to this.
=--

##References

* n-Forum: [Is every connection flat?](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=723&page=1)
* n-Forum: [functor](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=332&Focus=7304#Comment_7304)
* Eric\'s web: [[ericforgy:Shape Dependence in Commutative Diagrams]]

[[!redirects Shape Dependence in Commutative Diagrams]]


alternative magmoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
#### Categorification
+-- {: .hide}
[[!include categorification - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Just as a [[groupoid]] is the [[oidification]] of a [[group]] and a [[ringoid]] is the oidification of a [[ring]], an alternative magmoid should be the oidification of a [[alternative magma]].

## Definition

An __alternative magmoid__ is a [[magmoid]] $Q$ such that for every two objects $a,b \in Ob(Q)$ and for every morphism $f:a \to b$ and endomorphisms $g:a \to a$ and $h:b \to b$,

$$
  f \circ (g \circ g) = (f \circ g) \circ g
  \,
$$

and

$$
  h \circ (h \circ f) = (h \circ h) \circ f
  \,
$$

## Examples

* Every [[category]] is an alternative magmoid.

* A one-object alternative magmoid is a [[alternative magma]].

* A [[Mod]]-enriched alternative magmoid is called a alternative [[linear magmoid]].

## Related concepts

[[!include oidification - table]]


Alvaro Restuccia

* [InSpire page](https://inspirehep.net/authors/991923)

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=194082)

## Selected writings

On the [[super 2-brane in 4d]]-[[D=4 supergravity]]:

* [[Maria P. Garcia del Moral]], J. M. Pena, [[Alvaro Restuccia]], _$\mathcal{N}=1$ 4D Supermembrane from $11D$_, JHEP0807 039 (2008) [[arXiv:0709.4632](http://arxiv.org/abs/0709.4632)]

Lift of T-duality from [[string theory]] to a [[SL(2,Z)]]-[[U-duality]] acting on the [[M2-brane]]-[[Green-Schwarz action functional|sigma-model]]:

* [[Maria P. Garcia del Moral]], I. Martin, [[Alvaro Restuccia]], *Nonperturbative $SL(2,\mathbb{Z})$ $(p,q)$-strings manifestly realized on the quantum M2* [[arXiv:0802.0573](https://arxiv.org/abs/0802.0573)]

* [[Maria P. Garcia del Moral]], J. M. Pena, [[Alvaro Restuccia]], *Aspects of the T-duality construction for the Supermembrane theory*, J. Phys.: Conf. Ser. **720** (2016) 012025 [[arXiv:1504.06907](https://arxiv.org/abs/1504.06907), [doi:10.1088/1742-6596/720/1/012025](https://doi.org/10.1088/1742-6596/720/1/012025)]

On [[gauged supergravity|gauged]] [[D=9 supergravity]] from [[SL(2,Z)]]-[[U-duality]] acting on the [[M2-brane]]:

* [[Maria P. Garcia del Moral]], J. M. Pena, [[Alvaro Restuccia]], _Supermembrane origin of type II gauged supergravities in 9D_, JHEP 1209 (2012) 063 [[arXiv:1203.2767](http://arxiv.org/abs/1203.2767)]

* [[Maria P. Garcia del Moral]], C. Las Heras, P. Leon, J. M. Pena, [[Alvaro Restuccia]], _Fluxes, Twisted tori, Monodromy and $U(1)$ Supermembranes_,  J. High Energ. Phys. **2020** 97 (2020) [[arXiv:2005.06397](https://arxiv.org/abs/2005.06397), doi:10.1007/JHEP09(2020)097]

On [[massive type IIA supergravity]] via [[M2-branes]]:

* [[Maria P. Garcia del Moral]], [[Alvaro Restuccia]], *10D Massive Type IIA Supergravities as the uplift of Parabolic M2-brane Torus bundles*, Forthsch. d. Phys. **64** 4-5 (2016) [[arXiv:1511.04784](https://arxiv.org/abs/1511.04784), [doi:10.1002/prop.201500087](https://doi.org/10.1002/prop.201500087)]

* [[Maria P. Garcia del Moral]], P. Leon, [[Alvaro Restuccia]], *Wordsheet description of a* massive *type IIA superstring in 10D*, J. High Energ. Phys. **2023** 104 (2023) [[arXiv:2306.16620](https://arxiv.org/abs/2306.16620), doi:10.1007/JHEP11(2023)104]

category: people


Amalendu Krishna

* [Wikipedia entry](https://de.wikipedia.org/wiki/Amalendu_Krishna)

* [institute home page](http://www.tifr.res.in/People_Finder/compcode.php?param1=39)

## related $n$Lab entries

* [[algebraic K-theory]]

category: people


amalgamation

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Traditionally:

* **in [[group theory]]**

  1. the term *[[amalgamated free product of groups]]* refers to what really are *[[pushouts]]* in the [[category]] [[Grp]], while

  1. the plain term "[[free product]]" really refers to [[coproducts]] in the [[category]] [[Grpd]];

therefore it makes sense to generally agree that:

* **in [[category theory]]** the term "amalgamated sum" [[Gabriel & Zisman (1967), p. 1](pushout#GabrielZisman67)] and variants like "*amalgamated coproduct*" or just "*amalgamation*" refers to [[pushouts]] (seen, if you will, as [[coproducts]] in a [[coslice category]]).

Beware though, while the terminology "amalgamation" indeed neatly captures the nature of [[pushouts]] in general [[categories]], traditionally it is not widely used in this generality.

Moreover, but only vaguely related:

* **in [[model theory]]** there is a notion of *[[amalgamation property]]* of [[structures in model theory]].

Finally, the term *[[amalgamation theory]]* is used here and there...

category: disambiguation

[[!redirects amalgamations]]

[[!redirects amalgamated coproduct]]
[[!redirects amalgamated coproducts]]

[[!redirects amalgam]]
[[!redirects amalgams]]


amalgamation property

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## Idea

A class of [[structures in model theory]] has the __amalgamation property__ if for any three structures $A,B,C$ and embeddings $f_B: A\hookrightarrow B$, $f_C: A\hookrightarrow C$, there exist embeddings $g_B:B\hookrightarrow D$ and $g_C: C\hookrightarrow D$ such that $g_B\circ f_B = g_C\circ f_C$.

One of the simplest cases is when the free amalgam of structures $B\oplus_A C$ exists.

## Related entries

* [[amalgamation]]

* [[Galois type]]

* [[Ore condition]]

## References

The amalgamation method for generating strongly minimal
theories is introduced in

* [[Ehud Hrushovski]], _A new strongly minimal set_. Ann. Pure Appl. Logic,
62(2):147--166, 1993. Stability in model theory, III (Trento, 1991).

Recent references include

* [[Ehud Hrushovski]], _Groupoids, imaginaries and internal covers_, [arxiv/math.LO/0603413](http://arxiv.org/abs/math/0603413)
* Uri Andrews, _Amalgamation constructions and recursive model theory_, thesis, [pdf](http://www.math.wisc.edu/~andrews/andrewsthesis.pdf)
* John T. Baldwin, Alexei Kolesnikov, [[Saharon Shelah]],
_The amalgamation spectrum_, J. Symbolic Logic __74__:3 (2009) 914-928, [MR2548468](http://www.ams.org/mathscinet-getitem?mr=2548468), [doi](http:/dx.doi.org/10.2178/jsl/1245158091[euclid](http://projecteuclid.org/euclid.jsl/1245158091)
* J D Brody, _Model theory of graphs_, thesis, [pdf](http://tarski.fandm.edu/brody/mainthesis.pdf)

category: model theory

[[!redirects amalgamation properties]]


amalgamation theory

## Related concepts

* [[amalgamation]]

* [[amalgamated free product of groups]]

* [[amalgamation property]]

[[!redirects amalgamation theoies]]


Amanda Peet

* [webpage](http://ap.io/home/about-prof-peet/)

## Selected writings

On [[S-branes]]:

* {#KruczenskiMyersPeet02} [[Martin Kruczenski]], [[Robert Myers]], [[Amanda Peet]], _Supergravity S-Branes_, JHEP 0205 (2002) 039 ([arXiv:hep-th/0204144](https://arxiv.org/abs/hep-th/0204144))

On [[D-brane polarization]] into [[supertubes]]:

* [[Martin Kruczenski]], [[Robert Myers]], [[Amanda Peet]], David J. Winters, _Aspects of supertubes_, JHEP 0205:017, 2002 ([arXiv:hep-th/0204103](https://arxiv.org/abs/hep-th/0204103))

category: people


Amar Hadzihasanovic

* [website](https://www.irif.fr/~ahadziha/)

## Selected writings

On [[string diagrams]] and [[quantum entanglement]]:

* [[Amar Hadzihasanovic]], _The algebra of entanglement and the geometry of composition_, ([arXiv:1709.08086](https://arxiv.org/abs/1709.08086))

On [[rewriting]] in [[higher categories]]

* [[Amar Hadzihasanovic]], _Diagrammatic sets and rewriting in weak higher categories_, ([arXiv:2007.14505](https://arxiv.org/abs/2007.14505))

On the [[tensor product]] of [[PROs]] as the [[smash product]] of [[pointed spaces]]:

* [[Amar Hadzihasanovic]], _The smash product of monoidal theories_, ([arXiv:2101.10361](https://arxiv.org/abs/2101.10361))

category: people


Amaria Javed

member of [[CQTS]]

* [GoogleScholar page](https://scholar.google.ae/citations?user=NTzY1lUAAAAJ&hl=en)

## Selected writings

on [[solitons]] in [[photonic crystals]]:

* [[Amaria Javed]], Alaa Shaheen U. Al Khawaja, *Amplifying optical signals with discrete solitons in waveguide arrays*, Physics Letters A **384** 26 (2020) 126654 [[doi:10.1016/j.physleta.2020.126654](https://doi.org/10.1016/j.physleta.2020.126654)]

* Alaa Shaheen, [[Amaria Javed]], U. Al Khawaja, *Adding binary numbers with discrete solitons in waveguide arrays*, Phys. Scr. **95** (2020) 085107 [[arXiv:2108.01406](https://arxiv.org/abs/2108.01406), [doi:10.1088/1402-4896/aba2b2](https://doi.org/10.1088/1402-4896/aba2b2)]

category: people


amazing right adjoint

[[!redirects fractional exponential]]
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## Definition#

[[William Lawvere]]'s definition of an _atomic_ [[infinitesimal space]] is as an [[object]] $\Delta$ in a [[topos]] $\mathcal{T}$ such that the [[inner hom]] [[functor]] $(-)^\Delta : \mathcal{T} \to \mathcal{T}$ has a [[right adjoint]] (is an [[atomic object]]).

Notice that by definition of [[inner hom]], $(-)^\Delta$ always has a [[left adjoint]]. A [[right adjoint]] can only exist for very particular objects. Therefore the term **amazing right adjoint**.

## Right adjoints to representable exponentials

Assume $\mathcal{T} = Sh(C)$ is a [[Grothendieck topos]], that the [[Grothendieck topology]] on the [[site]] $C$ is [[subcanonical coverage|subcanonical]]. Let $\Delta \in C \hookrightarrow Sh(C)$ be a [[representable functor|representable object]].
Then, $(-)^\Delta: Sh(C)\to Sh(C)$ has a [[right adjoint]], hence $\Delta$ is an atomic [[infinitesimal space]],  precisely if $(-)^\Delta: C\to C$ preserves [[colimit]]s.
This is a special case of the general [[adjoint functor theorem#in_toposes]].
For if $(-)^\Delta$ preserves colimits, its [[right adjoint]] is

$$
 (-)_\Delta : (Y \in Sh(C)) \mapsto (U \mapsto Sh_C(U^\Delta, Y))
  \,.
$$

The $Y_\Delta$ defined this way is indeed a sheaf, due to the assumption that $(-)^\Delta$ preserves colimits. So this is indeed a [[right adjoint]].

## Related concepts

* A [[topos]] $\mathcal{X}$ is a [[local topos]] (over [[Set]]) if its [[global section]] functor $\Gamma = Hom(1_{\mathcal{X}}, -)$ admits a [[right adjoint]]. This is hence an "external" version of the amazing right adjoint, exhibiting $1_{\mathcal{X}}$ as "atomic".

* The topic of amazing right adjoints appears to have been studied mostly for toposes, but a related definition has been given for any category (with products): if $O$ is an [[exponentiable object]] of a category $\mathcal{C}$, then $O$ being a [[tiny object]] in the sense of [Definition 0.1](#Yetter1987) means that the [[internal hom|endofunctor]] $(-)^O$ has a right adjoint. (This adjoint is sometimes denoted $(-)^{1/O}$, c.f. [Lawvere, p.269](#Lawvere2002), and called 'fractional exponential')

In this situation, one then has, symbolically

$$
  (-)\times O \quad \dashv\quad (-)^O \quad \dashv\quad  (-)^{1/O}
$$

## References#

The ubiquity of right adjoints to exponential functors in the context of [[synthetic differential geometry]] was first pointed out in Lawvere (1980). Lawvere (2004) suggests to augment [[lambda calculus]] with such fractional operators. Thorough discussion of the concept is in Yetter (1987) and Kock&Reyes (1999). Moerdijk&Reyes (1991) have a succinct overview in the context of SDG as does Lawvere (1997).

* [[William Lawvere]], _Toward the Description in a Smooth Topos of the Dynamically Possible Motions and Deformations of a Continuous Body_, Cah.Top.Géom.Diff.Cat. **21** no.4 (1980) pp.377-392. ([pdf](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1980__21_4/CTGDC_1980__21_4_377_0/CTGDC_1980__21_4_377_0.pdf))

* [[William Lawvere]], _[[Toposes of laws of motion]]_, 1997.

* {#Law04} [[William Lawvere]], _Left and right adjoint operations on spaces and data types_ , Theor. Comp. Sci. **316** (2004) pp.105-111.

* [[Ieke Moerdijk]], [[Gonzalo Reyes]], [[Models for Smooth Infinitesimal Analysis]] , Springer Heidelberg 1991. (appendix 4)

* [[Anders Kock]], [[Gonzalo E. Reyes]], _Aspects of Fractional Exponent Functors_ , TAC **5** (1999) pp.251-265. ([pdf](http://www.tac.mta.ca/tac/volumes/1999/n10/n10.pdf))

* {#Yetter1987} [[David Yetter]], _On Right Adjoints of Exponential Functors_ , JPAA **45** (1987) pp.287-304. (Corrections in JPAA **58** (1989) pp.103-105)

* {#Lawvere2002} [[William Lawvere]], Categorical algebra for continuum micro physics, Journal of Pure and Applied Algebra 175 (2002) 267–287


Ambidexterity in K(n)-Local Stable Homotopy Theory

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This page collects some links related to

* [[Michael Hopkins]], [[Jacob Lurie]]:

  **Ambidexterity in $K(n)$-Local Stable Homotopy Theory**

  (2013)

  [pdf](http://www.math.harvard.edu/~lurie/papers/Ambidexterity.pdf)

  also: talk at _Notre Dame Graduate Summer School on Topology and Field Theories_ and _Harvard lecture_ 2012 ([video part 1](http://www.youtube.com/watch?v=eQayYLDw1VA), [part 2](http://www.youtube.com/watch?v=OEShrQyvmS4), [part 3](http://www.youtube.com/watch?v=nOIcdn1iUR4) [part 4](http://www.youtube.com/watch?v=ZwnClYedaYM), [pdf lecture notes](http://www.math.northwestern.edu/~celliott/notre_dame_notes/Lurie_notes.pdf) by Chris Elliott)

on [[biproduct|bilimits]] in [[K(n)-local stable homotopy theory]] and generally on [[ambidextrous adjunctions]] and their un-twisted [[Wirthmüller isomorphisms]] for [[(∞,1)-module bundles|∞-module bundles]] in [[semiadditive (∞,1)-categories]].

(The untwisted [[Wirthmüller isomorphism]] is the map $\mu$ in Construction 4.0.7 and then the [[norm map]] in Remark 4.1.12. The induced [[integration]] map considered in Construction 4.0.7, Notation 4.1.6 there is also discussed (for the general twisted case) in the article _[[schreiber:Quantization via Linear homotopy types]]_, see section 4.3 there for details.)

The discussion in the article is apparently motivated as part of what it takes to make precise the discussion of [[quantization]] in sections 3 and 8 of

* [[Dan Freed]], [[Mike Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]], _[[Topological Quantum Field Theories from Compact Lie Groups]]_ in P. R. Kotiuga (ed.) A celebration of the mathematical legacy of Raoul Bott AMS (2010) ([arXiv:0905.0731](http://arxiv.org/abs/0905.0731))

For the 2014 installment of UOregon's Moursund Lecture Series, Jacob Lurie gave three (video recorded) lectures on [[spectral algebraic geometry]], one of which is

* [[Jacob Lurie]], [Ambidexterity](http://media.uoregon.edu/channel/2014/05/23/moursund-mathematics-lecture-series-dr-jacob-lurie-lecture-2-ambidexterity/)

#Contents#
* table of contents
{:toc}

## 1. Multiplicative aspects of Dieudonne Theory

## 2. The Morava $K$-theory of Eilenberg-MacLane Spaces

* [[Morava K-theory]]

## 3. Alternating powers of $p$-Divisible groups

## 4. Ambidexterity

### 4.1 Beck-Chevalley fibrations and Norm maps

* [[Beck-Chevalley condition]]

* [[pointwise (∞,1)-Kan extension]]

### 4.2 Properties of the norm

### 4.3 Local systems

* [[local system]]

* [[ambidextrous space]]

### 4.4 Examples

* [[semiadditive (∞,1)-category]]

## 5. Ambidexterity of $K(n)$-Local stable homotopy theory

### 5.1 Ambidexterity and duality

### 5.2 The main theorem

* [[K(n)-local stable homotopy theory]]

### 5.3 Cartier duality

* [[Cartier duality]]

### 5.4 The global sections functor

category: reference