CombModCat

[[!redirects Ho(CombModCat)]]

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###Context###
#### Model category theory
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[[!include model category theory - contents]]
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#### Homotopy theory
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[[!include homotopy - contents]]
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#### $(\infty,1)$-Category theory
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[[!include quasi-category theory contents]]
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=--

#Contents#
* table of contents
{:toc}

## Idea

The concept of _[[model categories]]_ is one way of formulating the concept of certain classes of _[[homotopy theories]]_ or _[[(∞,1)-categories]]_. One way to make this precise while staying strictly within the context of [[1-category|1-]][[category theory]] is to consider the [[homotopy category]] of the ([[very large category|very large]]) category of [[model categories]] of (left) [[Quillen functors]] between them, hence its [[localization of a category]] at the [[Quillen equivalences]].

This should be particularly well-behaved for the sub-category $CombModCat$ of [[combinatorial model categories]]. Due to [[Dugger's theorem]], it should be true that

$$
  Ho(CombModCat)
  \;\coloneqq\;
  CombModCat\big[QuillenEquivs^{-1}\big]
  \;\simeq\;
  Ho(Pr(\infty,1)Cat)
$$

is [[equivalence of categories|equivalent]] to the [[homotopy category of an (infinity,1)-category|homotopy category]] of [[Pr(∞,1)Cat]], the [[(∞,1)-category]] of [[locally presentable (∞,1)-categories]] and [[(∞,1)-colimit]]-[[preserved limit|preserving]] [[(∞,1)-functors]] between them.

A proof for this statement, not just for [[homotopy categories]] but for the full homotopy theories ([[(infinity,1)-categories|$(\infty,1)$-categories]]), is now claimed in [Pavlov 2021](#Pavlov21).

An anlogous equivalence, but with presentable [[derivators]] and just at the level of [[homotopy 2-categories]], is due to  [Renaudin 06](#Renaudin06), see Corollary \ref{EquivalenceToHoPrDer} below.

## Definition and 2-localization
 {#Details}

\begin{definition}
\label{2CategoryOfModelCategories}
**([[2-category]] of [[combinatorial model category|combinatorial]] [[model categories]])**

Write

1. $ModCat$ for the [[2-category]] whose [[objects]] are [[model categories]], whose [[1-morphisms]] are [[left Quillen functors]] and [[2-morphisms]] are [[natural transformations]].

1. $\Delta ModCat$ for the [[2-category]] whose [[objects]] are [[simplicial model categories]] ([[classical model structure on simplicial sets|$sSet_{Qh}$]]-[[enriched model categories]]), whose [[1-morphisms]] are [[simplicial Quillen adjunction|simplicial]] [[left Quillen functors]] and [[2-morphisms]] are [[natural transformations]].

1. $CombModCat \subset ModCat$ and $\Delta CombModCa \subset \Delta ModCa$ for the [[full sub-2-categories]] on the [[left proper model categories|left proper]] [[combinatorial model categories]],

1. $LPropCombModCat \subset CombModCat$ and $LPropCombModCat \subset CombModCat$ for the further [[full sub-2-categories]] on the [[left proper model categories|left proper]] [[combinatorial model categories]].

\end{definition}

\begin{remark}
\label{LocalPresentationOfCombinatorialModelCategories}
**(local presentation of combinatorial model categories)**
\linebreak
By [[Dugger's theorem]], we may choose for every $\mathcal{C} \in CombModCat$ (Def. \ref{2CategoryOfModelCategories}) an [[sSet-category]] $\mathcal{S}$ and a [[Quillen equivalence]]

$$
  \mathcal{C}^p
  \;\coloneqq\;
  [\mathcal{S}^{op}, sSet]_{proj,loc}
  \overset{\simeq_{Qu}}{\longrightarrow}
  \mathcal{C}
$$

from the local projective [[model structure on sSet-enriched presheaves]] over $\mathcal{S}$. The latter is still a [[combinatorial model category]] but is also a [[left proper model category|left proper]] [[simplicial model category]].
\end{remark}

\begin{prop}
\label{Homotopy2CategoryOf2CatOfCombinatorialModelCategories}
**(the [[homotopy 2-category]] of [[combinatorial model categories]])**

The [[2-localization of a 2-category]]

$$
  LPropCombModCat\big[QuillenEquivs^{-1}\big]
$$

of the [[2-category]] of [[left proper model category|left proper] [[combinatorial model categories]] (Def. \ref{2CategoryOfModelCategories}) at the [[Quillen equivalences]] exists. Up to [[equivalence of 2-categories]], it has the same [[objects]] as $CombModCat$ and for any $\mathcal{C}, \mathcal{D} \in CombModCat$ its [[hom-category]] is the [[localization of categories]]

$$
  LPropCombModCat\big[QuillenEquivs^{-1}\big](\mathcal{C}, \mathcal{D})
  \;\simeq\;
  ModCat( \mathcal{C}^p, \mathcal{D}^p )\big[\{QuillenHomotopies\}^{-1}\big]
$$

of the category of [[left Quillen functors]] and [[natural transformations]] between local presentations $\mathcal{C}^p$ and $\mathcal{D}^p$ (Remark \ref{LocalPresentationOfCombinatorialModelCategories}) at those [[natural transformation]] that on [[cofibrant objects]] have components that are [[weak equivalences]] ("Quillen homotopies").
\end{prop}

This is the statement of [Renaudin 06, theorem 2.3.2](#Renaudin06).[^1]

## Relation to derivators

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

There is an [[equivalence of 2-categories]]

$$
  LPropCombModCat\big[ QuillenEquivs^{-1} \big]
  \;\simeq\;
  PresentableDerivators
$$

between the [[homotopy 2-category]] of [[combinatorial model categories]] (Prop. \ref{Homotopy2CategoryOf2CatOfCombinatorialModelCategories}) and the 2-category of presentable [[derivators]] with left adjoint morphisms between them.
=--

This is the statement of [Renaudin 06, theorem 3.4.4](#Renaudin06).[^1]

For $\mathcal{C}$ a [[2-category]] write

1. $\mathcal{C}_1$ for the 1-category obtained by discarding all [[2-morphisms]];

1. $\pi_0^{iso}(\mathcal{C})$ for the [[1-category]] obtained by identifying isomorphic [[2-morphisms]].

+-- {: .num_prop}
###### Proposition
**([[localization]] of $CombModCat$ at the [[Quillen equivalences]])**

The composite 1-functor

$$
  LPropCombModCat_1
   \longrightarrow
  \pi_0^{iso}(CombModCat)
    \overset{\pi_0^{iso}(\gamma)}{\longrightarrow}
  \pi_0^{iso}( CombModCat[QuillenEquivs^{-1}] )
$$

induced from the [[2-localization]] of Prop. \ref{Homotopy2CategoryOf2CatOfCombinatorialModelCategories} exhibits the ordinary [[localization of a category]] of the 1-category $CombModCat$ at the [[Quillen equivalences]], hence [[Ho(CombModCat)]]:

$$
  Ho(CombModCat)
  \;\coloneqq\;
  CombModCat_1\big[ QuillenEquivs^{-1} \big]
  \simeq
  \pi_0^{iso}( CombModCat[QuillenEquivs^{-1}] )
  \,.
$$

Moreover, this localization inverts precisely (only) the [[Quillen equivalences]].
=--

This is the statement of [Renaudin 06, cor. 2.3.8 with prop. 2.3.4](#Renaudin06).

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

There is an [[equivalence of categories]]

$$
  Ho(CombModCat)
  \;\simeq\;
  Ho(PresentableDerivators)
$$

between the [[homotopy category]] of [[combinatorial model categories]] and that of presentable [[derivators]] with left adjoint morphisms between them.
=--

## Related concepts

* [[formal (infinity,1)-category theory|formal $\infty$-category theory]]

* [[2-model category]]

* [[Ho(Cat)]]

* [[2-category of model categories]]

* [[double category of model categories]]

* [[Quillen adjoint triple]]

* [[PrCat]], [[Pr(∞,1)Cat]]

* [[combinatorial simplicial model category]]

[[!include categories of categories - contents]]

[[!include locally presentable categories - table]]

## References
 {#References}

The equivalence of the [[homotopy 2-category]] of [[combinatorial model categories]] with that of presentable [[derivators]] is due to:

* {#Renaudin06} [[Olivier Renaudin]], *Plongement de certaines théories homotopiques de Quillen dans les dérivateurs*, Journal of Pure and Applied Algebra Volume 213, Issue 10, October 2009, Pages 1916-1935
([arXiv:math/0603339](https://arxiv.org/abs/math/0603339), [doi:10.1016/j.jpaa.2009.02.014](https://doi.org/10.1016/j.jpaa.2009.02.014))

  > Beware that, for the time being, the entry [above](#Details) is referring to the numbering in the arXiv version of [Renaudin 2006](#Renaudin06), which differs from that in the published version.

The equivalence of the full homotopy theory (in particular the [[homotopy 2-category]]) of [[combinatorial model categories]] with [[presentable (infinity,1)-categories|presentable $\infty$-categories]] is due to

* {#Pavlov21} [[Dmitri Pavlov]], *Combinatorial model categories are equivalent to presentable quasicategories* ([arXiv:2110.04679](https://arxiv.org/abs/2110.04679))

[^1]: The condition of left properness does not appear in the arXiv version of [Renaudin 2006](#Renaudin06), but is added in the published version. While  [[Dugger's theorem]] (Rem. \ref{LocalPresentationOfCombinatorialModelCategories}) ensures that every combinatorial model category is Quillen equivalent to a left proper one, it is not immediate that every [[zig-zag]] of Quillen equivalences between left proper combinatorial model categories may be taken to pass through only left proper ones.

[[!redirects HoCombModCat]]

[[!redirects 2Ho(CombModCat)]]


comma category

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### Context
#### Category theory
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[[!include category theory - contents]]
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#### Limits and colimits
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[[!include infinity-limits - contents]]
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#Contents#
* table of contents
{:toc}

## Idea ##

The _comma category_ of two [[functors]] $f : C \to E$ and $g : D \to E$ is a [[category]] like an [[arrow category]] of $E$ where all arrows have their [[source]] in the image of $f$ and their [[target]] in the image of $g$ (and the morphisms between arrows keep track of how these sources and targets are in these images). It can also be seen a kind of [[2-limit]]: a [[directed homotopy theory|directed]] refinement of the [[homotopy pullback]] of two functors between [[groupoids]].

## Definition

We discuss three equivalent definitions of comma categories

* [Explicitly in components](#InComponents)

* [As a fiber product](#AsAFiberProduct)

* [As a 2-limit](#AsA2Limit)

\begin{rmk} The terminology "comma category" is a holdover from the original notation $(f,g)$ for such a category, which generalises $(x,y)$ or $C(x,y)$ for a [[hom-set]]. This is rarely used any more. More common modern notations for the comma category are $(f/g)$, which we will use on this page, and $(f\downarrow g)$. \end{rmk}

### Via components: the objectwise definition
 {#InComponents}

\begin{defn}\label{CommaCategory}
If $f\colon C\to E$ and $g \colon D\to E$ are [[functors]], their **comma category** is the [[category]] $(f/g)$ whose

* [[objects]] are [[triples]] $(c,d,\alpha)$ where $c\in C$, $d\in D$, and

  \[
    \label{MorphismMakingAnObjectInTheComma}
    \alpha \colon f(c) \to g(d)
  \]

  is a [[morphism]] in $E$,

and whose

* [[morphisms]] from $(c_1,d_1,\alpha_1)$ to $(c_2,d_2,\alpha_2)$ are pairs $(\beta,\gamma)$, where $\beta:c_1\to c_2$ and $\gamma:d_1\to d_2$ are morphisms in $C$ and $D$, respectively, such that $\alpha_2 . f(\beta) = g(\gamma) . \alpha_1$.

$$
  \array{
    f(c_1)
    &\stackrel{f(\beta)}{\rightarrow}&
    f(c_2)
    \\
    \downarrow^{\alpha_1}
    &&
    \downarrow^{\alpha_2}
    \\
    g(d_1)
    &\stackrel{g(\gamma)}{\to}&
    g(d_2)
    \\
    \\
    (c_1,d_1, \alpha_1)
    &\stackrel{(\beta,\gamma)}{\to}&
    (c_2,d_2, \alpha_2)
  }
$$

* [[composition]] of morphisms is given on components by composition in $C$ and $D$.

\end{defn}

\begin{remark}
In addition, there are two canonical [[forgetful functors]] defined on the comma category:

* there is a functor $H_C\colon (f/g)\rightarrow C$ which sends each object $(c,d,\alpha)$ to $c$, and each pair $(\beta,\gamma)$ to $\beta$.

* there is a functor $H_D\colon (f/g)\rightarrow D$ which sends each object $(c,d,\alpha)$ to $d$, and each pair $(\beta,\gamma)$ to $\gamma$.

Furthermore:

* there is a [[natural transformation]] $\theta : f \circ H_C \to g\circ H_D$ defined by $\theta_{(c,d,\alpha)} = \alpha$.

These functors and natural transformation together give the comma category a [[2-category theory|2-category theoretic]] [[universal property]]; see [this section](#AsA2Limit) for more.
\end{remark}

\begin{remark}\label{IsoCommaCategory}
  Alternatively, requiring the morphisms (eq:MorphismMakingAnObjectInTheComma) to be [[isomorphisms]] yields the notion of an **iso-comma category** (an [[iso-comma object]] in [[Cat]]). This has the [[universal property]] of a non-lax "[[2-pullback]]".
\end{remark}

### Via fiber products in the 1-category Cat
 {#AsAFiberProduct}

Let $I = \{a \to b\}$ be the (directed) [[interval category]] and $E^I = Funct(I,E)$ the [[functor category]].

The comma category is the [[pullback]]

$$
  \array{
    (f/g)
    &\to&
    E^I
    \\
    \big\downarrow
    & \scriptsize{(pb)} &
    \big\downarrow
    \mathrlap{^{\mathrlap{(F\mapsto F(a))\times(F\mapsto F(b))}}}
    \\
    C \times D
    &
      \underset{f \times g}{\longrightarrow}
    &
    E \times E
  }
$$

in the standard sense of pullback of morphisms in the 1-category [[Cat]] of categories.

Compare this with the construction of [[homotopy pullback]] ([here](homotopy+pullback#HomotopyPullbackByFactorizationLemma)), hence with the definition of [[loop space object]] and also with [[generalized universal bundle]].

### Via 2-category theory: as a 2-limit
 {#AsA2Limit}

The comma category is the [[comma object]] of the [[cospan]] $C\overset{f}{\rightarrow}E\overset{g}{\leftarrow}D$ in the [[2-category]] $Cat$.  This means it is an appropriate [[weighted limit|weighted]] 2-categorical [[2-limit|limit]] (in fact, a [[strict 2-limit]]) of the diagram

$$
  \array{
    && C
    \\
    && \downarrow^f
    \\
    D &\stackrel{g}{\to}& E
  }
$$

Specifically, it is the universal [[span]] making the following square commute up to a specified [[natural transformation]] (such a universal square is in general called a [[comma square]]):

$$
  \array{
    (f/g) &\overset{H_C}{\to}& C
    \\
    \mathllap{{}^{H_D}} \downarrow &\swArrow& \downarrow^{\mathrlap{f}}
    \\
    D &\stackrel{g}{\to}& E
  }
$$

(Sometimes this is called a "lax pullback", but that terminology properly refers to something else; see [[comma object]] and [[2-limit]].)

Notably, the forgetful functors $H_C$ and $H_D$ from the "objectwise" definition are thus recovered via a categorical construction: they are the projections from the summit of the "appropriate" 2-categorical limit.

In terms of the imagery of [[loop space objects]], the comma category is the category of [[interval object|directed paths]] in $E$ which start in the image of $f$ and end in the image of $g$.

## Examples

* If $f$ and $g$ are both the identity functor of a category $C$, then $(f/g)$ is the category $C ^{\mathbf{2}} $ of arrows  in $C$.

* If $f$ is the identity functor of $C$ and $g$ is the inclusion $1\to C$ of an object $c\in C$, then $(f/g)$ is the [[over category|slice category]] $C/c$.

* Likewise if $g$ is the identity and $f$ is the inclusion of $c$, then $(f/g)$ is the [[under category|coslice category]] $c/C$.

* A [[natural transformation]] $\tau \colon F \to G$ with $F,G :\colon C\to D$ may be regarded as a [[functor]] $T \colon C\to (F/G)$ with $T(c)=(c,c,\tau_c)$ and $T(f)=(f,f)$. Conversely, any such functor $T$ such that the two projections from $(F/G)$ back to $C$ are both [[left inverses]] for $T$ yields a corresponding natural transformation.  This is an expression of the universal property of $(F/G)$ as a [[comma object]].

## Properties

### Completeness and cocompleteness

If $C$ and $D$ are [[cocomplete category|cocomplete]] and $f: C \to E$ is [[cocontinuous functor|cocontinuous]] and $g: D \to E$ is an arbitrary functor (not necessarily cocontinuous) then the comma category $(f/g)$ is cocomplete.
Similarly, as $(f/g)^{op}\cong (g^{op}/f^{op})$, if $C$ and $D$ are [[complete category|complete]] and $g: D \to E$ is [[continuous functor|continuous]] and $f: C \to E$ is an arbitrary functor (not necessarily continuous) then the comma category $(f/g)$ is complete.

For a proof, see

* Rydeheard, David E., and Rod M. Burstall. Computational category theory. Vol. 152. Englewood Cliffs: Prentice Hall, 1988.  Section 5.2: colimits in comma categories.

### Functors and comma categories

* [[functors and comma categories]]

## Related concepts

* [[Quillen's theorem A]]
* [[comma object]]

## References
 {#References}

The notion of comma categories was introduced (in order to [characterize](adjoint+functor#InTermsOfCommaCategories)  [[adjoint functors]], but without giving them a name) in:

* [[F. W. Lawvere]], *[[Functorial Semantics of Algebraic Theories]]*, Ph.D. thesis, Columbia University (1963)

* Consolato Pellegrino, _Creazione di limiti nelle categorie comma_, Rivista di Matematica della Università di Parma 3 (1974): 201-204.

* Consolato Pellegrino, _Un teorema di completezza per le categorie comma, sue applicazioni agli n-grafi_, Atti del Seminario Matematico e Fisico dell'Università di Modena 23 (1974): 223-230.

Textbook accounts:

* [[Saunders MacLane]], §II.6 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* [[Francis Borceux]], Section 1.6 in: *[[Handbook of Categorical Algebra]]* Vol. 1: *Basic Category Theory* [[doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858)]

See also:

* [[Toby Bartels]], _[Comma categories](http://tobybartels.name/notes/#comma)_

[[!redirects comma categories]]

[[!redirects isocomma category]]
[[!redirects isocomma categories]]

[[!redirects iso-comma category]]
[[!redirects iso-comma categories]]

[[!redirects iso-comma-category]]
[[!redirects iso-comma-categories]]


comma category > 2cell

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comma double category

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

#Contents#
* table of contents
{:toc}

## Idea ##

A comma double category is a generalization of a [[comma category]] to [[double categories]] and [[virtual double categories]].

There are two apparently-maximal levels of generality of this construction, which intersect nontrivially but do not coincide.  The first is an ordinary [[comma object]] in the 2-category of virtual double categories (which includes as a full sub-2-category the 2-category of pseudo double categories and *lax* functors), which produces another virtual double category.  The second is a [[colax/lax comma object]] relative to the 2-monad whose algebras are pseudo double categories.  The two constructions intersect in the case of the comma object of a pseudo double functor over a lax one, between pseudo double categories.

## Definitions

### For virtual double categories ###

For virtual double categories, the comma virtual double category has the universal property of a [[comma object]] in the 2-category of virtual double categories, functors and vertical transformations.

Explicitly, it is constructed as follows. Let $C, D, E$ be virtual double categories and $F : C \to E, G : D \to E$ be functors of virtual double categories. The comma double category $F / G$ is defined as

1. Its vertical category is the ordinary comma category $F_v/G_v$ of the vertical components of the functors. We write an object as $A : F A_C \to G A_D$.
2. A horizontal arrow $R$ from $A$ to $B$ consists of horizontal arrows $R_C : A_C \to B_C$ and $R_D : A_D \to B_D$ and a 2-cell in $E$ from $F R_C$ to $G R_D$ along $A,B$.
3. A 2-cell $\alpha$ from $R_1,\ldots,R_n$ to $S$ consists of a pair of 2-cells $\alpha_C : (R_{1C},\ldots,R_{nC}) \to S_C$ and $\alpha_D : (R_{1D},\ldots,R_{nD}) \to S_D$ such that $G(\alpha_D) \circ (R_1,\ldots,R_n) = S \circ F(\alpha_C)$.

### The oplax/lax case

Let $K = Cat^{\rightrightarrows}$ be the 2-category of [[directed graphs]] [[internalization|internal]] to [[Cat]].  There is a [[2-monad]] $T$ on $K$ whose algebras are (pseudo) double categories, and whose lax and colax morphisms are lax and colax double functors respectively.  The oplax/lax comma double category is then an [[oplax/lax comma object]] for this 2-monad.

## Structures on a virtual comma double category ##

Next, we consider what properties are required of the input data (in the virtual case) to determine that a comma virtual double category has units and composites. An analogy with the double category case gives some guidance. Since functors of virtual double categories correspond to *lax* functors of double categories, we don't have any requirements for the functor $G$ on top of $D$ having composites or units. On the other hand, for $F$ to be "oplax", we require that it be normal for units or furthermore strong for composites.

+-- {: .un_prop}
###### Proposition

If $C$, $D$ and $E$ have units and $F$ is a normal functor, then $F / G$ has units.
=--

+-- {: .un_prop}
###### Proposition

If $C$, $D$ and $E$ have composites and $F$ is a strong functor, then $F / G$ has composites.
=--

Next, the comma has restrictions whenever the constituent categories do and the functors preserve them.

+-- {: .un_prop}
###### Proposition

If $C$, $D$ and $E$ have restrictions and $F,G$ preserve them, then $F / G$ has restrictions.
=--

In practice this proof burden can be reduced if we are interested in virtual equipments (i.e. having units and restrictions) because in that case restrictions are automatically preserved. We summarize this as follows:

+-- {: .un_prop}
###### Corollary
If $C, D$ and $E$ are [[virtual equipments]] and $F$ is a normal functor, then $F/G$ is a virtual equipment.
=--

## Examples ##

1. A monad $T$ in the horizontal bicategory of a double category $C$ is equivalent to a lax functor $T : 1 \to C$ from the terminal double category. In this case we might call $Id_C / T$ the [[slice double category]].

2. [[generalized multicategories]] can be constructed using a slice category when the monad $T$ is a [[polynomial monad]].  Specifically, let $C$ be the double category of [[polynomial functors]] in some locally cartesian closed category $E$; then a polynomial monad $T$ on $E$ can be identified with a horizontal monad in $C$ on the [[terminal object]] $1$.  The slice $Id_C/T$ is then equivalent to the "horizontal Kleisli category" presented in Cruttwell-Shulman; $T$-multicategories are then monads in that comma double category.

3. The double category of [[decorated cospans]] is naturally constructed as a comma double category. Given a symmetric lax monoidal functor $F : (C,+) \to (D,\otimes)$, there is an associated lax double functor from $F' : Cospan(C) \to BD$ where $BD$ is the [[delooping]] of $D$ into a double category whose horizontal category is $D$ and vertical category is the terminal category. Then there is a colax (pseudo even) double functor $* : 1 \to BD$ that picks out the unique object of $BD$. Then the double category of decorated cospans is $*/F'$.

4. [[poset-valued sets]] given by an endofunctor $F$ on $Rel$ and a poset $P$ can be viewed as the comma double category from $F$ to $P$, since a poset is a monad in $Rel$, and $F$ is a colax endofunctor of $Rel$.  The "morphisms" of poset-valued sets are the *horizontal* morphisms in the resulting comma double category.

5. The [[Dialectica construction]] associated to an internal poset $\Omega$ in a monoidal category $C$ with pullbacks can be obtained as a comma double category.  Let $Span(C)$ be the double category whose horizontal morphisms are spans in $C$, regard $C\times C^{op}$ as a double category in the horizontal direction, and let $F: C\times C^{op} \to Span(C)$ be the colax functor defined on objects by $(A,B) \mapsto A\otimes B$ and taking the pair $(f,g) : (A_1,B_1) \to (A_2,B_2)$ (so that $f:A_1\to A_1$ and $g:B_2\to B_1$) to the span $A_1\otimes B_1 \xleftarrow{1\otimes g} A_1\otimes B_2 \xrightarrow{f\otimes 1} A_2\otimes B_2$.  The internal poset $\Omega$ is a monad in $Span(C)$, so we have a comma double category $F/\Omega$, whose horizontal category is the Dialectica construction $Dial(C,\Omega)$.

6. The [[double gluing]] construction relative to a pair of functors $L:C\to E$ and $K:C\to E^{op}$ can be phrased as a comma double category of the cospan $C \xrightarrow{(L,K)} \mathbb{C}hu(E,1) \leftarrow Chu(E,1)$, where $C$ and $Chu(E,1)$ are regarded as vertically discrete double categories and $\mathbb{C}hu(E,1)$ is the [[double Chu construction]].  To obtain the relevant monoidal structures we can consider this instead to be a comma object in [[double polycategories]].

##References

An explicit description of the comma double category from an oplax to a lax functor is given in

* [[Kenny Courser]], _A bicategory of decorated cospans_ ([arxiv](https://arxiv.org/abs/1605.08100))

[[!redirects comma double categories]]
[[!redirects comma virtual double category]]
[[!redirects comma virtual double categories]]


comma object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### 2-Category theory
+-- {: .hide}
[[!include 2-category theory - contents]]
=--
#### Limits and colimits
+-- {: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

The notion of _comma object_ or _comma square_ is a generalization of the notion of [[pullback]] or _pullback square_ from [[category theory]] to [[2-category]] [[higher category theory|theory]]: it is a special kind of _[[2-limit]]_.

Where a [[pullback]] involves a [[commuting diagram|commuting square]], for a comma object this square is filled by a [[2-morphism]].

## Definition

\begin{definition}\label{CommaObject}
**(comma object)**
\linebreak
The *comma object* of a [[pair]] of [[1-morphisms]] $f \colon A\to C$ and $g \colon B\to C$ in a [[2-category]] is an [[object]] $(f/g)$ equipped with [[projections]] $p \colon (f/g)\to A$ and $q \colon (f/g)\to B$ and a [[2-morphism]] of this form:

$$
\array{
  (f/g)
  &
    \overset{p}{\longrightarrow}
  &
  A
  \\
  \mathllap{\scriptsize{q}} \big\downarrow
  &
  \swArrow \alpha
  &
  \big\downarrow   \mathrlap{\scriptsize{f}}
  \\
  B
  &
  \underset{g}{\longrightarrow}
  &
  C
}
$$

which is [[universal property|universal]] in the sense of a [[2-limit]].
\end{definition}
\begin{remark}\label{RelationToLaxLimits}
**(terminology)**
\linebreak
Comma objects are also sometimes called **lax pullbacks**, but this term more properly refers to the [[lax limit]] of a [[cospan]], which is a slightly different notion.
\end{remark}

\begin{remark}
**(in components)** More concretely:

Part of Def. \ref{CommaObject} is the statement that for any object $D$, 1-morphisms $p' \colon D\to A$, $q' \colon D\to B$ and 2-morphisms  $\sigma \colon f p'\Rightarrow g q'$ there is a 1-morphism $u \colon D\to(f/g)$ and isomorphisms $p u\cong p'$, $q u\cong q'$ such that modulo these isomorphisms, we have $\sigma=\alpha u$.

In addition there is the "2-dimensional universality" saying that given $u \colon D\to (f/g)$ and $v \colon D\to (f/g)$ and 2-morphisms $\mu \colon p u \to p v$ and $\nu \colon q u \to q v$ such that $\alpha v. f \mu = g\nu . \alpha u$, there exists a unique 2-morphism $\beta \colon u\to v$ such that $p\beta = \mu$ and $q \beta = \nu$.

Notice that the 2-dimensional property implies that in the 1-dimensional property, [[generalized the|the]] 1-morphism $u$ is unique up to unique isomorphism.  A square containing a 2-cell with this property is sometimes called a **comma square**.
\end{remark}

\begin{remark}\label{IsoCommaObjects}
By an **iso-comma object** one means the analogous notion as in Def. \ref{CommaObject}, now subject to the requirement that the [[2-morphism]] is a [[2-isomorphism]] and subject to the relevant universal property.

For more on this case see at *[[2-pullback]]*.
\end{remark}

\begin{remark}
The notion of a  **strict comma object** is analogous to that of Def. \ref{CommaObject} but has the [[universal property]] of a [[strict 2-limit]].  This means that given $p'$, $q'$, and $\sigma$ as above, there exists a _unique_ $u:D\to (f/g)$ such that $p u = p'$, $q u = q'$, and $\sigma u = \alpha$.  Note that any strict comma object is a comma object, but the converse is not in general true.

When combining this with the constraint in Rem. \ref{IsoCommaObjects}, one also has the notion of *strict iso-comma objects*
\end{remark}

## Properties

### Construction

The comma object $f/g$ can be constructed by means of [[pullbacks]] and [[cotensors]]:
$$
\array{
f/g & \to & P & \to & A \\
\downarrow & & \downarrow & & \downarrow \mathrlap{\scriptsize{f}} \\
Q & \to & C^{\mathbf{2}} & \underset{dom}{\to} & C \\
\downarrow & & \downarrow \mathrlap{\scriptsize{cod}} \\
B & \underset{g}{\to} & C
}
$$
where $C^{\mathbf{2}}$ is the cotensor of $C$ with the arrow category $\mathbf{2} = \bullet \to \bullet$.

### Pasting lemma

Suppose given a diagram
$$
\array{
  P & \to & Q & \to & A \\
  \downarrow & & \mathllap{\scriptsize{p}} \downarrow & \swArrow & \downarrow   \mathrlap{\scriptsize{f}} \\
  D & \underset{h}{\to} & C & \underset{g}{\to} & B
}
$$
where the right-hand square is a comma square.  Then the following are equivalent:

* the whole diagram is a comma square
* the left-hand square is a (2-)pullback square

The proof is analogous to that at [[pullback]].

## Examples

* In [[Cat]], a [[comma category]] is a comma object (in fact a strict one, as normally defined); these give their name to the general notion.

* In the 2-category of [[virtual double categories]], a comma object is a [[comma double category]].  If the virtual double categories are (pseudo) [[double categories]] and the domain functor $f$ in $f/g$ is strong (while $g$ might be only lax), then the comma object is also a pseudo double category and the comma object lives in the 2-category of pseudo double categories and lax functors.

## Related pages

* [[oplax/lax comma object]]

[[!include notions of pullback -- contents]]

[[!redirects comma object]]
[[!redirects comma objects]]
[[!redirects comma square]]
[[!redirects comma squares]]
[[!redirects lax pullback]]
[[!redirects lax pullbacks]]

[[!redirects strict comma object]]
[[!redirects strict comma objects]]

[[!redirects iso-comma object]]
[[!redirects iso-comma objects]]

[[!redirects isocomma object]]
[[!redirects isocomma objects]]

[[!redirects iso-comma-object]]
[[!redirects iso-comma-objects]]

[[!redirects strict iso-comma object]]
[[!redirects strict iso-comma objects]]

[[!redirects strict isocomma object]]
[[!redirects strict isocomma objects]]

[[!redirects strict iso-comma-object]]
[[!redirects strict iso-comma-objects]]

[[!redirects bi-iso-comma object]]
[[!redirects bi-iso-comma objects]]

[[!redirects bi-iso-comma-object]]
[[!redirects bi-iso-comma-objects]]


comma object > 2cell

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</svg>


CommAlg

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

#Contents#
* table of contents
{:toc}

## Definition

_$CommAlg$_ or _$CAlg$_denotes a [[category]] of commutative [[associative algebra]]s.

Usually this is understood implicitly as algebras over some [[ground field]] $k$. In this case

$$
  CommAlg \simeq k/CRing
$$

is the [[slice category]] of [[CRing]] -- the category of commutative [[ring]]s -- under $k$.

## Properties

* Via [[measure coalgebras]] $CommAlg$ is naturally [[enriched category|enriched]] over the category [[Coalg]] of [[cocommutative coalgebra]]s.

category: category

[[!redirects CAlg]]


This page provides some links related to the article

* [[William Lawvere]],

  _Comments on the development of topos theory_

  in Jean-Paul Pier (ed.)

  _Development of mathematics 1950-2000_,

  Birkhäuser, 2000

  reprinted as

  Reprints in Theory and Applications of Categories, No. 24 (2012) pp. 1-22

  ([TAC](http://www.tac.mta.ca/tac/reprints/articles/24/tr24abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/24/tr24.pdf))

on the history of [[topos theory]] with special amphasis on [[William Lawvere]]'s contributions and point of view, notably ending with discussion of foundations of [[physics]] in topos theory, see also his _[[Categories in Continuum Physics]]_.

category: reference


Community > history


commutant

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

#Contents#
* table of contents
{:toc}

## Definition

In an [[associative algebra]] $A$, the **commutant** of a [[set]] $B \subset A$ of elements of $A$ is the set

$$
  B' = \{a \in A | \forall b \in B:  a b = b a \}
$$

of elements in $A$ that commute with all elements in $B$.

## Properties

The operation of taking a commutant is a [[contravariant functor|contravariant]] map $P(A) \to P(A)$ that is adjoint to itself in the sense of [[Galois connection|Galois connections]]. In other words, we have for any two subsets $B, C \subseteq A$ the equivalence

$$B \subseteq C' \qquad iff \qquad C \subseteq B'.$$

Hence $B \subseteq B''$ and also $B' = B'''$.

## Related entries

* [[bicommutant theorem]]

* [[centralizer]]

[[!redirects commutants]]


Commutative Algebra

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

This entry is about the article

* [[Jacob Lurie]], _Commutative Algebra_ ([arXiv](http://arxiv.org/abs/math/0703204))

on [[higher algebra]] and in particular [[E-infinity ring]]s.

Meanwhile this has become part of the book

* _[[Higher Algebra]]_

category: reference


commutative algebra

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

#Contents#
* table of contents
{:toc}

## Idea

A __commutative $k$-algebra__ (with $k$ a [[field]] or at least a [[commutative ring]]) is an [[associative unital algebra]] over $k$ such that the multiplicative operation is [[commutative]].
Equivalently, it is a [[commutative ring]] $R$ equipped with a ring [[homomorphism]] $k \to R$.

There is a generalization of commutativity when applied to [[finitary monads]] in $Set$, that is [[generalized rings]], as studied in [[Nikolai Durov|Durov's]] thesis.

__Commutative algebra__ is the subject studying commutative algebras. It is closely related and it is the main algebraic foundation of [[algebraic geometry]]. Some of the well-known classical theorems of commutative algebra are the [[Hilbert basis theorem]] and [[Nullstellensatz]] and [[Krull's theorem]], as well as many results pertaining to [[syzygy|syzygies]], [[resultant|resultants]] and [[discriminant|discriminants]].

## Related entries

* [[commutative magma]]

* [[abelian group]]

* [[commutative ring]]

* [[commutative monoid]]

* [[commutative C-star-algebra]]

* [[graded commutative algebra]]

* [[supercommutative algebra]]

## References

* [[Michael Atiyah]], [[Ian G. Macdonald]], _Introduction to commutative algebra_, (1969, 1994) $[$[pdf](http://math.univ-lyon1.fr/~mathieu/CoursM2-2020/AMD-ComAlg.pdf), [ISBN:9780201407518](https://www.routledge.com/Introduction-To-Commutative-Algebra/Atiyah/p/book/9780201407518)$]$

* H. Matsumura, _Commutative algebra_, 2 vols.; see also the online summary notes by D. Murfet, [Matsumura.pdf](http://therisingsea.org/notes/Matsumura.pdf), [Matsumura-Part2.pdf](http://therisingsea.org/notes/Matsumura-Part2.pdf)

* [[D. Eisenbud]], _Commutative algebra: with a view toward algebraic geometry_, Grad. Texts in Math. 150, Springer-Verlag 1995.
* James Milne, _A primer of commutative algebra_, (online notes in progress) [webpage](http://www.jmilne.org/math/xnotes/ca.html), [pdf](http://www.jmilne.org/math/xnotes/CA.pdf)

Discussion of commutative algebra with [[constructive mathematics|constructive]] methods:

* Henri Lombardi, Claude Quitté, _Commutative algebra: Constructive methods. Finite projective modules_ ([arXiv:1605.04832](https://arxiv.org/abs/1605.04832))
[[!redirects commutative algebras]]
[[!redirects commutative unital algebra]]


commutative algebra in a symmetric monoidal (infinity,1)-category > history

* [[commutative monoid in a symmetric monoidal (infinity,1)-category]]


commutative algebraic theory

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

#Contents#
* table of contents
{:toc}

## Idea

An _[[algebraic theory]]_ is said to be _commutative_ if its operations are algebra homomorphisms under any interpretation, generalizing the familiar case of the theory of [[commutative monoid|commutative monoids]].

A more general notion is that of _[[monoidal monads]]_.

## Definition

Two operations, $\alpha$ and $\beta$, of an [[algebraic theory]] are said to __commute__ if for any [[matrix]] $M$ of elements, with the number of rows given by the arity of $\alpha$ and the number of columns by the arity of $\beta$, one gets the same result whether one

1. applies $\alpha$ to each column of $M$ and then $\beta$ to the resulting row, or
1. applies $\beta$ to each row of $M$ and then $\alpha$ to the resulting column.

(We formulate this notion in an element-free way below.)

Note that an operation of arity $0$ or $1$ always commutes with itself; this is not necessarily the case for higher arities. Commuting nullary operations are necessarily equal.

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=--

The operations that commute with a given set of operations in an algebraic theory form a subtheory. The __centre__ of an algebraic theory is given by the operations that commute with all the operations of the theory. An algebraic theory is __commutative__ if every pair of its operations commute. Another way of describing the centre is to say that it consists of those operations which are also [[homomorphisms]]; an algebraic theory is commutative if all of its operations are homomorphisms.

This concept may also be formalized in the language of [[monads]], where it naturally generalizes to the concept of [[commutative monad]]. Suppose given an [[algebraic theory]] that admits the construction of a free algebra on any set, so that the theory may be equivalently described by a monad $T$ on $Set$, the category of sets. The monad $T$ is automatically strong, i.e., carries a left strength

$$\sigma_{A,B} \,\colon\, A \times (T B) \longrightarrow T(A \times B)$$

(by [this example](strong+monad#example-in-set)), from which a right strength

$$\tau_{A,B} \,\colon\, (T A) \times B \longrightarrow T(A \times B)$$

can be derived by exploiting the symmetry ([[braiding]]) of the cartesian product.

In this notation, we have the following definition.
\begin{definition}\label{CommutativeMonad}
**([[commutative monads]] on $Set$)**
\linebreak
$T$ is called _[[commutative monad|commutative]]_ if there is [[equality]] $\alpha = \beta$ between the two maps

$$
  \alpha
  ,
  \beta
    \;\colon\;
  T A \times T B
    \rightrightarrows
  T(A \times B)
  \,,
$$

where

* $\alpha$ is the composite

  \[
    \label{Alpha}
    \alpha_{A,B}
    \;\colon\;
    T A \times T B
      \overset{
        \sigma_{A, T B}
      }{\longrightarrow}
    T(A \times T B)
      \overset{
        T(\tau_{A, B})
      }{\longrightarrow}
    T T(A \times B)
      \overset{
        m(A \otimes B)
      }{\longrightarrow}
    T(A \times B)
    \,.
  \]

* $\beta$ is the composite

  \[
    \label{Beta}
    \beta_{A,B}
    \;\colon\;
    T A \times T B
      \overset{
        \tau_{T A, B}
      }{\longrightarrow}
    T(T A \times B)
      \stackrel{
        T(\sigma_{A, B})
      }{\longrightarrow}
    T T(A \times B)
      \stackrel{
        m(A \times B)
      }{\longrightarrow}
    T(A \times B)
    \,.
  \]

\end{definition}

It is worth checking what this description gives more explicitly. Starting with a pair of elements in $T A \times T B$,

$$\langle (\omega; a_1, \ldots, a_m), (\chi; b_1, \ldots, b_n)\rangle$$

(and here we should be considering equivalence classes of such formal operations), the map $\alpha$ sends this to

$$(\omega(\chi, \ldots, \chi); (a_1, b_1), \ldots (a_1, b_n)), \ldots, (a_m, b_1), \ldots, (a_m, b_n))$$

where $\omega(\chi, \ldots, \chi)$ is the evident operation of arity $m n = n + \ldots + n$. In more detail, $\alpha(\langle (\omega; \vec{a}), (\chi; \vec{b} \rangle)$ is the end result of the sequence

$$\langle (\omega; \vec{a}), (\chi; \vec{b}) \rangle \stackrel{\sigma}{\mapsto} (\omega; (a_1, (\chi; \vec{b})), \ldots, (a_m, (\chi; \vec{b}))) \stackrel{T\tau}{\mapsto} (\omega; (\chi; (a_1, \vec{b})), \ldots, (\chi; (a_m, \vec{b}))) \stackrel{m}{\mapsto} (\omega(\chi, \ldots, \chi); (a_1, \vec{b}), \ldots, (a_m, \vec{b})).$$

Similarly, the map $\beta$ sends the pair $\langle (\omega; \vec{a}), (\chi; \vec{b}) \rangle$ to

$$(\chi(\omega, \ldots, \omega); (\vec{a}, b_1), \ldots, (\vec{a}, b_n))$$

where $\chi(\omega, \ldots, \omega)$ is the evident operation of arity $n m = m + \ldots + m$.

### Relation to commutativity of operations

A classical example of a commutative algebraic theory is the theory of [[Abelian groups]]. This is a slightly misleading example: it is not true in general that the operations of a commutative algebraic theory are always commutative in the sense that they are invariant under permutation of their operands. That this is true for Abelian groups follows from the [[Eckmann–Hilton argument]], since a binary operation $\otimes$ commutes with itself when
\[(a \otimes b) \otimes (c \otimes d) = (a \otimes c) \otimes (b \otimes d)\]
and, since the theory of groups is associative and unital, we can take $a = d = 1$ and deduce that
\[b \otimes c = c \otimes b.\]
However, if we suppose *only* a binary operation that commutes with itself, then the operation is not invariant under a transposition of its arguments, since there are models of that theory like $(a, b) \mapsto a \otimes b \coloneqq a$ where such invariance is violated.

There is a fairly general situation in which commutativity of an operator with itself implies that the operation is invariant under permutation of arguments.

\begin{proposition}
Let $f$ and $g$ be $n$-ary operations and let $1$ be a constant that is a unit for both $f$ and $g$, in the sense that $x \vdash f(1, \ldots, 1, x, 1, \ldots, 1) = x$ and similarly for $g$. If $f$ and $g$ commute, then $f = g$ and $f$ is invariant under permutation of its operands.
\end{proposition}

\begin{proof}
Let $\sigma$ be a permutation of $\{ 1, \ldots, n \}$. Let $y_1, \ldots, y_n$ be variables.
Define a matrix $\{ x_{ij} \}_{ij}$ where $x_{i \sigma_i} = y_i$ and $x_{ij} = 1$ otherwise. Then commutativity of $f$ and $g$ implies that
\[f(g(x_{11}, \ldots, x_{1n}), \ldots, g(x_{n1}, \ldots, x_{nn})) = g(f(x_{11}, \ldots, y_{n1}), \ldots, f(y_{1n}, \ldots, y_{nn}))\]
and hence that
\[f(g(1, \ldots, 1, y_1, 1, \ldots, 1), \ldots, g(1, \ldots, 1, y_n, 1, \ldots, 1)) = g(f(1, \ldots, 1, y_{\sigma_1}, 1, \ldots, 1), \ldots, f(1, \ldots, 1, y_{\sigma_n}, 1, \ldots, 1))\]
hence
\[f(y_1, \ldots, y_n) = g(y_{\sigma_1}, \ldots, y_{\sigma_n})\]
Taking $\sigma$ to be the identity permutation, we have that $f = g$. It then follows that $f$ is invariant under permutation of its operands.
\end{proof}

This implies, for instance, that the $\otimes$ of an Abelian group is commutative.

\begin{corollary}
Suppose a given algebraic theory is commutative and every operation has a unit. Then every operation is invariant under permutation of its operands, and there is at most one $n$-ary operation for each $n \in \mathbb{N}$.
\end{corollary}

\begin{proof}
In a commutative algebraic theory, there is at most constant, so the units for each operator are equal. The result then follows from the previous proposition, since every operation commutes with every other operation.
\end{proof}

### Abstract formulation

The category $\mathbf{Th}$ of Lawvere theories is endowed with a symmetric monoidal product (called _[[tensor product theory|Kronecker product]]_; see Freyd's article in the references),

$$\otimes \colon \mathbf{Th} \times \mathbf{Th} \to \mathbf{Th},$$

whereby $(S \otimes T)$-algebras are $S$-algebras internal to $T$-algebras, or equally well $T$-algebras internal to $S$-algebras. A commutative theory is tantamount to a commutative monoid in the symmetric monoidal category $\mathbf{Th}$.

If $S$ and $T$ are commutative theories, then their coproduct in the category of commutative theories is $S \otimes T$.

### Commutative theory as monoidal monad

Let $T$ be the $Set$-monad of a commutative theory. Then the map (eq:Alpha)

$$\alpha_{A, B}: T A \times T B \to T(A \times B)$$

can be shown to be the structure map for a monoidal structure on $T$, i.e., making $T$ a (symmetric) [[lax monoidal functor]], and in fact the monad multiplication and unit become [[monoidal natural transformations]].

In other words, we get a monad in the [[2-category]] of symmetric monoidal categories, lax symmetric monoidal functors, and monoidal transformations: a [[monoidal monad]].

In fact, it may be shown that commutative Lawvere theories on $Set$ are precisely the same things as (finitary) symmetric monoidal monad structures on $(Set, \times)$, as shown by Anders Kock. For more on this, see [[monoidal monad]].

### As commutative monoid in a duoidal category

The commutativity of a theory can also be expressed as an abstract property of a monoid in a [[duoidal category]], specialized to the duoidal category of finitary endofunctors.

## Properties

### Closed monoidal structure on algebras
 {#ClosedMonoidalStructureOnAlgebras}

We discuss that the category of [[algebras for an algebraic theory]] over a commutative algebraic theory is canonically a [[closed monoidal category|closed]] [[symmetric monoidal category]] ([Keigher 78](#Keigher78), [Seal 12](#Seal12)).

If $f_1,\ldots , f_n$ are homomorphisms $A \to B$ of models (algebras) of a commutative algebraic theory, and $\omega$ is an $n$-ary operation of it, then the function $A \to B$ given by sending $a \in A$ to $\omega(f_1(a),\ldots ,f_n(a)) \in B$ is again a homomorphism, which is naturally called $\omega(f_1,\ldots ,f_n)$. In this way $Hom(A,B)$ is enriched as a model of the algebraic theory, and we have a [[closed category]] of models and homorphisms. Furthermore, this [[internal hom|internal]] $Hom$ has a [[left adjoint]] $\otimes$ for which the free model on one generator is a unit, so we have a [[closed monoidal category]], in fact a closed [[symmetric monoidal category]].

The monoidal structure $\otimes$ can be extracted by a straightforward generalization of the usual [[tensor product of abelian groups]] (or of [[commutative monoids]]), where "[[bilinear map|bilinearity]] conditions" = "linearity in separate variables" is replaced by "$T$-homomorphicity in separate variables", where $T$ is the [[monad]] of the algebraic theory.

In slightly more detail, if $A$ and $B$ are $T$-algebras, the tensor product $A \otimes B$ ought to be $T(A \times B)$ modulo equivalences which we may write suggestively as

$$\omega(a_1, \ldots, a_m) \otimes \chi(b_1, \ldots, b_n) \sim (\omega(\chi, \ldots, \chi); a_1 \otimes b_1, \ldots, a_m \otimes b_n)$$

where the left side is represented by a composite

$$T A \times T B \stackrel{\xi_A \times \xi_B}{\to} A \times B \stackrel{u(A \times B)}{\to} T(A \times B)$$

(the $\xi$'s are $T$-algebra structures), and the right side by the monoidal structure map on $T$,

$$\alpha_{A, B} \colon T A \times T B \to T(A \times B).$$

In more detail still, $A \otimes B$ is the following [[coequalizer]] in $Alg_T$:

$$\array{
T(T A \times T B) & \stackrel{T\alpha}{\to} & T T(A \times B) & \\
 & ^\mathllap{T(\xi_A \times \xi_B)} \searrow & \downarrow^\mathrlap{m} & \\
 & & T(A \times B) & \to & A \otimes B
}$$

([Seal 12, section 2.2 and theorem 2.5.5](#Seal12))

This construction carries over to the wider context of monoidal monads, see [[tensor product of algebras over a commutative monad]].

## Examples

See (more generally) [[commutative monad|examples of commutative monads]].

## References

The notion of commutative algebraic theory was introduced by [[Fred Linton]]:

* [[Fred Linton]], _Autonomous equational categories_, Journal of Mathematics and Mechanics 15.4 (1966), 637-642.

It was formulated in terms of [[monads]] by [[Anders Kock]].

* [[Anders Kock]], _Monads on symmetric monoidal closed categories_, Arch. Math. 21 (1970), 1--10.

The [[closed category]]-structure on the EM-category of monoidal monads was studied in

* [[Anders Kock]], _Strong functors and monoidal monads_, Arhus Universitet, Various Publications Series No. 11 (1970).  [PDF](http://home.imf.au.dk/kock/SFMM.pdf).

* [[Anders Kock]], _Closed categories generated by commutative monads_ ([[KockMonoidalMonads.pdf:file]])

and the [[monoidal category]]-structure in

* {#Keigher78} [[William Keigher]], _Symmetric monoidal closed categories generated by commutative adjoint monads_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 19 no. 3 (1978), p. 269-293  ([NUMDAM](http://www.numdam.org/numdam-bin/fitem?id=CTGDC_1978__19_3_269_0), [pdf](http://archive.numdam.org/article/CTGDC_1978__19_3_269_0.pdf))

* {#Seal12} Gavin J. Seal, _Tensors, monads and actions_ ([arXiv:1205.0101](http://arxiv.org/abs/1205.0101))

There is [related MO discussion](http://mathoverflow.net/a/75929/381).

The Kronecker product of theories was introduced in an article of Freyd:

* [[Peter Freyd]], _Algebra valued functors in general and tensor products in particular_, Colloq. Math. 14 (1966), 89-106.

Recently [[Nikolai Durov]] rediscovered that notion for the purposes of geometry (under the name __commutative algebraic monad__), constructed their spectra (generalizing the [[spectrum (geometry)|spectrum]] of Grothendieck) and theory of [[generalized scheme after Durov|generalized schemes on this basis]]. There is a generalized version of the [[Eckmann–Hilton argument]] concerning commutative finitary monads. Much detail including many examples and further constructions are in his thesis

* [[Nikolai Durov]], _New approach to Arakelov geometry_, [arXiv:0704.2030](http://arXiv.org/abs/0704.2030)

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commutative C-star-algebra

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#### Operator algebra
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# Contents
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## Idea

A _commutative $C^\ast$-algebra_ is a [[C-star algebra]] whose underlying [[associative algebra]] is a [[commutative algebra]].

## Properties

### Relation to topological spaces

Commutative $C^\ast$-algebras are those identified by [[Gelfand duality]] with algebras of [[continuous functions]] on [[compact Hausdorff topological spaces]].

### Relation to KK-theory

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###### Example

Every commutative C*-algebra is a [[nuclear C*-algebra]].

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## Related concepts

* [[poset of commutative subalgebras]]

* [[KK-bootstrap category]]

* [[noncommutative topology]]

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commutative diagram

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### Context
#### Category theory#
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# Commutative diagrams
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## Idea

In [[category theory]], a _commutative diagram_ is a [[free diagram]] in which all [[parallel morphisms]] obtained by [[composition|composing]] morphisms in the diagram agree.

For example that a square diagram of the form

$$
  \array{& X & \overset{f}\rightarrow & Z & \\
          g & \downarrow &&\downarrow & g'\\
          &Y & \underset{f'}\rightarrow& W & \\
}$$

commutes is to say that $g' \circ f = f' \circ g$ (see also at _[[commuting square]]_).

## Definitions

### Slick definition

For our purposes, a __[[free diagram]]__ $D$ in a [[category]] $C$ consists of a [[free category]] $J$ and a [[functor]] to $C$ from the [[free category]] on $J$:
$$ J \overset{D}\to C ,\; J\;\text{a free category}. $$
Then this diagram $D$ __commutes__ if this functor $D$ factors (up to [[natural isomorphism]]) through a [[poset]] $P$:
$$ J \to P \to C \;\cong\; J \to C ,\; P\;\text{a poset} ;$$
or equivalently (treating $C$ as a [[strict category]]) if the functor factors up to [[equality]] through a [[proset]] $Q$:
$$ J \to Q \to C \;\cong\; J \to C ,\; Q\;\text{a proset} .$$
In the above, we are identifying [[posets]], and [[prosets]] with certain categories in the usual ways.

### Elementary definition

Recall that a __[[free category]]__ $J$ consists of a [[set]] $V$ of _vertices_, a set $E$ of _edges_, and two [[functions]] $s,t\colon E \to V$.  Given a category $C$, a __free diagram__ $D$ of shape $J$ in a category $C$ is consists of a map from $V$ to the [[objects]] of $C$ and a map from $E$ to the [[morphisms]] of $C$, both denoted $F$, such that $F(s(e)) = S(F(e))$ and $F(t(e)) = T(F(e))$ for each edge $e$, where $S,T$ are the [[source]] and [[target]] maps in $C$.

Recall that a __[[path]]__ $p$ in $J$ consists of a [[list]] $(v_0,v_1,\ldots,v_n)$ of vertices and a list $(e_1,\ldots,e_n)$ of edges such that $s(e_i) = v_{i-1}$ and $t(e_i) = v_{i}$ for each $i$, where $n$ is any [[natural number]] (possibly zero).  We say that $v_0$ is the __source__ of the path and that $v_n$ is its __target__.  Given a path $p$ and a diagram $D$, the __composite__ of $p$ under $D$ is the [[composite]] $F(e_1);\ldots;F(e_n)\colon F(v_0) \to F(v_n)$ in $C$. (Note that the paths of length zero are composed to the identity $\id_{F(v_0)}\colon F(v_0) \to F(v_0)$.)

A diagram $D$ __commutes__ if, given any two vertices $x,y$ in $J$ and any two paths $p,p'$ with source $x$ and target $y$, the composites of $p$ and $p'$ under $D$ are equal in $C$.

## Related entries

* [[diagram]]

* [[internal diagram]]

* [[free diagram]]

## References

* [Wikipedia](http://en.wikipedia.org/wiki/Commutative_diagram)
* [Mathworld](http://mathworld.wolfram.com/CommutativeDiagram.html)

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