PlanetMath: comma category

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (232)
Orphanage
Unclass'd
Unproven (519)
Corrections (23)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

comma category

(Definition)

Naive notion of a comma category

Let $\mathcal{C}$ be a category and two subcategories

and

of $\mathcal{C}$ . We form a category

as follows:

The objects of are of the form , where is an object in , is an object in , and is a morphism (in $\mathcal{C}$ ) from to :
$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{a \ar[d]_f \\ b} } \end{xy}$
The morphisms of are of the form $(x,y):(a,b,f)\to (c,d,g)$ , where $x:a\to c$ and $y:b\to d$ are morphisms, such that
$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{a \ar[d]_f \ar[r]^x & c \ar[d]^g \ b \ar[r]^y & d} } \end{xy}$
is a commutative diagram: .

It is easy to check that is indeed a category. For example, given object , is the corresponding identity morphism. Furthermore, if we have the following two commutative diagrams

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{a \ar[d]_f \ar[r]^x & c \ar[d]^g ... ...qquad \xymatrix{c \ar[d]_g \ar[r]^i & m \ar[d]^h \ d \ar[r]^j & n} } \end{xy}$

we may combine them and form the following commutative diagram

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{a \ar[d]_f \ar[r]^x & c \ar[d]_g \ar[r]^i & m \ar[d]^h \ b \ar[r]^y & d \ar[r]^j & n} } \end{xy}$

which shows that

, so that $(ix,jy)\in (D_1,D_2)$ is the composition of

and

Definition. , so constructed, is called a comma category (the comma between and ), or a slice category.

Examples. In the following examples, identity morphisms and compositions of morphisms are implicitly assumed to be included in the subcategories.

and each consists of a single object of $\mathcal{C}$ , say , then is just $\hom(a,b)$ . The objects of $\hom(a,b)$ are just morphisms $a\to b$ , and the morphism of $\hom(a,b)$ is . $\hom(a,b)$ is a discrete category.
consists of an object in $\mathcal{C}$ and $D_2= \mathcal{C}$ . Then the comma category, written $(a,\mathcal{C})$ or $(a\downarrow \mathcal{C})$ , may be visualized as a “cone” with apex and base $\mathcal{C}$ .
Similarly, we can form an “inverted cone” $(\mathcal{C},b)$ or $(\mathcal{C}\downarrow b)$ .
Taking $D_1=D_2=\mathcal{C}$ , then the comma category $(\mathcal{C}, \mathcal{C})$ is the arrow category of $\mathcal{C}$ whose objects are morphisms of $\mathcal{C}$ and morphisms can be identified with commutative squares in $\mathcal{C}$ .

Formal definition of a comma category

The diagrams

, can be joined to the original category $\mathcal{C}$ via the inclusion functors

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{D_1 \ar[r]^{I_1} & \mathcal{C} & D_2 \ar[l]_{I_2} } } \end{xy}$

which suggests that a comma category may be more generally defined in terms of a pair of categories $\mathcal{A},\mathcal{B}$ , and a pair of functors

into a certain given category $\mathcal{C}$ . Specifically, let $\mathcal{A}$ and $\mathcal{B}$ be categories and $F: \mathcal{A}\to \mathcal{C}$ and $G:\mathcal{B}\to \mathcal{C}$ be functors into a specific category $\mathcal{C}$ . A comma category of the diagram

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{\mathcal{A} \ar[r]^{F} & \mathcal{C} & \mathcal{B} \ar[l]_{G} } } \end{xy}$

written

or $(F\downarrow G)$ , consists of the following:

objects have the form , where
1. is an object of $\mathcal{A}$ ,
2. is an object of $\mathcal{B}$ , and
3. $f:F(a)\to G(b)$ is a morphism in $\mathcal{C}$ ;
  $\displaystyle \begin{xy} *!C\xybox{ \xymatrix{F(a) \ar[d]_f \\ G(b)} } \end{xy}$
morphisms from to have the form , where
1. $x:a\to c$ is a morphism of $\mathcal{A}$ ,
2. $y:b\to d$ is a morphism of $\mathcal{B}$ , such that
3. the following diagram
  $\displaystyle \xymatrix@+=2cm{F(a) \ar[d]_f \ar[r]^{F(x)} & F(c) \ar[d]^g \ G(b) \ar[r]^{G(y)} & G(d)}$
  is commutative: .
morphism composition in $(F\downarrow G)$ is given by $(x_2,y_2)\circ (x_1,y_1):=(x_2\circ x_1, y_2\circ y_1)$ , where $(x_1,y_1):(a_1,b_1,f_1)\to (a_2,b_1,f_2)$ and $(x_2,y_2):(a_2,b_2,f_2)\to (a_3,b_3,f_3)$ .

It is an easy exercise to verify that indeed, $(F\downarrow G)$ is a category. For example, the identity morphism on

is provided by the morphism

Remark. If $\mathcal{A}$ and $\mathcal{B}$ happen to be subcategories of $\mathcal{C}$ and are the inclusion functors, then we may write as $(\mathcal{A},\mathcal{B})$ or $(\mathcal{A}\downarrow \mathcal{B})$ .