commutative von Neumann algebra

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#### Algebra
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#### Measure and probability theory
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[[!include measure theory - contents]]
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#Contents#
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## Idea

A [[von Neumann algebra]] that is [[commutative]].

## Properties

The [[category]] of commutative von Neumann algebras
is a [[full subcategory]] of that of [[von Neumann algebras]]
and has many special properties.

## Duality

The following five categories are [[equivalence of categories|equivalent]]:

* the [[opposite category]] of commutative von Neumann algebras;
* [[compact topological space|compact]] strictly localizable enhanced [[measurable spaces]];
* [[measurable locales]];
* [[hyperstonean locales]];
* [[hyperstonean spaces]].

This result can be seen as a [[measure theory|measure-theoretic]] counterpart of the [[Gelfand duality]] between commutative unital [[C*-algebras]] and [[compact Hausdorff topological spaces]].

## Properties

From <https://mathoverflow.net/questions/384346/is-the-opposite-category-of-commutative-von-neumann-algebras-a-topos/384357#384357>:

The [[opposite category]] of commutative von Neumann algebras is not a [[topos]] because [[Cartesian product|categorical products]] with a fixed object do not always [[preserved colimit|preserve]] [[small colimit|small]] [[colimits]].
See Theorem 6.4 in Andre Kornell's Quantum Collections.

## Closed monoidal structure

The [[opposite category]] of commutative von Neumann algebras admits a non-[[cartesian monoidal category|cartesian]] [[closed monoidal category|closed monoidal structure]], where the [[tensor product|monoidal product]] corresponds to the spatial product of measurable spaces.

## Related concepts

* [[von Neumann algebra]]

* [[Hilbert W*-module]]

* [[W*-representation]]

* [[duality between geometry and algebra]]

## References

* [[Jacques Dixmier]],
_Sur certains espaces considérés par [[M. H. Stone]]_.
Summa Brasiliensis Mathematicae 2 (1951), 151–182.
[PDF](https://dmitripavlov.org/scans/dixmier.pdf).

* [[Irving E. Segal]],
_Equivalences of measure spaces_.
American Journal of Mathematics 73:2 (1951), 275–313.
[doi:10.2307/2372178](https://doi.org/10.2307/2372178).

* [[Dmitri Pavlov]]
_Gelfand-type duality for commutative von Neumann algebras_.
[2005.05284](https://arxiv.org/abs/2005.05284)

[[!redirects commutative von Neumann algebras]]


commutativity of homotopy limits and colimits

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#### Limits and colimits
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#Contents#
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## Idea

…

## Definition

…

## Universal commutativity

In a locally $\lambda$-presentable [[(∞,1)-category]], $\lambda$-filtered (∞,1)-colimits commute with $\lambda$-small (∞,1)-limits.

In particular, in a locally finitely presentable [[(∞,1)-category]], filtered (∞,1)-colimits commute with finite (∞,1)-limits.

In an algebraic (∞,1)-category, i.e., one that is equivalent to the category of finite product-preserving functors from a small (∞,1)-category with finite products to the (∞,1)-category of spaces,
sifted (∞,1)-colimits commute with finite (∞,1)-products.

## Special commutativity: model ∞-categories

From a [MathOverflow answer](https://mathoverflow.net/questions/287091/why-do-we-need-model-categories/287234#287234):

Model categories provide a powerful framework for commuting (homotopy) limits and colimits,
and, more generally, for commuting left adjoint functors and (homotopy) limits,
as well as right adjoint functors and (homotopy) colimits.
(In what follows, I omit the adjective “homotopy” before limits and colimits.)

In finitely presentable ∞-categories filtered colimits commute with finite limits
and in finitely presentable ∞-categories with a set of compact projective generators
sifted colimits commute with finite products.
However, many other situations of interest are not covered by such statements.
For instance, one might want to commute a sifted colimit past an infinite product, a pullback,
or a cosifted limit (e.g., a cosimplicial totalization).
One might also want to commute sifted colimits past finite products
in ∞-categories that do not have a set of compact projective generators,
e.g., the ∞-category of small ∞-categories.

In all such situations the relevant statement is false at least for some diagrams,
so whatever criterion we devise must analyze the specific diagrams at question.
This is precisely what model categories achieve.

For example, one might want to commute $K$-indexed colimits (for some small diagram $K$)
past some right adjoint functor $F: C\to B$.
(For instance, one can take $F$ to be
$$lim_L: Fun(L,B)\to B,$$
the limit functor for $L$-indexed diagrams,
which will allows us to commute $K$-indexed colimits past $L$-indexed limits.)

We would like to devise a condition on a diagram $D: K\to C$ that would guarantee that the canonical
comparison map
$$colim_K F(D) \to  F(colim_K D)$$
is an equivalence in the ∞-category $B$.
This is achieved by the following creative procedure.
First, consider the relative ∞-category (i.e., ∞-category equipped with a class of maps (weak equivalences) closed under composition)
$Fun(K,C)$ whose weak equivalences are created by the functor $colim_K$ (i.e., a natural transformation of functors $K\to C$
is a weak equivalence if its $K$-colimit is an equivalence in the ∞-category $C$).
The ∞-category $Fun(K,B)$ is turned into a relative ∞-category in the same way.
The functor
$$Fun(K,F): Fun(K,C)\to Fun(K,B)$$
is a functor between ∞-categories that need not preserve weak equivalences.

Now comes the (potentially) creative part: equip the relative ∞-categories $Fun(K,C) and Fun(K,B)$
with model structures such that $Fun(K,F)$ is a right Quillen functor.
In many situations of interest this can be done immediately using existing tools.

Our criterion now says that if $D: K\to C$ is a _fibrant_ diagram,
then the comparison map
$$colim_K F(D) \to  F(colim_K D)$$
is an equivalence in the ∞-category B,
i.e., the colimit of $D$ over $K$ commutes with $F$.
Different choices of model structures give us different criteria.

For instance, one can take $K=\Delta^{op}$, $B=spaces$ (alias ∞-groupoids),
$C=Fun(L,B)$ (i.e., $L$-indexed diagrams of spaces)
and
$$F=lim_L: Fun(L,B)\to B,$$
the $L$-indexed limit functor,
where $L$ can be taken to be the pullback diagram or the infinite discrete category
and the model structure can be taken to be the projective model structure.
In this case we obtain a criterion for commuting
$\Delta^{op}$-indexed colimits past $L$-indexed limits.
This recovers, for example, the traditional methods
for computing homotopy pullbacks of simplicial sets (replace one of the legs by a fibration),
infinite homotopy products of simplicial sets (fibrantly replace all terms), etc.

There are many variations on the above theme, for instance,
one can consider *weighted* limits (in the sense of enriched category theory)
and then derive the limit functor with respect to both the functor and the weight,
which yields even more powerful computational tools etc.

Many classical results on model categories fit in the above framework.
For instance, if $M$ is a simplicial model category,
$X$ is a cofibrant object,
and $Y$ is a fibrant object,
then the simplicial mapping space $Map(X,Y)$ from $X$ to $Y$
computes the mapping space in the ∞-localization of $M$ with respect to its weak equivalences.
We can fit this in the above framework by taking
$$F=Map(-,Y): Fun(\Delta^op,M)\to Fun(\Delta^op,Spaces),$$
and observing that a cofibrant object $X$ has a cofibrant cosimplicial resolution $X\otimes\Delta^n$.

Textbook accounts include Chapter 7 of [Cisinski](#Cisinski).

## Related concepts

* [[commutativity of limits and colimits]]

* [[π-Kan condition]]

## References

### Universal commutativity

* [[Jacob Lurie]], *[[Higher Topos Theory]]* (2009)

### Special commutativity

On conditions that [[base change]] via [[homotopy pullback]] commutes with [[geometric realization of simplicial topological spaces|realization]] (cf. *[[π-Kan condition]]* and [this Prop.](geometric+realization+of+simplicial+topological+spaces#SufficientConditionsForRealizationToPreserveHomotopyPullback)):

* {#BousfieldFriedlander78} [[Aldridge Bousfield]], [[Eric Friedlander]], _Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets_, Springer Lecture Notes in Math. **658** Springer (1978) 80-130 [[pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/bousfield-friedlander.pdf), [[BousfieldFriedlanderSpectra.pdf:file]]]

* {#Lurie11} [[Jacob Lurie]], *Simplicial spaces*, Lecture 7 of: *[Algebraic L-theory and Surgery](https://www.math.ias.edu/~lurie/287x.html)* (2011) [[pdf](https://www.math.ias.edu/~lurie/287xnotes/Lecture7.pdf)]

* {#Rezk14} [[Charles Rezk]], *When are homotopy colimits compatible with homotopy base change?* (2014) [[i-hate-the-pi-star-kan-condition.pdf](https://faculty.math.illinois.edu/~rezk/i-hate-the-pi-star-kan-condition.pdf), [[RezkHomotopyColimitsBaseChange.pdf:file]]]

* [[Jacob Lurie]], §A.5, esp. Def. A.5.2.1, Thms. A.5.3.1, A.5.4.1, A.5.6.1 in  *[[Spectral Algebraic Geometry]]* (2018) [[pdf](https://math.harvard.edu/~lurie/papers/SAG-rootfile.pdf)]

{#ForATreatment} For a treatment without ∞-categories, see also [BEBP19, Def. 3.2, Ex. 3.10, Thm. 3.17](#BEBP19) below.

An even larger class of maps of so-called _weak Kan fibrations_ is studied in Section 3 of

* {#BEBP19} [[Daniel Berwick-Evans]], [[Pedro Boavida]], [[Dmitri Pavlov]], _Classifying spaces of infinity-sheaves_, Algebraic & Geometric Topology  [[arXiv:1912.10544](https://arxiv.org/abs/1912.10544)]

The general theory of [[model (∞,1)-categories]] that allows one to commute homotopy colimits past homotopy limits is developed in a series of papers by Aaaron Mazel-Gee:

* [[Aaron Mazel-Gee]], *Model ∞-categories I: some pleasant properties of the ∞-category of simplicial spaces* [[arXiv:1412.8411](https://arxiv.org/abs/1412.8411)]

* [[Aaron Mazel-Gee]], *Model ∞-categories II: Quillen adjunctions*, New York Journal of Mathematics **27** (2021) 508-550.  [[arXiv:1510.04392](https://arxiv.org/abs/1510.04392), [nyjm:27-21](http://nyjm.albany.edu/j/2021/27-21.html)]

* [[Aaron Mazel-Gee]], *Model ∞-categories III: the fundamental theorem*, New York Journal of Mathematics **27** (2021) 551-599 [[arXiv:1510.04777](https://arxiv.org/abs/1510.04777), [nyjm:27-22](http://nyjm.albany.edu/j/2021/27-22.html)]

and in

* {#Cisinski} [[Denis-Charles Cisinski]], Chapter 7 of: *[[Higher Categories and Homotopical Algebra]]*, Cambridge Studies in Advanced Mathematics **180**, Cambridge University Press (2019) [[doi:10.1017/9781108588737](https://doi.org/10.1017/9781108588737), [pdf](https://cisinski.app.uni-regensburg.de/CatLR.pdf)]

In particular, see Propositions 7.5.5, 7.7.4, 7.7.5.

[[!redirects commutativity of homotopy colimits and limits]]


commutativity of limits and colimits

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## Idea

In general, [[limit]]s and [[colimit]]s do not commute.  It is therefore of interest to list the special conditions under which certain limits do commute with certain colimits. (See also at *[[permuting limits and colimits]]*.)

## Definition

Let $C$ and $D$ be (usually [[small category|small]]) categories, and $E$ a category that has both $C$-colimits and $D$-limits.  Then for any functor $F \colon C \times D \to E$, there is a canonical morphism

$$
  \label{FormulaForCommutativityOfLimitsOverColimits}
   colim_C lim_D F
  \to
   lim_D colim_C F.
$$

We say that **$C$-colimits commute with $D$-limits in $E$** if this is an [[isomorphism]] for all such $F$.  This is equivalent to both of the statements:

* The functor $colim_C : [C,E] \to E$ [[preserved limit|preserves]] (i.e. "commutes with") $D$-limits.
* The functor $lim_D : [D,E] \to E$ preserves $C$-colimits.

## Examples

### Preservation by functor categories and localizations

If $C$-colimits commute with $D$-limits in $E$, then the same is true in any [[functor category]] $[J,E]$, since limits and colimits in the latter are both pointwise in $E$.

Also, if $C$-colimits commute with $D$-limits in $E$, and if $E'$ is a [[reflective subcategory]] of $E$ with a reflector $L$ that preserves $D$-limits, then $C$-colimits also commute with $D$-limits in $E'$.  This follows because the functor $colim_C : [C,E'] \to E'$ factors as the composite $[C,E'] \hookrightarrow [C,E] \xrightarrow{colim_C} E \xrightarrow{L} E'$ in which all three functors preserve $D$-limits.

### Filtered colimits commute with finite limits
 {#FilteredColimitsCommuteWithFiniteLimits}

In [[Set]], [[filtered category|filtered]] [[colimits]] commute with [[finite limits]] (e.g. [MacLane (1971), §IX.2 Thm. 1 (p. 211)](#MacLane71)).

In fact, $C$ is a [[filtered category]] *if and only if* $C$-colimits commute with finite limits in $Set$.  More generally, filtered colimits commute with [[L-finite category|L-finite]] limits.

By the above remarks, it follows that filtered colimits commute with finite limits in any [[Grothendieck topos]].

### Sifted colimits commute with finite products
  {#SiftedColimitsCommuteWithFiniteProducts}

Again in [[Set]] (and hence also in any [[Grothendieck topos]]), [[sifted category|sifted colimits]] commute with [[finite products]]. In fact, this is usually taken to be the definition of a [[sifted category]], and then a theorem of [Gabriel-Ulmer 71](sifted+colimit#GabrielUlmer71) characterizes sifted categories as those for which the [[diagonal functor]] $C \to C \times C$ is a [[final functor]].

As a special case, [[categories with finite products are cosifted]].

For more on this see at _[[distributivity of products and colimits]]_.

### Taking orbits under the action of a finite group commutes with cofiltered limits

This means that if $G$ is a finite group, $C$ is a small cofiltered category and $F : C \to G Set$ is a functor, the canonical map

$$ (\lim F)/G \to \lim_{j \in F} (F(j)/G) $$

is an isomorphism. This fact is mentioned by André Joyal in _Foncteurs analytiques et espèces de structures_; a proof can be found [here](http://www.matem.unam.mx/~omar/notes/fingrpcomm.html).

### Coproducts commute with connected limits

+-- {: .num_example #CoproductsCommutingWithConnectedLimits}
###### Proposition

Let  $A$ be a [[set]], $C$ a [[connected category]], and $F \colon C\times A \longrightarrow Set$ a [[functor]]. Then the canonical morphism

$$
  \underset{a\in A}{\coprod} \underset{\longleftarrow}{\lim}_{c\in C} F(c,a)
    \longrightarrow
   \underset{\longleftarrow}{\lim}_{c\in C} \coprod_{a\in A}F(c,a)
$$

is an [[isomorphism]]. This remains true if [[Set]] is replaced by any [[Grothendieck topos]].

=--

More generally, if $\mathbf{H}$ is an [[(∞,1)-topos]], $A$ is an [[n-groupoid]], and $C$ is a small [[(∞,1)-category]] whose [[classifying space]] is [[n-connected]], then $C$-limits commute with $A$-colimits in $\mathbf{H}$. This follows from the fact that the colimit functor $\mathbf{H}^A\to\mathbf{H}$ induces an equivalence of (∞,1)-topoi $\mathbf{H}^A\simeq \mathbf{H}_{/A}$. For example, if $C$ is a [[cofiltered (∞,1)-category]] or even a [[cosifted (∞,1)-category]], then the classifying space of $C$ is weakly contractible and hence $C$-limits commute with $A$-colimits in $\mathbf{H}$ for any [[∞-groupoid]] $A$.

### Classes of limits and sound doctrines

In general, for any class of limits $\Phi$, one may consider the class of all colimits that commute with $\Phi$-limits and dually.  These classes of limits and colimits share many of the properties of the above examples, especially when $\Phi$ is a [[sound doctrine]].

## Relation to stability under base change
  {#ColimitsStableByBaseChange}

{#BewareThatUniversalColimitsAreNotStrictlySpeakingColimitsCommutingWithPullback} Stability of a colimit under pullback looks informally like a "commutativity" condition between colimits and pullbacks, but it is not actually in general an instance of the general notion of commutativity of limits and colimits, though it is an instance of [[distributivity of limits over colimits]].  See also [[pullback-stable colimit]] for more.

## Related concepts

* [[distributivity of limits over colimits]], [[permuting limits and colimits]]

* [[pullback-stable colimit]]

* [[absolute colimits]]

## References

* [[Peter Hilton]], *Commuting limits*, Cahiers de Topologie et Géométrie Différentielle Catégoriques 10.1 (1968): 127-138.

* [[Beno Eckmann]], [[Peter John Hilton]], *Commuting limits with colimits*, Journal of Algebra **11** 1 (1969) 116-144.

* Armin Frei, and John L. MacDonald, *Limits in categories of relations and limit-colimit commutation*, Journal of Pure and Applied Algebra **1** 2 (1971) 179-197

* {#MacLane71} [[Saunders MacLane]], §IX.2 in: *[[Categories for the Working Mathematician]]*, Graduate texts in mathematics, Springer (1971) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* François Foltz, *Sur la commutation des limites*, Diagrammes **5** (1981) F1-F33

* {#BJTS14} [[Marie Bjerrum]], [[Peter Johnstone]], [[Tom Leinster]], [[William F. Sawin]], *Notes on commutation of limits and colimits*, Theory and Applications of Categories **30** (2015), 527-532 [[arXiv:1409.7860](http://arxiv.org/abs/1409.7860), [tac:3015](http://www.tac.mta.ca/tac/volumes/30/15/30-15abs.html)]

[[!redirects commutativity of limits and colimits]]
[[!redirects commutativity of limits with colimits]]
[[!redirects commutativity of a limit with a colimit]]
[[!redirects commutativity of limits over colimits]]
[[!redirects commutativity of a limit over a colimit]]

[[!redirects commutativity of colimits and limits]]
[[!redirects commutativity of colimits with limits]]
[[!redirects commutativity of a colimit with a limit]]

[[!redirects limits commuting with colimits]]


commutator

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## Definition

### In rings and algebras

For $A$ a [[ring]] or [[associative algebra]], the **commutator** of two elements $x,y \in A$ is the element

$$
  [x,y] \coloneqq x y - y x
  \,.
$$

### In groups

For $G$ a [[group]], the **group commutator** of two elements $a,b \in G$ is the element

$$
  [a,b] \coloneqq a^{-1} b^{-1} a b
  \,.
$$

## Properties

* For $A$ an [[associative algebra]], the underlying [[vector space]] of $A$ equipped with the commutator bracket is a [[Lie algebra]] $(A,[-,-])$.

## Related concepts

* [[Lie bracket]]

* [[group commutator]]. [[commutator subgroup]]

* [[anti-commutator]]

* [[graded commutator]]

* [[associator]]

* [[commutator function]]

* [[canonical commutation relation]], [[Weyl algebra]]

## References

On commutators in [[operator algebra]]:

* [[Calvin R. Putnam]], *Commutation Properties of Hilbert Space Operators and Related Topics*, Ergebnisse der Mathematik und ihrer Grenzgebiete **36**, Springer (1967) [[doi:10.1007/978-3-642-85938-0](https://doi.org/10.1007/978-3-642-85938-0)]

[[!redirects commutators]]

[[!redirects commutator bracket]]
[[!redirects commutator brackets]]


commutator subgroup

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## Definition

For $G$ a [[group]] its _commutator subgroup_ $[G,G] \hookrightarrow G$ is the smallest [[subgroup]] containing all the [[group commutator]]
elements $[g,h] \coloneqq g^{-1} h^{-1} g h$.

This definition also makes sense for [[invertible semigroups]], for which they would be called 'commutator invertible subsemigroups'.

## Properties

* The commutator subgroup is a [[normal subgroup]]. Therefore the [[quotient group]] $G^{ab} \coloneqq G/[G,G]$ exists. This is an [[abelian group]], called the [[abelianization]] of $G$.

## Related concepts

* [[abelianization]]

[[!redirects commutator subgroups]]


Como

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This entry is about (the legendary conference Como 1990 and) the excellent proceedings volume

* A. Carboni, M. Pedicchio, G. Rosolini, _Category theory_ , Proceedings of the International Conference held in Como 1990, 213–236, Lecture Notes in Math. __1488__, Springer (1991).

which is one of the overall historical heights of the (quality content of the) proceedings within the tradition of conferences in pure [[category theory]]. Now it is even hard to find a publisher for contemporary category conference proceedings...

The preface is

* [[Bill Lawvere|F. W. Lawvere]], _[[Some Thoughts on the Future of Category Theory]]_

  (on [[cohesive toposes]]).

*Some* of the articles include the following.

* [[André Joyal|A. Joyal]], [[Myles Tierney|M. Tierney]], _Strong stacks and classifying spaces_, 213–236 ([pdf](http://www.pps.jussieu.fr/~mellies/semantics-operads-categories/joyal-tierney-strong-stacks.pdf))

   (on [[(2,1)-sheaves]] and [[2-sheaves]]).

* S. Kasangian, S. Vigna, _Trees in distributive categories_,  237--248

* [[Anders Kock|A. Kock]], _Algebras for the partial map classifier monad_,  262--278

* [[Bill Lawvere|F. W. Lawvere]], _Intrinsic co-Heyting boundaries and the Leibniz rule in certain toposes_,  279--281

  (see also at _[[co-Heyting algebra]]_)

* [[Dusko Pavlovic|Duško Pavlović]], _Categorical interpolation: descent and the [[Beck-Chevalley condition]] without direct images_, 306--325 ($n$lab remark: important clarification of the subject from logics point of view).

* A. J. Power, _An $n$-categorical pasting theorem_,  326--58

  (on [[pasting diagrams|pasting]] in [[n-categories]]).

* [[Gonzalo Reyes|G. E. Reyes]], H. Zolfaghari, _Topos-theoretic approaches to modality_,  359--378

  (on [[modal logic]] in [[topos theory]]).

* [[Stephen Schanuel|S. H. Schanuel]], _Negative sets have Euler characteristic and dimension_, 379--385

* M. Thiébaud, _Modular categories_,  386--400

* J. J. C. Vermeulen, _Some constructive results related to compactness and the (strong) Hausdorff property for locales_,  401--409

  (on [[locales]]).

* [[Andre Joyal|A. Joyal]], [[Ross Street|R. Street]], _An introduction to [[Tannaka duality]] and [[quantum groups]]_, 413–492

...

About the geographic place see wikipedia: [Como](http://en.wikipedia.org/wiki/Como) and the conference centre: [Villa Olmo](http://en.wikipedia.org/wiki/Villa_Olmo).

In 2018, a [school and conference](http://tcsc.lakecomoschool.org/) on topos theory was hosted in Como ([video playlist](https://www.youtube.com/playlist?list=PLh_3Q6ZRqWs0LBptMGClJ8OArR0fBT6Pp)).

category: reference


comodal type

[[!redirects anti-modal type]]

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#### Modalities, Closure and Reflection
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[[!include modalities - contents]]
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# Contents
* table of contents
{: toc}

## Idea

Given a [[modal operator]] $\bigcirc$, then a [[type]] $X$ may be called _$\bigcirc$-comodal_ or _$\bigcirc$-connected_ (largely now the preferred term) if $\bigcirc X \simeq \ast$ (the [[unit type]]).

Since a type $Y$ is called a _$\bigcirc$-[[modal type]]_ if $\bigcirc Y \simeq Y$, being comodal is in a sense the opposite extreme of being modal.

In as far as [[modal operators]] have [[categorical semantics]] as [[idempotent monads|idempotent (co-)-monads]]/[[idempotent (∞,1)-monads|idempotent (∞,1)-(co-)monads]] anti-modal types are familiar in [[homotopy theory]] as forming [[localizing subcategories]].

## Examples

* for $\bigcirc = n \coloneqq \tau_n$ the [[n-truncation]] modality, then _$n$-comodal_ is indeed _[[n-connected]]_, by [this proposition](n-connected+object+of+an+%28infinity%2C1%29-topos#ConnectednessViaTruncationComodalness).

* a comodal type for a [[reduction modality]] or [[infinitesimal shape modality]] is an [[anti-reduced type]], is an [[infinitesimally thickened point]].

## References

The term "anti-modal type" appears in

* [[Mike Shulman]], _All Modalities are HITs_, 2012 ([web](http://homotopytypetheory.org/2012/11/19/all-modalities-are-hits/))

[[!redirects anti-modal type]]
[[!redirects anti-modal types]]
[[!redirects anti modal type]]
[[!redirects anti modal types]]

[[!redirects antimodal type]]
[[!redirects antimodal types]]

[[!redirects comodal type]]
[[!redirects comodal types]]
[[!redirects co-modal type]]
[[!redirects co-modal types]]

[[!redirects comodal object]]
[[!redirects comodal objects]]

[[!redirects anti-modal object]]
[[!redirects anti-modal objects]]


comodule

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* table of contents
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## Idea

A _comodule_ is to a [[comonoid]] as a [[module]] is to a [[monoid]]. Where a [[module]] is equipped with an [[action]], a comodule is dually equipped with a _coaction_.

## Definition

Given a [[comonoid]] $C$ with comultiplication $\Delta_C: C\to C\otimes C$ and counit $\epsilon:C\to \mathbf{1}$ in a [[monoidal category]] $\mathcal{M}$, and an [[object]] $M$ in $\mathcal{M}$,
a __left $C$-coaction__ is

* a [[morphism]] $\rho: M\to C\otimes M$

* which is

  * co[[associative]] i.e. (for $\mathcal{M}$ nonstrict use the canonical isomorphism $C\otimes (C\otimes M)\cong (C\otimes C)\otimes M$ to compare the sides) $(\Delta_C\otimes\mathrm{id}_M)\circ\rho = (\mathrm{id}_C\otimes\rho)\circ\rho: M\to C\otimes C\otimes M$

  * and co[[unital]] i.e. $(\epsilon\otimes \mathrm{id}_M)\circ\rho = \mathrm{id}_M$ (in this formula, $\mathbf{1}\otimes M$ is identified with $M$).

In some monoidal categories, e.g. of (super)[[vector space]]s, and of [[Hilbert space]]s, one often says (left/right) [[corepresentation]] instead of (left/right) coaction.

## The category of comodules

Let $k$ be a [[commutative ring]] and let $\mathcal{M} = Mod(k)$.  Then one has the following properties:

* The category of comodules over a [[coalgebra]] is [[comonadic]] over $Mod(k)$.
* It is [[cocomplete]] and [[complete category|complete]].
* It has a [[cogenerator]].

See [Wischnewsky](#Wischnewsky).

## Examples

* [[Hopf algebra]], [[Hopf algebroid]]

* The category of [[linear representations]] of an [[affine group]] is equivalent to the category of comodules over a [[Hopf algebra]].

## Related concepts

* [[dg-comodule]]

* [[comodule spectrum]]

## References

Properties of the category of comodules over a [[coalgebra]] are studied in

* {#Wischnewsky} Manfred Wischnewsky, _On linear representations of affine groups, I_, Pacific Journal of Mathematics, Vol. 61, No. 2, 1975, [Project Euclid](http://projecteuclid.org/euclid.pjm/1102868049).

On [[locally presentable category|local presentability]] of [[CocommCoalg|categoris of coalgebras]] and their comodules:

* [[Hans-Eberhard Porst]], *On corings and comodules*, Archivum Mathematicum **42** 4 (2006) 419-425 [[dmlcz:108017](https://dml.cz/handle/10338.dmlcz/108017), [pdf](http://dml.cz/bitstream/handle/10338.dmlcz/108017/ArchMathRetro_042-2006-4_7.pdf)]

[[!redirects comodules]]
[[!redirects co-module]]
[[!redirects co-modules]]
[[!redirects coaction]]
[[!redirects coactions]]
[[!redirects co-action]]
[[!redirects co-actions]]


comodule algebra

Given a [[bimonoid]] $B$ and a [[monoid]] $A$ in a [[symmetric monoidal category]] $C$, a right [[coaction]] $\rho: A\to A\otimes B$ is a __right Hopf coaction__ if $\rho$ is a morphism of monoids where $A\otimes B$ has the standard structure of a tensor product of monoids in the symmetric monoidal category $C$.


comodule spectrum

[[!redirects comodule spectra]]

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#### Higher algebra
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=--
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#Contents#
* table of contents
{:toc}

## Idea

Dual to the concept of a [[module spectrum]] over a [[ring spectrum]] is a _comodule spectrum_ over a [[coring spectrum]], the analog in [[stable homotopy theory]] of the concept of [[comodules]] in [[algebra]] and [[homological algebra]].

## Properties

### Over suspension spectra
 {#OverSuspensionSpectra}

The [[suspension spectrum]] $\Sigma^\infty X = \mathbb{S}[X]$ of any [[∞-groupoid]] ([[homotopy type]] of a  [[topological space]]) $X$ is canonically a [[coring spectrum]] by the fact that every $X$, is uniquely a [[coalgebra object]] in the [[Cartesian monoidal (∞,1)-category]] [[∞Grpd]] via the [[diagonal]] ([here](cartesian+monoidal+infinity%2C1-category#CoalgebraObjects)), and using that $\Sigma^\infty$ is a [[strong monoidal functor]].

If $X$ is [[connected object in an (∞,1)-topos]] (the homotopy type of a [[connected topological space]]) then $\mathbb{S}[X]$-comodule spectra are equivalently [[module spectra]] over the [[∞-group ∞-ring]] $\mathbb{S}[\Omega X]$ of the [[loop space]] [[∞-group]] of $X$.

$$
   CoModSpectra_{\mathbb{S}[X]}
  \;\simeq\;
   ModSpectra_{\mathbb{S}[\Omega X]}
$$

([Hess-Shipley 14, theorem 1.2 with prop. 5.18](#HessShipley14))

See also at _[[A-theory]]_.

## Related concepts

* [[dg-comodule]]

## References

* {#HessShipley14} [[Kathryn Hess]], [[Brooke Shipley]], _Waldhausen K-theory of spaces via comodules_, Advances in Mathematics 290 (2016): 1079-1137 ([arXiv:1402.4719](https://arxiv.org/abs/1402.4719))


comonad

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#### Algebra
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[[!include higher algebra - contents]]
=--
=--
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#Contents#
* table of contents
{:toc}

## Definition

A __comonad__ (or *cotriple*) on a [[category]] $A$ is a [[comonoid]] in the [[monoidal category]] of endofunctors $A \to A$. More generally, a comonad in a [[2-category]] $E$ is a [[comonoid]] in the [[monoidal category]] $E(X,X)$ for some [[object]] $X\in K$.

Just as a monad may be defined for any 2-category, $E$, as a [[lax 2-functor]] from $\mathbf{1}$ to $E$, so a comonad in $E$ is an [[oplax 2-functor]] from $\mathbf{1} \to E$.

See at _[[monad]]_ for more.

## Properties

### Coalgebras

A **[[coalgebra over a comonad]]** (or **comodule**) over a comonad $C$ on a category $A$ is an object $a\in A$ with a map $a\to C a$ satisfying dual axioms to those for an [[algebra over a monad]].  The category of coalgebras is called its (co-)[[Eilenberg-Moore category]] and satisfies a universal property dual to that of the [[Eilenberg-Moore object]] for a monad; it can thereby be internalized to any [[2-category]].  The [[forgetful functor]] from the category of coalgebras to the category $A$ is called a [[comonadic functor]].  Similarly, a comonad also has a [[co-Kleisli category]].

### Comonadic homology and descent

Any comonad on $A$ induces an augmented [[simplicial object|simplicial]] [[endofunctor]] of $A$ consisting of its iterates.  If $A$ is an [[abelian category]] and the comonad is [[additive monad|additive]], then this is the basis of [[comonadic homology]].  [[algebra for a monad|Comodules]] (= coalgebras) over the comonad with underlying endofunctor $M_R\mapsto M_R\otimes_R S$ in $R$-$Mod$ for the  extension of rings $R\hookrightarrow S$ correspond to the [[descent]] data for that extension. [[zoranskoda:gluing categories from localizations|Gluing of categories from localizations]] may also be formalized via comonads.

### Mixed distributive laws

[[distributive law|Distributive laws]] between a monad and a comonad are so-called _mixed_ distributive laws; a special case has been rediscovered in physics under the name _entwining structures_ (Brzeziński, Majid 1997). Their theory is often studied in the connection with the theory of comonads in the bicategory of rings, modules and morphisms of modules, that is [[coring]]s. There is a homomorphism of bicategories from a bicategory of entwinings to a bicategory of corings ([Škoda 2008](http://front.math.ucdavis.edu/0805.4611)), which is an analogue of the 2-functor $comp$ (R. Street, _Formal theory of monads_, JPAA 1972) of strict 2-categories in the case of distributive laws of monads (recall also that a distributive law among monads corresponds to a monad in the 2-category of monads).

## Examples

* [[necessity]]

* [[writer comonad]]

* [[store comonad]]

* [[flat modality]], [[infinitesimal flat modality]], [[reduction modality]], [[bosonic modality]]

* [[jet comonad]]

## Related concepts

* [[model structure on coalgebras over a comonad]]

## References

### General

* {#ApplegateTierney70} [[Harry Applegate]], [[Myles Tierney]], _Iterated cotriples_, Lecture Notes in Math. **137** (1970) 56-99 ([doi:10.1007/BFb0060440](https://doi.org/10.1007/BFb0060440))

Some introductory material on [[comonads]], [[coalgebra over a comonad|coalgebras]] and [[co-Kleisli morphisms]] can be found in

* [[Paolo Perrone]], _Notes on Category Theory with examples from basic mathematics_, Chapter 5. ([arXiv](http://arxiv.org/abs/1912.10642))

### As contexts in computer science

On [[comonads in computer science]]:

* {#UustaluVene08}  [[Tarmo Uustalu]], [[Varmo Vene]], *Comonadic Notions of Computation*, Electronic Notes in Theoretical Computer Science **203** 5 (2008) 263-284 [[doi:10.1016/j.entcs.2008.05.029](https://doi.org/10.1016/j.entcs.2008.05.029)]

* {#POM13} [[Tomas Petricek]], [[Dominic Orchard]], [[Alan Mycroft]], *Coeffects: Unified Static Analysis of Context-Dependence*, in: *Automata, Languages, and Programming. ICALP 2013*, Lecture Notes in Computer Science **7966** Springer (2013) [[doi:10.1007/978-3-642-39212-2_35](https://doi.org/10.1007/978-3-642-39212-2_35)]

* {#GKOBU16} Marco Gaboardi, [[Shin-ya Katsumata]], [[Dominic Orchard]], Flavien Breuvart, [[Tarmo Uustalu]], *Combining effects and coeffects via grading*, ICFP 2016: Proceedings of the 21st ACM SIGPLAN International Conference on Functional Programming (2016) 476–489 [[doi:10.1145/2951913.2951939](https://doi.org/10.1145/2951913.2951939), [talk abstract](https://icfp16.sigplan.org/details/icfp-2016-papers/31/Combining-Effects-and-Coeffects-via-Grading), [video rec](https://www.youtube.com/watch?v=l1ZNMT3fQCo)]

* {#KRU20} [[Shin-ya Katsumata]], Exequiel Rivas, [[Tarmo Uustalu]], LICS (2020) 604-618 *Interaction laws of monads and comonads* [[arXiv:1912.13477](https://arxiv.org/abs/1912.13477), [doi:10.1145/3373718.3394808](https://doi.org/10.1145/3373718.3394808)]

[[!redirects comonads]]
[[!redirects cotriple]]
[[!redirects cotriples]]


comonadic functor

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* table of contents
{:toc}

## Idea

The notion of a _comonadic functor_ is the dual of that of a [[monadic functor]]. See there for more background.

## Definition

Given a pair $L\dashv R$ of [[adjoint functors]], $L\colon A \to B\colon R$, with [[counit of an adjunction|counit]] $\epsilon$ and [[unit of an adjunction|unit]] $\eta$, one forms a [[comonad]] $\mathbf{\Omega} = (\Omega, \delta, \epsilon)$ by $\Omega \coloneqq L \circ R$, $\delta \coloneqq L \eta R$. $\mathbf{\Omega}$-[comodules](comonad#coalgebras) (aka $\mathbf{\Omega}$-coalgebras) form a category $B_{\mathbf{\Omega}}$ and there is a natural [[comparison functor]] $K = K_{\mathbf{\Omega}}\colon A \to B_{\mathbf{\Omega}}$ given by $A \mapsto (L A, L A \stackrel{L(\eta_A)}\to L R L A)$.

A functor $L\colon A\to B$ is __comonadic__ if it has a [[right adjoint]] $R$ and the corresponding [[comparison functor]] $K$ is an [[equivalence of categories]].  The adjunction $L \dashv R$ is said to be a **comonadic adjunction**.

## Properties

Beck's [[monadicity theorem]] has its dual, comonadic analogue. To discuss it, observe that for every $\Omega$-comodule $(N, \rho)$,

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manifestly exhibits a [[split equalizer]] sequence.

...

## Examples

### Comonadic adjoints to monadic functors

If $T: Set \to Set$ is a monad on $Set$, with corresponding monadic functor $U: Set^T \to Set$, then the left adjoint $F: Set \to Set^T$ is comonadic provided that the unique algebra map $!: F(0) \to F(1)$ is a [[regular monomorphism]] and not an [[isomorphism]]. In particular, if $T$ is given by an [[algebraic theory]] with at least one constant and a model with more than one element, then the left adjoint is comonadic (if the initial algebra $F(0)$ is nonempty, then $!: F(0) \to F(1)$ is a split monomorphism since any constant $c: 1 \to F(0)$ induces an algebra map $F(1) \to F(0)$ that retracts $!$; on the other hand there is more than one algebra map $F(1) \to M$ if $M$ has more than one element, so $F(1)$ isn't itself initial).

More generally, let $A$ be a [[locally small category]] with small [[copower|copowers]], and suppose $a$ is an object such that $0 \to a$ is a regular monomorphism but not an isomorphism, then the copowering with $a$,

$$- \cdot a: Set \to A$$

(left adjoint to $A(a, -): A \to Set$) is comonadic. See proposition 6.13 and related results in this [paper](http://www.tac.mta.ca/tac/volumes/16/1/16-01abs.html) by Mesablishvili.

### Necessity
 {#Necessity}

In the context of the treatment of [[modal operators]] given at _[Possible worlds via first-order logic and type theory](necessity+and+possibility#InFirstOrderLogicAndTypeTheory)_, we find that [[base change]] along the terminal morphism from the type of worlds, $W$, and its [[right adjoint]],  [[dependent product]], form a comonadic adjunction. $W^{\ast} \dashv \prod_W$.

Let $\mathbf{H}$ be a [[topos]]. For a [[type]]/[[object]], $P$, in $\mathbf{H} \simeq \mathbf{H}_{/\ast}$ we have that

$$
  W^{\ast} (P) = P \times W \to W
$$

([[projection]] on the second component). And for $[Q \to W]$ in $\mathbf{H}_{/W}$, we have

$$
  \prod_W (Q) = \Gamma_W(Q)
$$
([[space of sections]]).

Hence the composite, a kind of [[necessity]] operator

$$
  \Box_W \colon \mathbf{H}_{/W}\stackrel{\prod_W}{\longrightarrow} \mathbf{H} \stackrel{W^\ast}{\longrightarrow} \mathbf{H}_{/W}
$$

(an analog of the [[jet comonad]]), sends $Q \to W$ to $\Gamma_W(Q) \times W \to W$.

If $W$ is an [[inhabited object]] so that $W \to \ast$ is an [[epimorphism]], then $W^\ast$ is a [[conservative functor]]. Since it moreover preserves all [[limits]] (having a further [[left adjoint]], the [[dependent sum]] operation) the [[monadicity theorem]] applies and says that the coalgebras for this comonad are the types in $\mathbf{H}$, or rather the types in $\mathbf{H}_{/W}$ which are in the image of $W^{\ast}$, hence of the form $P \times W$, and equipped with the map

$$
  (const, id)
  \;\colon\;
  P \times W
   \longrightarrow
  \Box_W (P \times W) \simeq P^W \times W
$$
over $W$ ($P^W$ denotes the [[internal hom]]),  where the first component of the map sends $p: P$ to the constant map at $p$.

To see this as an [[equalizer]], consider the two maps from $\Box_W (P \times W \to W)$ to $\Box_W  \Box_W (P \times W \to W)$. Over $W$, these are formed from the two ways of mapping $P^W$ to $P^{(W \times W)}$, by sending $f$ in $P^W$ to $g \colon (w, w') \mapsto f(w)$ and to $h \colon (w, w') \mapsto f(w')$.

The equalizer of these two maps is formed from the $f$ for which $g = h$ for all $w, w'$ in $W$, that is, when $f$ is constant. So the equalizer is equivalent to  $W^{\ast} P = P \times W \to W$.

## Related concepts

* [[monadic functor]], **comonadic functor**

* [[monadicity theorem]]

[[!redirects comonadic functor]]
[[!redirects comonadic functors]]
[[!redirects comonadicity theorem]]
[[!redirects comonadic adjunction]]
[[!redirects comonadic adjunctions]]


comonoid

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#Contents#
* table of contents
{:toc}

## Definition

A **comonoid** (or __comonoid object__) in a [[monoidal category]] $\mathcal{M}$ is a [[monoid in a monoidal category|monoid object]] $C$ in the [[opposite category]] $\mathcal{M}^{op}$ (which canonically becomes a monoidal category via the same [[tensor product]] operation as in $\mathcal{M}$).

With the usual definition of [[monoids]] as having a [[unit]], this means that a comonoid $C$ is equipped with a *counit*, which in [[string diagram]]-notation for $\mathcal{M}$ is of this form:

\begin{tikzcd}[row sep=15pt]
  C
  \ar[d, phantom, "\multimap"{rotate=-90, scale=1.8}, "{\scalebox{.7}{co-unit}}"{xshift=16pt, yshift=6pt}]
  \\
  {}
\end{tikzcd}

One speaks of a **unital comonoid** (or __unital comonoid object__; [[internalization|internal to]] [[Vect]] also called an *[[augmented algebra|augmented]] [[coalgebra]]*) if $C$ is in addition equipped with a morphism $\eta \colon I \rightarrow C$

\begin{tikzcd}[row sep=15pt]
  {}
  \ar[d, phantom, "\multimap"{rotate=90,, scale=1.8}, "{\scalebox{.7}{unit}}"{xshift=10pt, yshift=-4pt}]
  \\
  C
\end{tikzcd}

which verifies the properties that the [[unit]] would verify if $C$ was a [[bimonoid]], ie. in [[string diagrams]]:

<img src="/nlab/files/unital-comonoid.png"/>

## Examples
 {#Examples}

\begin{example}
**(coassociative coalgebras)**
A comonoid object in [[VectorSpaces]] (with its usual [[tensor product of vector spaces]]) is called a *[[coalgebra]]*.

(Beware though that the term "coalgebra" is overused in many ways, for instance in *[[coalgebras for endofunctor]]*. To be more specific to the linear-algebraic context one can say *[[coassociative coalgebra]]*.)
\end{example}

\begin{example}\label{InACartesianMonoidalCategory}
**(cartesian comonoids)**
\linebreak
Every [[set]] carries a *unique* [[structure]] of a comonoid in the category of [[Sets]] with respect to the usual [[cartesian product]].

Generally, every object $X \,\in\, \mathcal{C}$ in a [[cartesian monoidal category]] $(\mathcal{C}, \ast, \times)$ becomes (see also [there](cartesian+monoidal+category#ComonoidObjects)) a ([[cocommutative coalgebra|cocommutative]]) comonoid by taking the

* counit to be the [[terminal object|terminal morphism]] $ X \xrightarrow{\exists !} \ast$

* coproduct to be the [[diagonal morphism]] $diag_X \,\colon\, X \to X \times X$.

The analogous statement remains true for [[cartesian monoidal (infinity,1)-categories]] (see [there](https://ncatlab.org/nlab/show/cartesian+monoidal+infinity,1-category#CoalgebraObjects)).
\end{example}

Obvious as Exp. \ref{InACartesianMonoidalCategory} may be, it plays a somewhat profound role in various contexts:

\begin{example}
**([[suspension spectra|suspension]] [[coring spectra]])**
\linebreak
In the case of [[topological spaces]] or other models of  classical [[homotopy types]], and using that the [[suspension spectrum]]-construction is a [[strong monoidal functor]], Exp. \ref{InACartesianMonoidalCategory} implies the remarkable fact that [[suspension spectra]] carry [[coring spectrum]]-[[structure]] via [smash-monoidal diagonals](suspension+spectrum#SmashMonoidalDiagonals).

\end{example}

## Related concepts

* [[coalgebra]]

* [[co-associativity]]

* [[co-unitality]]

* [[gs-monoidal category]]

[[!include reader-writer (co)monads -- table]]

[[!redirects comonoid]]
[[!redirects comonoids]]
[[!redirects comonoid object]]
[[!redirects comonoid objects]]

[[!redirects co-monoid]]
[[!redirects co-monoids]]


comorphism

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## Idea

Given two [[topological space]]s $X,Y$ and a [[continuous map]] $f \colon X\to Y$, or equivalently the [[functor]] $f^{-1} \colon \mathrm{Open}(X)\to\mathrm{Open}(Y)$, forming the morphism of [[site]]s $\mathrm{Open}(Y)\to\mathrm{Open}(X)$, one can introduce two kinds of "morphisms" between [[sheaf|sheaves]] on $X$ and on $Y$: _morphisms_ and _comorphisms_ of sheaves over different base spaces. Depending on a motivation one or another is more natural.

1. *Morphisms.* Recall that for a fixed base space, say $X$, the category of [[etale space]]s over $X$ is equivalent to the category of sheaves over $X$. Etale spaces make a subcategory of the slice category $\mathrm{Bun}_X=\mathrm{Top}/X$ of spaces ("bundles") over $X$. Denote by $p_F:E(F)\to X$ the etale map corresponding to a sheaf $F$. Thus for different bases we can simply define morphisms of sheaves $F\to G$ in etale space language, as continuous maps between the etale spaces $\xi: E(F)\to E(G)$ over $f$, i.e. $p_G\circ\xi=f\circ p_F$. These maps restrict to set theoretic maps on the level of [[stalk]]s.

2. *Comorphisms.* The idea of comorphism is the idea of cospaces, rather than spaces; namely we like to replace a space, like in [[noncommutative geometry]], by an object in a dually behaved category (say commutative rings; however it does not need to be strictly the dual category). For example, consider the ring of smooth functions on a [[manifold]]. A smooth morphism of manifolds $M\to N$ amounts to a set theoretic map ($f$ above) of underlying sets together with a compatible family of maps of algebras $k_V :C^\infty(V)\to C^\infty(f^{-1} V)$ where $V\subset N$ runs through all open subsets of $N$. Here of course $k_V : g\mapsto g\circ f|_{f^{-1}V}$.

## Definition

Let $F$ be a $R$-valued [[presheaf]] over a category $C$ and $G$ a $R$-valued presheaf over a category $D$. A __comorphism__ of $R$-valued presheaves $F\to G$ over a functor $f^{-1}:D\to C$ (thought of as a morphism $f$ of sites in the opposite direction) is a morphism $k$ of $R$-valued presheaves over $C$: $k : G\to f_* F = F\circ f^{-1}$. Instead of saying that $k$ is a co(homo)morphism over $f$ (or $f$-comorphism), one often says that comorphism of presheaves is a pair $(f,k)$. If the presheaf is the structure sheaf of a [[ringed site]] one calls the comorphism a _morphism_ of ringed sites; the usual notation is then $(f,f^\sharp)$.

[\begin{tikzcd}
	& {C^{op}} \\
	{D^{op}} && R
	\arrow["{f^{-1}}", from=2-1, to=1-2]
	\arrow["F", from=1-2, to=2-3]
	\arrow[""{name=0, anchor=center, inner sep=0}, "G"', from=2-1, to=2-3]
	\arrow["k"', shorten <=3pt, Rightarrow, from=0, to=1-2]
\end{tikzcd}](https://q.uiver.app/?q=WzAsMyxbMSwwLCJDXntvcH0iXSxbMiwxLCJSIl0sWzAsMSwiRF57b3B9Il0sWzIsMCwiZl57LTF9Il0sWzAsMSwiRiJdLFsyLDEsIkciLDJdLFs1LDAsImsiLDIseyJzaG9ydGVuIjp7InNvdXJjZSI6MjB9fV1d)

In the case of a topological space, and sheaves of sets (or algebras of some kind), there is another description in terms of stalks, namely comorphisms may be described as certain [[multivalued function|multi-valued maps]] between stalks. More precisely, given a continuous map $f:X\to Y$ and sheaves $F$ over $X$ and $G$ over $Y$, a comorphism $(f,f^\sharp)$ is a collection of maps $f^\sharp_x : G_{f(x)}\to F_{x}$ for all $x\in X$ such that for any [[section]] $s\in G(V)$ where $V^{\mathrm{open}}\subset Y$ the map $x\mapsto f^\sharp_x(s(f(x)))$ is a section of $F$ over $f^{-1}(V)$. We can view all $f^{\sharp}_x$ where $x\in f^{-1}(y)$ for a fixed $y$ as a single multi-valued map of stalks $G_y\to \coprod_{x\in f^{-1}(y)} F_x$ satisfying the continuity constraint mentioned.

##Remarks##

1. Notice that by [[adjoint functor|adjunction]], a morphism $G\to f_* F$ of sheaves of sets over $Y$ corresponds to a unique morphism $f^* G\to F$ of sheaves of sets over $X$.

2. In the case of sheaves of sets, the $G$-component of the unit of adjunction $\eta_G :G\to f_* f^* G$ is clearly a comorphism $G\to f^* G$ over $f$. Given any comorphism $(f,k):G\to F$ (i.e. $k:G\to f_* F$) we can decompose it as a composition
$$
G\stackrel{\eta_G}\to f^* G\stackrel{k\circ\eta_G^{-1}}\to F
$$
where the first map is a (canonical) comorphism over $f$ and the second map is a morphism of sheaves over $X$. Both maps may be viewed as multi-valued maps of etale spaces with the corresponding continuity conditions. In fact this is the unique decomposition of a given $f$-comorphism $k$ of sheaves of sets $G\to F$ into a compositon of an $f$-comorphism of sheaves of sets $G\to f^*G$ and a morphism of sheaves of sets over $X$. There is also a unique decomposition of $k$ as
$
G\to f_* F\to F,
$
where $G\to f_*F$ is a morphism and $f_* F\to F$ is a comorphism over $f$, namely
$$
G\stackrel{i^{-1}\circ k}\to f_* F\stackrel{i}\to F,
$$
where $i$ is the $f$-comorphism which is given by the identity $f_*F\to f_* F$. In fact, this can be seen as an intermediate step in the adjunction $f^*\dashv f_*$:
$$
\mathrm{Hom}_X(f^*G,F)\cong \mathrm{cohom}_f(G,F)\cong\mathrm{Hom}_Y(G,f_* F).
$$

## Related concepts

* [[ringed space]], [[ringed topos]]

* [[locally ringed space]]

## References

In the generality of [[ringed toposes]]:

* [[Alexander Grothendieck]], [[Jean-Louis Verdier]], Def. 13.1 of *Morphisme de topos annelés*, Section 13 in Exposé iv in: [[M. Artin]] et al., *Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos*, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 ([[SGA 4]]), Lecture Notes in Mathematics __269__, Springer (1972) [[doi:10.1007/BFb0081551](https://link.springer.com/book/10.1007/BFb0081551), [pdf](http://www.cmls.polytechnique.fr/perso/laszlo/sga4/SGA4-1/sga41.pdf)]

See also the references at *[[ringed space]]* and *[[ringed topos]]*.

[[!redirects comorphism]]

[[!redirects comorphisms]]


comoving time

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## Idea

In [[cosmology]] the [[radiation]] and [[matter]]-content of the [[observable universe]] may typically be approximated (in particular before any relevant [[structure formation]] has taken place and [[FRW models]] serve as a good approximation) by a [[dust]] or [[fluid]] which fills all of [[space]]. The [[proper time]] seen by a [[particle]] in this (possibly idealized or even hypothetical) [[dust]]/[[fluid]] is then called _comoving time_, i.e. the [[time]] as measured by an [[observer]] (i.e. along a [[lightlike]] curve in the [[spacetime]] [[pseudo-Riemannian manifold]]) whose motion is at rest with respect to the ambient dust/fluid.

Often the adjective "comoving" is suppressed when the context is assumed to be clear. For instance whenever any event in primordial cosmology (such as [[baryogenesis]], [[nucleosynthesis]], etc.) is said to occur at a given "time after the [[big bang]]", then what is meant is always comoving time.

## Related concepts

* [[redshift]]

* [[proper time]]

* [[physical unit]]

* [[scale]]

## References

See also

* Wikipedia, _[Comoving and proper distances](https://en.wikipedia.org/wiki/Comoving_and_proper_distances)_

[[!redirects comoving times]]

[[!redirects co-moving time]]
[[!redirects co-moving times]]


compact closed 2-category

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## Idea

A compact closed 2-category (also called an *autonomous* symmetric monoidal 2-category) is the (weak) [[2-category]]-analog of the notion of _[[compact closed category]]_.  That is, it is a [[symmetric monoidal 2-category]] in which all objects have [[dual object|duals]].

## Examples

1. For $V$ a cocomplete [[closed monoidal category|closed]] [[symmetric monoidal category]], the bicategory $V Prof$ of small $V$-[[enriched categories]] and $V$-enriched [[profunctors]] is compact closed.  ([Day-Street](#DayStreet))

1. For $C$ a [[2-category]] with weak [[finite limits]], the [[(infinity,n)-category of spans|2-category of spans]] in $C$ is compact closed. ([Stay 13](#Stay13))

## Internalization

In line with the [[microcosm principle]], internal to a compact closed 2-category one can define a notion of *compact closed [[bicategory of maps|map]] [[pseudomonoid]]*, which specializes in $V Prof$ (at least in the [[Cauchy complete category|Cauchy complete]] case) to the usual notion of [[compact closed category|compact closed]] $V$-category.  ([Day-Street](#DayStreet))  To deal with non-Cauchy complete categories, one can instead talk about compact closed [[proarrow equipments]].

## Related concepts

* [[monoidal 2-category]]
* [[compact closed category]]

## References

* [[Mike Stay]], _Compact Closed Bicategories_ ([arXiv:1301.1053](http://arxiv.org/abs/1301.1053))
 {#Stay13}

* [[Brian Day]] and [[Ross Street]], _Monoidal bicategories and Hopf algebroids_
 {#DayStreet}

[[!redirects compact closed 2-categories]]

[[!redirects compact closed bicategory]]
[[!redirects compact closed bicategories]]


compact closed category

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## Definition

A **compact closed category**, or simply a **compact category**, is a [[symmetric monoidal category]] in which every object is [[dualizable object|dualizable]], hence a [[rigid monoidal category|rigid]] [[symmetric monoidal category]].

More generally, if we drop the symmetry requirement, we obtain a [[rigid monoidal category]], a.k.a. an *autonomous category*.  Thus a compact category may also be called a **rigid symmetric monoidal category** or a **symmetric autonomous category**.  A maximally clear, but rather verbose, term would be a **symmetric monoidal category with duals for objects**.

A compact closed category is a special case of the notion of [[compact closed pseudomonoid]] in a [[monoidal bicategory]], and similarly for the autonomous cases.

## Properties

### Internal hom and compact closure

A rigid symmetric monoidal category $(\mathcal{C}, \otimes)$ is in particular a [[closed monoidal category]], with the [[internal hom]] given by

$$
  [A,B]
    \;\simeq\;
  B \otimes A^*
$$

(where $A^*$ is the [[dual object]] of $A$), via the [[adjunction]] [[natural equivalence]] that defines [[dual objects]]

$$
  \mathcal{C}\big(C,[A,B]\big)
  \simeq
  \mathcal{C}\big(C, B \otimes A^\ast\big)
  \simeq
  \mathcal{C}\big(C \otimes A, B\big)
  \,.
$$

This is what the terminology "compact closed" refers to.

The inclusion from the category of compact closed categories into the category of closed symmetric monoidal categories also has a [[left adjoint]] ([Day 1977](#Day1977)).  Given a closed symmetric monoidal category $\mathcal{S}$, the free compact closed category $C(\mathcal{S})$ over $\mathcal{S}$ may be described as a [[localization]] of $\mathcal{S}$ by the maps

$$ \sigma : [A,B] \otimes C \to [A, B \otimes C] $$

corresponding to the [[tensorial strength]] of the functors $[A,-] : \mathcal{S} \to \mathcal{S}$.

### Relation to traced monoidal categories
 {#RelationToTracedMonoidalCategories}

Given a [[traced monoidal category]] $\mathcal{C}$, there is a [[free construction]] completion of it to a compact closed category $Int(\mathcal{C})$ [[Joyal, Street & Verity 1996](#JoyalStreetVerity96)]:

the objects of $Int(\mathcal{C})$ are pairs $(A^+, A^-)$ of objects of $\mathcal{C}$, a morphism $(A^+ , A^-) \to (B^+ , B^-)$ in $Int(\mathcal{C})$ is given by a morphism of the form $A^+\otimes B^- \longrightarrow A^- \otimes B^+$ in $\mathcal{C}$, and [[composition]] of two such morphisms $(A^+ , A^-) \to (B^+ , B^-)$ and $(B^+ , B^-) \to (C^+ , C^-)$ is given by [[trace|tracing out]] $B^+$ and $B^-$ in the evident way.

Every compact closed category is self-dual, i.e. equivalent to its opposite.

### Relation to star-autonomous categories

A compact closed category is a [[star-autonomous category]]: the [[tensor unit]] is a [[dualizing object]]. Thus it is also an [[mix rule|isomix]] category. (But note that, for example, the symmetric monoidal category of [[sup-lattices]] is star-autonomous, with dualizing object given by the unit, but not compact closed. In a compact closed category, the dualizing functor is additionally monoidal.)

### Incompatibility with distributivity

\begin{theorem}
If a compact closed category has binary products that distribute over binary coproducts, it is [[thin category|thin]].
\end{theorem}
\begin{proof}
By Lemma 4 of [Houston 06](#Houston06), whose proof only requires binary products and coproducts, for any objects $A$ and $B$ the canonical morphism
$$(A\times A)+(B\times B) \to (A+B)\times (A+B)$$
is invertible, which we can write as
$$ A^2 + B^2 \to (A+B)^2.$$
This map factors through
$$ A^2 + B^2 + 2\cdot A\times B$$
via the coproduct injection and a pair of distributivity maps.  Since the latter are isomorphisms, so is the former.  This means that for any object $X$, if there exists a morphism $A^2+B^2 \to X$, then there exists a unique morphism $2\cdot A\times B \to X$.

Now taking $B=X=A$, we observe that there is a morphism $A^2+A^2 \to A$.  Therefore, there is a unique morphism $2\cdot A^2 \to A$, and therefore a unique morphism $A^2 \to A$.  In particular, the two projections $\pi_1 : A\times A\to A$ and $\pi_2 : A\times A\to A$ are equal, which is to say that $A$ is [[subterminal]].  Since $A$ was arbitrary, the category is thin.
\end{proof}

## Examples

\begin{example}
**(finite-dimensional vector spaces)**
\linebreak
The category [[FinDimVect]] of [[finite-dimensional vector spaces]] is compact closed with respect to the usual [[tensor product of vector spaces]], see [there](finite-dimensional+vector+space#CompactClosure).

  (It is not compact closed with the [[direct sum]] as monoidal product.)
\end{example}

\begin{example}
A compact closed [[discrete category]] is just an [[abelian group]].
\end{example}

\begin{example}
The [[delooping]] $\mathbf{B}M$ of a [[commutative monoid]] $M$ is a compact closed category, and conversely, any compact closed category (or more generally, any [[closed monoidal category]]) with a single object must be isomorphic to the delooping of some commutative monoid.
\end{example}

## Related concepts

* [[compact closed 2-category]]

* [[compact closed dagger category]]

* [[linear logic]]

## References

The characterization of the free compact closed category over a closed symmetric monoidal category is described in

* {#Day1977} [[Brian Day]], *Note on compact closed categories*, J. Austral. Math. Soc. (Series A) **24**  309-311 (1977) [[doi:10.1017/S1446788700020334](https://doi.org/10.1017/S1446788700020334)]

Discussion of [[coherence]] in compact closed categories is due to:

* {#KellyLaplaza80} [[Max Kelly]], [[M. L. Laplaza]], *Coherence for compact closed categories*, Journal of Pure and Applied Algebra, **19** 193-213 (1980) [doi:10.1016/0022-4049(80)90101-2, [pdf](https://core.ac.uk/download/pdf/82696829.pdf)]

On the relation to [[traced monoidal categories]]:

* {#JoyalStreetVerity96} [[André Joyal]], [[Ross Street]], [[Dominic Verity]], _Traced monoidal categories_, Math. Proc. Camb. Phil. Soc. **119** (1996) 447-468 [[pdf](http://sci-prew.inf.ua/v119/3/S0305004100074338.pdf), [doi:10.1017/S0305004100074338](https://doi.org/10.1017/S0305004100074338)]

See also:

* Wikipedia, _[Compact closed category](http://en.wikipedia.org/wiki/Compact_closed_category)_

On the relation to [[quantum operations]] and [[completely positive maps]]:

* [[Peter Selinger]], *Dagger compact closed categories and completely positive maps*, Electronic Notes in Theoretical Computer Science **170** (2007) 139-163, [[doi:10.1016/j.entcs.2006.12.018](https://doi.org/10.1016/j.entcs.2006.12.018)]

On [[biproducts]]:

* [[Robin Houston]], *Finite products are biproducts in a compact closed category*, Journal of Pure and Applied Algebra **212** 2 (2008) 394-400 [[arXiv:math/0604542](https://arxiv.org/abs/math/0604542), [doi:10.1016/j.jpaa.2007.05.021](https://doi.org/10.1016/j.jpaa.2007.05.021)]

On compact closure in [[homotopical algebra]] and relating to the [Barrat-Priddy theorem](stable+cohomotopy#AsAlgebraicKTheoryOverTheFieldWithOneElement):

* [[Amit Sharma]], *Compact closed categories and Γ-categories (with an appendix by [[André Joyal]])*, Theory and Applications of Categories **37** 37 (2021) 1222-1261 [[arXiv:2010.09216](https://arxiv.org/abs/2010.09216), [tac:37-37](http://www.tac.mta.ca/tac/volumes/37/37/37-37abs.html)]

[[!redirects compact closed]]
[[!redirects compact closed categories]]
[[!redirects compact category]]
[[!redirects compact categories]]
[[!redirects symmetric monoidal category with duals for objects]]
[[!redirects symmetric monoidal categories with duals for objects]]
[[!redirects symmetric monoidal category with duals]]
[[!redirects symmetric monoidal categories with duals]]
[[!redirects compact closed monoidal category]]


compact closed dagger category

[[!redirects compact closed dagger categories]]

#Contents#
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## Definition ##
A **compact closed dagger category** is a [[symmetric monoidal dagger category]] $C$ with

* an object $A^*$ called the **dual** for every object $A:Ob(C)$

* for every object $A:Ob(C)$, a morphism $\iota^R_A: \Iota \to A \otimes A^*$ called the **right unit**

* for every object $A:Ob(C)$, a morphism $\iota^L_A: \Iota \to A^* \otimes A$ called the **left unit**

* for every object $A:Ob(C)$, a morphism $\eta^L_A: A \otimes A^* \to \Iota$ called the **left counit**

* for every object $A:Ob(C)$, a morphism $\eta^R_A: A^* \otimes A \to \Iota$ called the **right counit**

such that

* for all objects $A:Ob(C)$, ${\iota^R_A}^\dagger = \eta^L_A$

* for all objects $A:Ob(C)$, ${\iota^L_A}^\dagger = \eta^R_A$

* for all objects $A:Ob(C)$, $\iota^L_{A}A \circ A\eta^L_{A}$

* for all objects $A:Ob(C)$, $\iota^R_{A^*}A^* \circ A\eta^R_{A^*}$

* for all objects $A:Ob(C)$, $A\iota^R_{A} \circ \eta^R_{A}A$

* for all objects $A:Ob(C)$, $A^*\iota^R_{A^*} \circ \eta^R_{A^*}A^*$

## Examples ##

* The [[dagger category]] [[Hilb]] of [[Hilbert spaces]] and [[continuous map|continuous]] [[linear maps]] is a compact closed dagger category.

## See also ##

* [[dagger category]]

* [[monoidal dagger category]]

* [[symmetric monoidal dagger category]]

* [[compact closed category]]

## References ##

* [[Chris Heunen]], [[Andre Kornell]], *Axioms for the category of Hilbert spaces* ([arXiv:2109.07418](https://arxiv.org/abs/2109.07418))


compact connected space

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## Definition ##

By a *compact connected space* one will typically mean a [[topological space]] which is both [[compact topological space|compact]] and [[connected topological space|connected]].

Beware that in [[constructive mathematics]] there is a definition of "compact connectedness" which is really just one version of "connectedness": see at *[[connected topological space]]* the section *[Constructive definition](connected+space#ConstructiveDefinition)*.

## References

See the references at *[[topology]]*.

[[!redirects compact connected spaces]]


compact double category

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## Idea

A **compact double category** is a (usually pseudo) [[double category]] which is [[monoidal double category|symmetric monoidal]], and in which every object has an assigned [[dualizable object|dual]] in the "proarrow" direction (as in a [[compact closed category]]).  The basic example is the double category $\underline{Cat}$ of categories, functors, and [[profunctors]], in which the dual of a category is its [[opposite category]].

On this page, we will draw the "functor" direction horizontally and the "profunctor" direction vertically.  Thus, in a compact double category, every object has a vertical dual.

Note that if a compact double category is a [[proarrow equipment]], then its [[vertical bicategory]] is a [[symmetric monoidal bicategory]], which is moreover an [[autonomous bicategory]].  However, saying that a double category is compact says more than this: it says that the vertical duals are assigned in a horizontally functorial way.  Just as duals in a monoidal category are characterized up to isomorphism, duals in an autonomous bicategory are characterized up to equivalence, which in this case would mean *vertical* equivalence.  But equivalence in the bicategory $Prof$ of categories and profunctors is a weaker notion than [[equivalence of categories]], so just saying that $C^{op}$ is a dual of $C$ in $Prof$ does not characterize $C^{op}$ up to equivalence, only up to [[Morita equivalence]] (i.e. equivalence of [[Cauchy completion]]s).  The extra structure (not merely extra properties) of the compact double category $\underline{Cat}$ includes the specification of $C^{op}$ up to equivalence (i.e., horizontal equivalence), along with all the attendant structure.

Likewise, a **compact virtual double category** is a [[virtual double category]] which is compact in a similar sense: it has a monoidal structure and functorially assigned duals.  Note that in order to match with our conventions on this page, the role of vertical and horizontal cells in a virtual double category is flipped from the choices made at [[virtual double category]], and the cells must be transposed and have their multi-sources on the left, rather than the top.

By contrast, in a **virtually compact virtual double category**, the monoidal and compact structure has also been "virtualized": rather than coming equipped with a horizontally functorial tensor product, in such a virtual double category there are also horizontal arrows with multi-sources that are finite lists of objects with variance, such as $(A,B^{op},C)$.  We can also have a **virtually compact double category** which is an honest double category, but whose monoidal and compact structure is only virtual.

Finally, if a compact double category or virtual double category is a [[proarrow equipment]] or a [[virtual equipment]], we of course call it a **compact (virtual) proarrow equipment**.

## Definition

Let $T$ be the "free category" monad on the [[virtual equipment]] $Span(Quiv)$ of [[spans]] in the category of [[quivers]], and let $T' = Mod(T)$ be the resulting monad on the virtual equipment $Mod(Span(Quiv)) = Cat(Quiv)$ whose objects are internal quivers in $Cat$, or equivalently internal categories in $Quiv$.  Then a (pseudo) $T'$-algebra is precisely a (pseudo) double category, while a [[virtual T-algebra]] is precisely a virtual double category.  In particular, the virtual equipment $Vdc = nKMod(Cat(Quiv),T')$ consists of virtual double categories, functors, and profunctors between them.

Let $G$ be the [[pseudomonad]] on $Cat(Quiv)$ defined as follows.  We may consider an object of $Cat(Quiv)$ as like a double category, but with only horizontal composition: there are no vertical composites or identities.  If $C$ is such, then:

* The objects of $G(C)$ are finite ordered lists of objects of $C$ with variance, such as $(x,y^{op},z,x^{op})$.  Of course, we include the empty list.

* There can exist a horizontal arrow of $G(C)$ from one list $\vec{x}$ to another $\vec{y}$ only if the two lists have the same length, say $n$.  In this case, such an arrow is given by a permutation $\sigma \in S_n$ such that the variance of $x_i$ matches the variance of $y_{\sigma (i)}$, together with for each $i$ a horizontal arrow $x_i \to y_{\sigma(i)}$ in $C$.  Composition of these is defined in the evident way using composition in $C$ and multiplying permutations.

* A vertical arrow from $\vec{x}$ to $\vec{y}$ in $G(C)$ is given by a "graph" from $\vec{x}$ to $\vec{y}$ labeled by vertical arrows of $C$, together with an ordered list of endo-vertical-arrows of $C$ (called "loops").  To be precise, by such a "graph" we mean a fixed-point-free involution of $\vec{x} \sqcup \vec{y}^{op}$ which reverses variance.  Here $\vec{y}^{op}$ means reverse the variance all through $\vec{y}$; thus if $x_i$ is matched with $y_j$, they must have the *same* variance, while if $x_i$ is matched with $x_j$, or $y_i$ with $y_j$, they must have opposite variances.

  To define a vertical arrow in $G(C)$, together with such a graph we also require, for each matched pair, a vertical arrow in $C$, according to the following rules.  If $x_i$ is matched with $y_j$ and neither is "opped," then we require a vertical arrow $x_i\to y_j$, while if both are "opped," we require instead a vertical arrow $y_j\to x_i$.  And if $x_i$ is matched with $x_j^{op}$, we require a vertical arrow $x_i\to x_j$, while if $y_i$ is matched with $y_j^{op}$, we require a vertical arrow $y_j\to y_i$.  (And, in addition to all this, we also require an ordered list of loops.)

* The squares in $G(C)$ are defined in a straightforward way, incorporating two graphs which are related by a pair of permutations, and a collection of labeling squares from $C$ with appropriately chosen boundaries.

With this definition, $G$ is evidently an endofunctor of the category of $Cat$-quivers.  We can extend it to the virtual equipment $Cat(Quiv)$ in a straightforward way, mimicking the definition above for horizontal arrows and squares to define it on the proarrows (which are like [[double profunctors]] but, again, without vertical composites).

Note that the action of $G$ on vertical arrows is very much like that of the monad on $Cat$ whose algebras are [[compact closed categories]].  The main difference is that in the latter, rather than an *ordered* list of loops (an element of the [[free monoid]] on endomorphims), we have an element of the free *commutative* monoid on the endomorphisms.  This change is what will make $G$ be only a pseudomonad, rather than a monad, but it is also crucial for the applications.

We next extend $G$ to a pseudomonad in the [[Gray-category]] of [[virtual equipments]].  The unit $C\to G(C)$ is easy to define and strictly natural: an object $x$ goes to the unary list $(x)$, horizontal arrows are labeled with the unique permutation of one element, and vertical arrows are labeled with the unique graph between two such unary lists.

The multiplication is somewhat trickier....

The multiplication defined in this way is still strictly natural, and it satisfies the laws relating it to the unit transformation strictly, but its "associativity" law is only satisfied up to isomorphism, making $G$ into a fairly strict sort of pseudomonad.

We now claim that there is a [[distributive law]] relating $T'$ to $G$, and therefore $G$ has the structure of a pseudomonad on the object $(Cat(Quiv),T')$ in the category of monads-on-virtual-equipments....

It follows by the functoriality of the construction of [[generalized multicategories]] that $G$ induces a pseudomonad $G' = nKMod(G,T')$  on the virtual equipment $Vdc = nKMod(Cat(Quiv),T')$ of virtual double categories.  Moreover, we can verify that $G'$ preserves pseudo double categories, and induces a monad $G''$ on the virtual equipment of double categories and [[double profunctors]].  Finally, we can define:

* A **compact double category** is a pseudo $G''$-algebra.  By general nonsense about distributive laws, this should be the same as a pseudo $G T'$-algebra, where $G T'$ is the composite pseudomonad on $Cat(Quiv)$ resulting from the distributive law.

* A **compact virtual double category** is a pseudo $G'$-algebra.

* A **virtually compact virtual double category** is a virtual $G T'$-algebra (in $Cat(Quiv)$).  I *don't* think this is quite the same as a virtual $G'$-algebra in $Vdc$.

* A **virtually compact double category** is a virtual $G''$-algebra.

In each of the above cases, we can replace "double category" by [[proarrow equipment]] if the (virtual) double category in question is additional a ([[virtual equipment|virtual]]) equipment.

## Examples

The basic example is $\underline{Cat}$, in which the objects are categories, the horizontal arrows are functors, and the vertical arrows are profunctors.  This is a compact proarrow equipment.  There are similar examples $V \underline{Prof}$ for any Benabou [[cosmos]] $V$.  In fact, as long as $V$ is any symmetric [[multicategory]], we can define a virtually compact virtual equipment $V \underline{Prof}$.

## Extraordinary 2-cells

Let $C$ be a virtually compact virtual double category with units (it could be an equipment).  Note that the source of a general 2-cell in $C$ is a graph whose edges are labeled by composable strings of vertical arrows in $C$.  Suppose also that $f\colon \vec{x} \to z$ and $g\colon \vec{y}\to z$ are horizontal arrows in $C$.  An **extraordinary 2-cell** in $C$ is defined to be a 2-cell whose target is the unit/identity $U_z$, and whose source is a loop-free graph whose edges are all labeled by empty strings (or, equivalently, by identities).  One can verify that in $\underline{Cat}$, this reproduces the usual notion of [[extraordinary natural transformation]].

## Structures in a compact double category

In a (possibly virtual) compact double category, we can define internal notions of "closed category," "closed monoidal category," and so on.

...

## Representability

...

## References

* [Blog post](http://golem.ph.utexas.edu/category/2010/03/extraordinary_2multicategories.html) about extraordinary 2-multicategories and their ilk.

[[!redirects compact virtual double category]]
[[!redirects virtually compact double category]]
[[!redirects virtually compact virtual double category]]
[[!redirects compact double categories]]
[[!redirects compact virtual double categories]]
[[!redirects virtually compact double categories]]
[[!redirects virtually compact virtual double categories]]
[[!redirects compact closed double category]]
[[!redirects compact closed double categories]]
[[!redirects compact equipment]]
[[!redirects compact proarrow equipment]]
[[!redirects compact virtual equipment]]
[[!redirects virtually compact equipment]]
[[!redirects virtually compact proarrow equipment]]
[[!redirects virtually compact virtual equipment]]

[[!redirects autonomous double category]]
[[!redirects autonomous virtual double category]]
[[!redirects virtually autonomous double category]]
[[!redirects virtually autonomous virtual double category]]
[[!redirects autonomous double categories]]
[[!redirects autonomous virtual double categories]]
[[!redirects virtually autonomous double categories]]
[[!redirects virtually autonomous virtual double categories]]
[[!redirects autonomous equipment]]
[[!redirects autonomous proarrow equipment]]
[[!redirects autonomous virtual equipment]]
[[!redirects virtually autonomous equipment]]
[[!redirects virtually autonomous proarrow equipment]]
[[!redirects virtually autonomous virtual equipment]]


compact element

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(0,1)$-Category theory
+-- {: .hide}
[[!include (0,1)-category theory - contents]]
=--
#### Compact objects
+-- {: .hide}
[[!include compact object - contents]]
=--
=--
=--

# Compact/finite elements
* table of contents
{: toc}

## Definition

Let $P$ be a [[poset]] such that every [[directed set|directed]] [[subset]] of $P$ has a [[join]]; that is, $P$ is a [[dcpo]]. A __compact element__, or __finite element__, of $P$ is a [[compact object]] in $P$ regarded as a [[thin category]]; that is, homs out of it commute with these [[directed joins]].

In other words, $c \in P$ is compact precisely if for every directed subset $\{d_i\}$ of $P$ we have
$$
  (c \leq \bigvee_i d_i )
  \Leftrightarrow
  \exists_i (c \leq d_i)
  \,.
$$
Of course, the $\Leftarrow$ part of this is automatic, so the real condition is the $\Rightarrow$ part.  In more elementary terms:

*  If $c \leq \bigvee D$ for $D$ a directed subset, then $c \leq d$ for some $d \in D$.

In the case where $P$ has a [[top element]] $1$, we say that $P$ is *compact* if $1$ is a compact element.

## Examples

*  Given a [[set]] $X$, the finite elements of its [[power set]] are precisely the (Kuratowski)-[[finite set|finite]] [[subsets]] of $X$.  (This is the origin of the term 'finite element'.)

*  Given a [[topological space]] (or [[locale]]) $X$, the compact elements of its [[frame of open subspaces]] are precisely the [[compact space|compact]] [[open subspaces]] of $X$.  (This is the origin of the term 'compact element'.)

* If $R$ is a (not necessarily commutative) [[ring]], the lattice $Idl(R)$ of two-sided [[ideals]] of $R$ is compact. Indeed, the top element is the ideal generated by $1$, the multiplicative identity, and $1 \in \bigvee_i I_i$ implies $1 \in I_i$ for some index $i$.

[[!redirects compact element]]
[[!redirects compact elements]]
[[!redirects finite element]]
[[!redirects finite elements]]


compact element in a lattice

## Idea

This notion is a variant of a notion of a compact space (every cover has a finite subcover) and more specifically a special case of [[compact element]] for orders.
However this entry is written in parallel and with a view toward another generalization -- compact elements in [[quantale]]s.

## Definition

Let $L$ be a [[complete lattice]]. An element $c\in L$ is __compact__ iff any of the following equivalent conditions hold:

(i) for every $S\subset L$, such that $c\leq sup(S)$, there is a finite subset $F\subset S$ such that $c\leq sup(F)$

(ii) for every [[directed set|directed subset]] $S\subset L$ such that $c\leq S$
there is $s\in S$ with $c\leq s$.

## Frames with properties concerning compact elements

A [[frame]] $L$ is __algebraic__ if every element of $L$ is the sup of some set of compact elements. (Cf. also entry [[algebraic lattice]]).
In that case, one can take the set of all elements below or equal the given element.  The latter definition is accepted also for [[quantale]]s.

$L$ is __coherent__ if it is algebraic and the finite inf of a set of compact elements
is compact.

__Theorem.__ A frame $L$ is coherent iff $L$ is isomorphic to the frame of ideals in some [[distributive lattice]].

## Literature

A [[quantale]] is __precoherent__ if it is algebraic and binary products of compact elements are compact. Precoherent quantale is __coherent__ if the truth is a compact element.

* [[Kimmo Rosenthal|K. I. Rosenthal]], _Quantales and their applications_, Longman 1990

* [[Peter Johnstone]], [[Stone spaces]]

[[!redirects coherent frame]]
[[!redirects algebraic frame]]
[[!redirects coherent quantale]]
[[!redirects compact element in a complete lattice]]


compact Hausdorff rings are profinite

# Contents
* table of contents
{:toc}

## Introduction

The goal of this article is to prove that a [[compactum|compact Hausdorff]] [[ring]] $R$ is a [[profinite space|profinite]] ring, i.e., is canonically isomorphic as a [[topological ring]] to its [[profinite completion]] $\widehat{R}$, topologized as an inverse limit of [[finite set|finite]] rings with their [[discrete topology|discrete topologies]].

Further commentary on this result can be found in a Café [blog post](#Lein) of Tom Leinster, in connection with some work by his student Barry Devlin.

## Preliminaries

All topological rings considered are $T_0$ (see [[separation axiom]]). As is well-known from the study of [[uniform structures]], this implies that all our rings are [[Hausdorff space|Hausdorff]] and even [[Tychonoff spaces]]. In particular, finite rings are Hausdorff and therefore discrete. If $S$ is a finite ring and $f: R \to S$ is a [[continuous map|continuous]] ring homomorphism, this means the [[kernel]] of $f$ must be an [[open set|open]] [[ideal]] of $R$ (for us, "ideal" always means two-sided ideal), since $\{0\}$ is open in $S$.

In the converse direction, if $R$ is a compact ring and $I$ is an open ideal, then the ring $R/I$ with the [[quotient topology]] coming from $R$ is discrete (every point is open), and compact since $R$ is compact, and so $R/I$ must then be finite.

Let $\mathcal{O}(R)$ denote the [[poset]] of open ideals of $R$, ordered by [[subset|inclusion]]. This poset is [[directed set|codirected]] by taking finite [[intersections]] of ideals. Each inclusion $I \subseteq J$ induces a [[surjection]] of finite rings $R/I \to R/J$, and so we get a [[functor]] $\mathcal{O}(R) \to Ring$ whose (projective) [[limit]] is the profinite completion $\widehat{R}$ of $R$; the quotient maps $proj_I: R \to R/I$ of course induce a canonical map $\pi: R \to \widehat{R}$.

+-- {: .num_prop #surj}
###### Proposition
The map $\pi: R \to \widehat{R}$ is surjective.
=--

+-- {: .proof}
###### Proof
Since $R$ is compact and $\widehat{R}$ is Hausdorff, the image $\pi(R)$ is closed in $\widehat{R}$, so it suffices to show the image is a [[dense subspace]].

So, we are to show that if $U \subseteq \widehat{R}$ is open and [[inhabited set|inhabited]], then there exists $x \in R$ with $\pi(x) \in U$. Let $\pi_I: \widehat{R} \to R/I$ denote a typical component map of the limit cone. A [[basis of a topology|basis]] for the topology of $\widehat{R}$ consists of inhabited finite intersections of the form $U = \pi_{I_1}^{-1}(x_1) \cap \ldots \cap \pi_{I_n}^{-1}(x_n)$ where each $x_k$ is a point of $R/I_k$, so WLOG assume $U$ is of this form.

Let $y = \langle y_I \rangle_{I \in \mathcal{O}(R)} \in \widehat{R}$ be any point belonging to $U$ (so $y_{I_i} = x_i$ for $i = 1, \ldots, n$), and put $J = \bigcap_{i=1}^n I_i$. Then for each $x_i$ the coordinate $y_J$ maps to $x_i$ under the map $R/J \to R/I_i$, precisely because of the compatibility conditions on the coordinates imposed by the limit. Then, letting $y' \in R$ be an element that maps to $y_J$ under $R \to R/J$, the element $\pi(y')$ lies in $U$ and we are done.
=--

## Compact rings are totally disconnected

A key technical result is the following (see this $n$-Category Café [comment](#Trim)):

+-- {: .num_theorem #totdisc}
###### Theorem
Every compact Hausdorff ring $R$ is [[connected space|totally disconnected]].
=--

+-- {: .proof}
###### Proof
Let $R^\ast$ be the [[Pontryagin duality|Pontryagin dual]] of (the additive group of) $R$; as a space $R^\ast$ is discrete. As $R$ and $R^\ast$ are [[locally compact Hausdorff space|locally compact Hausdorff]], they are [[exponentiable space|exponentiable]] as spaces, and it follows quickly that the functorial maps

$$Hom(R, R) \to Hom(R^\ast, R^\ast), \qquad Hom(R^\ast, R^\ast) \to Hom(R^{\ast\ast}, R^{\ast\ast}) \cong Hom(R, R)$$

are continuous homomorphisms of topological groups. Using naturality of the isomorphism $R \cong R^{\ast\ast}$, these maps are mutually inverse, and so they give an [[isomorphism]] of topological groups.

The multiplication map $R \times R \to R$ [[currying|transforms]] to a continuous injection $i: R \to Hom(R, R) \cong Hom(R^\ast, R^\ast)$. As $R$ is compact and $Hom(R^\ast, R^\ast)$ is Hausdorff, the injection $i$ maps $R$ homeomorphically onto its image in $Hom(R^\ast, R^\ast)$, i.e., is a subspace embedding. But $Hom(R^\ast, R^\ast)$ is manifestly a subspace of a product of discrete spaces and hence totally disconnected, so $R$ is totally disconnected as well. (In more detail: if $C \subseteq R$ is connected, then the image of $C$ under the composite

$$C \subseteq R \stackrel{i}{\to} Hom(R^\ast, R^\ast) \hookrightarrow \prod_{x \in R^\ast} R^\ast \stackrel{proj_x}{\;\;\to\;\;} R^\ast$$

is also connected and hence a one-point space for each $x \in R^\ast$, so that $i(C)$ and therefore $C$ are one-point spaces.)
=--

## Compact rings have enough open ideals

By Theorem \ref{totdisc}, Pontryagin duality implies compact rings are totally disconnected, which readies us for a next wave of results.

+-- {: .num_lemma #compopen}
###### Lemma
Let $X$ be a compact Hausdorff totally disconnected space. For each $x \in X$ and open neighborhood $V$ of $x$, there exists a compact open set $U$ such that $x \in U \subseteq V$.
=--

+-- {: .proof}
###### Proof
In a compact Hausdorff space $X$, the [connected component](/nlab/show/connected+space#connected_components) of a point $x$ equals its quasi-component; an online proof may be found [here](http://math.stackexchange.com/a/11423/43208). So if $X$ is also totally disconnected, then $x$ is also the intersection of all [[clopen set|clopens]] containing it. Now let $V$ be an open [[neighborhood]] of $x$, so $\bigcap_{x \in K\; clopen} K \subseteq V$ and therefore $\neg V \subseteq \bigcup_{x \in K\; clopen} \neg K$. As $\neg V$ is compact, finitely many such clopens $\neg K_1, \ldots, \neg K_n$ cover $\neg V$, and so $U \coloneqq \bigcap_{i=1}^n K_n \subseteq V$. This $U$ is compact and open and contains $x$, which completes the proof.
=--

+-- {: .num_lemma #opengroup}
###### Lemma
Let $A$ be a compact Hausdorff totally disconnected abelian group. For each open neighborhood $V$ of $0 \in A$, there exists an open subgroup $O$ such that $O \subseteq V$.
=--

+-- {: .proof}
###### Proof
By Lemma \ref{compopen}, there is a compact open $W$ with $0 \in W \subseteq V$. Now let $\alpha: W \times W \to A$ be the restriction of the addition operation $+: A \times A \to A$. Since $\alpha$ is continuous, there is for each $x \in W$ an open neighborhood $V_x \times U_x \subseteq \alpha^{-1}(W)$ of $(x, 0)$ where we may assume $U_x$ is *symmetric* (i.e., $U_x = -U_x$). Finitely many of the $V_x$ cover $W$, say $V_{x_1}, \ldots, V_{x_n}$. Then putting $U = U_{x_1} \cap \ldots \cap U_{x_n}$, we have that $\alpha(W \times U) \subseteq W$, or $W + U \subseteq W$ for short. This $U$ is again symmetric: $U = -U$.

Hence if $U^n = U + \ldots + U$ denotes the set of all sums of $n$ elements belonging to $U$, then $U^n$ is open and $W + U^n \subseteq W$ by induction, whence $U^n \subseteq W$. Since $U$ is symmetric, the union $O = \bigcup_n U^n$ is the subgroup generated by $U$; it is open and contained in $W$ and therefore also in $V$, which completes the proof.
=--

+-- {: .num_lemma #openideal}
###### Lemma
Let $R$ be a compact Hausdorff totally disconnected ring. For each open neighborhood $V$ of $0$, there is an open ideal $I$ such that $I \subseteq V$.
=--

+-- {: .proof}
###### Proof
By Lemma \ref{opengroup}, there is an open additive subgroup $O \subseteq V$. Let $I = \{x \in R: R x R \subseteq O\}$. It is evident that $0 \in I$ and $I$ is an ideal of $R$. Now hold $x \in I$ fixed. For each $(a, b) \in R \times R$, continuity of multiplication implies there are open neighborhoods $W_{(a, b)}$ of $a$, $W_{(a, b)}'$ of $b$, and $V_{(a, b)}$ of $x$ such that $W_{(a, b)} V_{(a, b)} W_{(a, b)}' \subseteq O$. By compactness of $R \times R$, finitely many sets of the form $W_{(a, b)} \times W_{(a, b)}'$ cover $R \times R$. The corresponding finite intersection $\bigcap_{i=1}^n V_{(a_i, b_i)}$ is an open neighborhood of $x$ that is contained in $I$, thus completing the proof.
=--

## Compact rings are profinite

Now for our final result.

+-- {: .num_theorem}
###### Theorem
For $R$ a compact ring, the canonical map $\pi: R \to \widehat{R}$ is [[monomorphism|injective]].
=--

This and Proposition \ref{surj} taken together imply that $\pi$ is a ring isomorphism. It is also a [[homeomorphism]] because it is a continuous [[bijection|bijective]] map from a compact space to a Hausdorff space. Therefore $\pi$ is an isomorphism of topological rings.

+-- {: .proof}
###### Proof
As we have an inclusion $\widehat{R} \hookrightarrow \prod_{I \in \mathcal{O}(R)} R/I$ of the profinite completion as projective limit, we see the kernel of $\pi$ is the same as the kernel of

$$\langle proj_I \rangle_{I \in \mathcal{O}(R)}: R \to \prod_{I \in \mathcal{O}(R)} R/I$$

which is precisely $\bigcap_{I \in \mathcal{O}(R)} I$. Since by Lemma \ref{openideal} every neighborhood of $0 \in R$ contains an open ideal $I \in \mathcal{O}(R)$, this intersection is contained in the intersection of all open neighborhoods of $0$, which is $\{0\}$ since $R$ is Hausdorff. Thus the kernel of $\pi$ is trivial, as was to be shown.
=--

## Related concepts

* [[profinite group]]

## References

* [[Tom Leinster]], _Holy Crap, Do You Know What A Compact Ring Is?_, blog post, $n$-Category Café (August 20, 2014). ([link](https://golem.ph.utexas.edu/category/2014/08/holy_crap_do_you_know_what_a_c.html))
 {#Lein}

* [[Todd Trimble]], comment on _Holy Crap, Do You Know What A Compact Ring Is?_, blog post, $n$-Category Café (August 24, 2014). ([link](https://golem.ph.utexas.edu/category/2014/08/holy_crap_do_you_know_what_a_c.html#c047108))
 {#Trim}

* L. Ribes and P. Zalesskii, _Profinite groups_, volume 40 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge (2000), Springer-Verlag,
Berlin.


compact Hausdorff spaces are normal

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

#Contents#
* table of contents
{:toc}

## Statement

+-- {: .num_defn #CompactHausdorffSpacesAreNormal}
###### Proposition

Every [[compact Hausdorff topological space]] is a [[normal topological space]].

=--

In fact a stronger statement holds: [[paracompact Hausdorff spaces are normal]].

To prove this, consider the following lemma:

+-- {: .num_lemma #SeparationByNeighbourhoodsOfPointsFromCompactSubsetsInHausdorffSpaces}
###### Lemma
**(separation by neighbourhoods of points from compact subspaces in Hausdorff spaces)**

Let

1. $(X,\tau)$ be a [[Hausdorff topological space]];

1. $Y \subset X$ a [[compact topological space|compact]] [[subspace]].

Then for every $x \in X \backslash Y$ there exists

1. an [[open neighbourhood]] $U_x \supset \{x\}$;

1. an open neighbourhood $U_Y \supset Y$

such that

* they are still disjoint: $U_x \cap U_Y = \emptyset$.

=--

+-- {: .proof}
###### Proof
of lemma \ref{SeparationByNeighbourhoodsOfPointsFromCompactSubsetsInHausdorffSpaces}

By the assumption that $(X,\tau)$ is Hausdorff, we find for every point $y \in Y$ disjoint open neighbourhoods $U_{x,y} \supset \{x\}$ and $U_y \supset \{y\}$. By the nature of the [[subspace topology]] of $Y$, the restriction of all the $U_y$ to $Y$ is an [[open cover]] of $Y$:

$$
  \left\{
    (U_y \cap Y) \subset Y
  \right\}_{y \in Y}
  \,.
$$

Now by the assumption that $Y$ is compact, there exists a finite subcover, hence a [[finite set]] $S \subset Y$ such that

$$
  \left\{
    (U_y \cap Y) \subset Y
  \right\}_{y \in S \subset Y}
$$

is still a cover.

But the finite intersection

$$
  U_x \coloneqq \underset{s \in S \subset Y}{\bigcap} U_{x,s}
$$

of the corresponding open neighbourhoods of $x$ is still open, and by construction it is disjoint from all the $U_{s}$, hence in particular from their union

$$
  U_Y \coloneqq \underset{s \in S \subset Y}{\bigcup}  U_s
  \,.
$$

Therefore $U_x$ and $U_Y$ are two open subsets as required.

=--

+-- {: .proof}
###### Proof
of prop. \ref{CompactHausdorffSpacesAreNormal}

First we claim that $(X,\tau)$ is [[regular topological space|regular]]. To show this, we need to find for each point $x \in X$ and each disjoint closed subset $Y \in X$ dijoint open neighbourhoods $U_x \supset \{x\}$ and $U_Y \supset Y$. But since [[closed subspaces of compact spaces are compact]], the subset $Y$ is in fact compact, and hence this is in fact the statement of lemma \ref{SeparationByNeighbourhoodsOfPointsFromCompactSubsetsInHausdorffSpaces}.

Next to show that $(X,\tau)$ is indeed normal, we apply the idea of the proof of lemma \ref{SeparationByNeighbourhoodsOfPointsFromCompactSubsetsInHausdorffSpaces} once more:

Let $Y_1, Y_2 \subset X$ be two disjoint closed subspaces. By the previous statement then for every point $y_1 \in Y$ we find disjoint open neighbourhoods $U_{y_1} \subset \{y_1\}$ and $U_{Y_2,y_1} \supset Y_2$. The union of the $U_{y_1}$ is a cover of $Y_1$, and by compactness of $Y_1$ there is a finite subset $S \subset Y$ such that

$$
  U_{Y_1} \coloneqq \underset{s \in S \subset Y_1}{\bigcup} U_{y_1}
$$

is an open neighbourhood of $Y_1$ and

$$
  U_{Y_2} \coloneqq \underset{s \in S \subset Y}{\bigcap} U_{Y_2,s}
$$

is an open neighbourhood of $Y_2$, and both are disjoint.

=--

## Related statements

* [[Hausdorff spaces are sober]]

* [[compact subspaces of Hausdorff spaces are closed]]

* [[closed subspaces of compact Hausdorff spaces are equivalently compact subspaces]]

* [[a CW-complex is a Hausdorff space]]

* [[Hausdorff implies sober]]

* [[maps from compact spaces to Hausdorff spaces are closed and proper]]

* [[continuous images of compact spaces are compact]]

* [[paracompact Hausdorff spaces equivalently admit subordinate partitions of unity]]

## References

* TopoSpaces, _[Compact Hausdorff implies normal](https://topospaces.subwiki.org/wiki/Compact_Hausdorff_implies_normal)_