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Comma category

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A '''comma category''' (also sometimes called a '''slice category''') is a construction in category theory, a branch of mathematics. It provides another way of looking at morphisms: instead of simply relating objects of a Category (mathematics) category to one another, they become objects in their own right. This notion was introduced in 1963 by F. W. Lawvere, although the technique did not become generally known until many years later. Today, it has become particularly important to mathematicians, because several important mathematical concepts can be treated as comma categories. There are also certain guarantees about the existence of Limit (category theory) limits and colimits in the context of comma categories. The name comes from the notation originally used by Lawvere, which involved the comma punctuation mark. Although standard notation has changed, the use of a comma as an operator being potentially confusing, the name persists.

Definition
The most general comma category construction involves two converging functors. Typically, one of these will be a "selection" or "constant" functor: many accounts of category theory consider these special cases only, but the term is actually much more general. (A selection functor maps every object in the domain category to the same, fixed object in the codomain category, and every domain morphism to the identity morphism of that fixed object. Often, the choice of domain category is not relevant; typically, the discrete category having only one object is used.)

General form
Suppose that $\backslash mathcal\{A\}$, $\backslash mathcal\{B\}$, and $\backslash mathcal\{C\}$ are categories, and $T$ and $S$ are functors :Image:CommaCategory-04.png We can form the comma category $(T\; \backslash downarrow\; S)$ as follows: *The objects are triples $(\backslash alpha,\; \backslash beta,\; f)$, with $\backslash alpha$ an object in $\backslash mathcal\{A\}$, $\backslash beta$ an object in $\backslash mathcal\{B\}$, and $f\; :\; T(\backslash alpha)\backslash rightarrow\; S(\backslash beta)$ a morphism in $\backslash mathcal\{C\}$. *The morphisms from $(\backslash alpha,\; \backslash beta,\; f)$ to $(\backslash alpha\text{'},\; \backslash beta\text{'},\; f\text{'})$ are pairs $(g,\; h)$ where $g\; :\; \backslash alpha\; \backslash rightarrow\; \backslash alpha\text{'}$ and $h\; :\; \backslash beta\; \backslash rightarrow\; \backslash beta\text{'}$ are morphisms in $\backslash mathcal\; A$ and $\backslash mathcal\; B$ respectively, such that the following diagram commutative diagram commutes: Image:CommaCategory-03.png center|Morphisms in (T ↓ S) Morphisms are composed by taking $(g,\; h)\; \backslash circ\; (g\text{'},\; h\text{'})$ to be $(g\; \backslash circ\; g\text{'},\; h\; \backslash circ\; h\text{'})$, whenever the latter expression is defined.

Category of objects under A
The first special case occurs with $T$ being a selection functor, and $S$ an identity functor (so $\backslash mathcal\{B\}\; =\; \backslash mathcal\{C\}$). (Then $T(\backslash alpha)\; =\; A$ for some ''fixed'' $A$ in $\backslash mathcal\{C\}$ and every $\backslash alpha$ in $\backslash mathcal\{A\}$). We then have the category of ''objects under $A$'', sometimes called ''objects co-over $A$'', written $(A\; \backslash downarrow\; \backslash mathcal\{C\})$. This is also known as the ''coslice category'' with respect to $A$. The objects $(\backslash alpha,\; \backslash beta,\; f)$ can be simplified to $(\backslash beta,\; f)$, since fixing $A$ makes $\backslash alpha$ irrelevant; and $f\; :\; T(\backslash alpha)\; \backslash rightarrow\; S(\backslash beta)$ simplifies to $f\; :\; A\; \backslash rightarrow\; \backslash beta$ - often, $f$ is called something like $i\_\backslash beta$, to indicate injection. In a similar way, morphisms like $(g,\; h)\; :\; (B,\; i\_B)\; \backslash rightarrow\; (B\text{'},\; i\_\{B\text{'}\})$ reduce to simply $h\; :\; B\; \backslash rightarrow\; B\text{'}$, as $g$ is just the identity morphism on $A$. The following must be a commutative diagram:

Image:CommaCategory-02.png

Category of objects over A
Similarly, $T$ might be an identity functor and $S$ a selection functor: this is the category of ''objects over $A$'' (where $A$ is the object of $\backslash mathcal\{C\}$ selected by $S$), written $(\backslash mathcal\{C\}\; \backslash downarrow\; A)$. This is also known as the ''slice category'' over $A$. It is the Dual (category theory) dual concept to objects-under-$A$. The objects are pairs $(\backslash beta,\; \backslash pi\_\backslash beta)$ with $\backslash pi\_\backslash beta\; :\; \backslash beta\; \backslash rightarrow\; A$; the $\backslash pi$ stands for projection onto $A$. Given $(B,\; \backslash pi\_B)$ and $(B\text{'},\; \backslash pi\_\{B\text{'}\})$, a morphism in the comma category is a map $g\; :\; B\; \backslash rightarrow\; B\text{'}$ making the following diagram commute:

Image:CommaCategory-01.png

Other variations
In either of these two cases, the identity functor may be replaced with some other functor; this yields a family of categories particularly useful in the study of adjoint functors. For example, if $S$ is the forgetful functor mapping an abelian group to its underlying set, and $t$ is the set selected by $T$, then $(t\; \backslash downarrow\; S)$ is a comma category whose objects are maps from $t$ to certain sets. This relates to the left adjoint of $S$, which is the functor that maps a set to the free abelian group having that set as its basis: some of the objects of $(t\; \backslash downarrow\; S)$ will be sets underlying such groups. Another special case occurs when both $S$ and $T$ are selection functors. If $S$ selects $A$ and $T$ selects $B$, then the comma category produced is equivalent to the set of morphisms between $A$ and $B$. (Strictly, it is a discrete category - all the morphisms are identity morphisms - which may be identified with the set of its objects.)

Examples of use


Some notable categories
Several interesting categories have a natural definition in terms of comma categories. * The category of pointed sets is a comma category, $(\backslash bull\; \backslash downarrow\; \backslash mathbf\{Set\})$ with $\backslash bull$ being (a functor selecting) any singleton set, and $\backslash mathbf\{Set\}$ (the identity functor of) the category of sets. Each object of this category is a set, together with a function selecting some element of the set: the "basepoint". Morphisms are functions on sets which map points to points. In a similar fashion one can form the category of pointed spaces $(\backslash bull\; \backslash downarrow\; \backslash mathbf\{Top\})$. * The category of Graph (mathematics) graphs is $(\backslash mathbf\{Set\}\; \backslash downarrow\; D)$, with $D\; :\; \backslash mathbf\{Set\}\; \backslash rightarrow\; \backslash mathbf\{Set\}$ the functor taking a set $s$ to $s\; \backslash times\; s$. The objects $(a,\; b,\; f)$ then consist of two sets and a function; $a$ is an indexing set, $b$ is a set of nodes, and $f\; :\; a\; \backslash mapsto\; (b\; \backslash times\; b)$ chooses pairs of elements of $b$ for each input from $a$. That is, $f$ picks out certain edges from the set $b\; \backslash times\; b$ of possible edges. A morphism in this category is made up of two functions, one on the indexing set and one on the node set. They must "agree" according to the general definition above, meaning that $(g,\; h)\; :\; (a,\; b,\; f)\; \backslash mapsto\; (a\text{'},\; b\text{'},\; f\text{'})$ must satisfy $f\text{'}\; \backslash circ\; g\; =\; S(h)\; \backslash circ\; f$. In other words, the edge corresponding to a certain element of the indexing set, when translated, must be the same as the edge for the translated index. * Many "augmentation" or "labelling" operations can be expressed in terms of comma categories. Let $T$ be the functor taking each graph to the set of its edges, and let $A$ be (a functor selecting) some particular set: then $(T\; \backslash downarrow\; A)$ is the category of graphs whose edges are labelled by elements of $A$. This form of comma category is often called ''objects $T$-over $A$'' - closely related to the "objects over $A$" discussed above. Here, each object takes the form $(B,\; \backslash pi\_B)$, where $B$ is a graph and $\backslash pi\_B$ a function from the edges of $B$ to $A$. The nodes of the graph could be labelled in essentially the same way. * A category is said to be ''locally cartesian closed'' if every slice of it is cartesian closed (see above for the notion of ''slice''). Locally cartesian closed categories are the classifying category classifying categories of dependent type theory dependent type theories.

Limits and universal morphisms
Colimits in comma categories may be "inherited". There is a theorem which says that if $\backslash mathcal\{A\}$ and $\backslash mathcal\{B\}$ are cocomplete, $T\; :\; \backslash mathcal\{A\}\; \backslash rightarrow\; \backslash mathcal\{C\}$ is a cocontinuous functor, and $S\; :\; \backslash mathcal\{B\}\; \backslash rightarrow\; \backslash mathcal\{C\}$ another functor (not necessarily cocontinuous), then the comma category $(T\; \backslash downarrow\; S)$ produced will also be cocomplete. For example, in the above construction of the category of graphs, the category of sets is cocomplete, and the identity functor is cocontinuous: so graphs are also cocomplete - all (small) limits exist. This result is much harder to obtain directly. See Limit (category theory) for more information on the terminology used in this example. The notion of a Universal property universal morphism to a particular colimit, or from a limit, can be expressed in terms of a comma category. Essentially, we create a category whose objects are cones, and where the limiting cone is a terminal object; then, each universal morphism for the limit is just the morphism to the terminal object. This works in the dual case, with a category of cocones having an initial object. For example, let $\backslash mathcal\{C\}$ be a category with $F\; :\; \backslash mathcal\{C\}\; \backslash rightarrow\; \backslash mathcal\{C\}\; \backslash times\; \backslash mathcal\{C\}$ the functor taking each object $c$ to $(c,\; c)$ and each arrow $f$ to $(f,\; f)$. A universal morphism from $(a,\; b)$ to $F$ consists, by definition, of an object $(c,\; c)$ and morphism $\backslash rho\; :\; (a,\; b)\; \backslash rightarrow\; (c,\; c)$ with the universal property that for any morphism $\backslash rho\text{'}\; :\; (a,\; b)\; \backslash rightarrow\; (d,\; d)$ there is a unique morphism $\backslash sigma\; :\; c\; \backslash rightarrow\; d$ with $F(\backslash sigma)\; \backslash circ\; \backslash rho\; =\; \backslash rho\text{'}$. In other words, it is an object in the comma category $((a,\; b)\; \backslash downarrow\; F)$ having a morphism to any other object in that category; it is initial. This serves to define the coproduct in $\backslash mathcal\{C\}$, when it exists.

Adjunctions
Lawvere showed that the functors $F\; :\; \backslash mathcal\{C\}\; \backslash rightarrow\; \backslash mathcal\{D\}$ and $G\; :\; \backslash mathcal\{D\}\; \backslash rightarrow\; \backslash mathcal\{C\}$ are adjoint functors adjoint if and only if the comma categories $(F\; \backslash downarrow\; \backslash mathcal\{D\})$ and $(\backslash mathcal\{C\}\; \backslash downarrow\; G)$ are isomorphic, and equivalent elements in the comma category can be projected onto the same element of $\backslash mathcal\{C\}\; \backslash times\; \backslash mathcal\{D\}$. This allows adjunctions to be described without involving sets, and was in fact the original motivation for introducing comma categories. Category:Category theory de:Kommakategorie

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