PlanetMath: comma category

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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. $x:F(a)\to F(c)$ is a morphism of $\mathcal{C}$ ,
2. $y:F(b)\to F(d)$ is a morphism of $\mathcal{C}$ ,
3. the following diagram
  $\displaystyle \begin{xy} *!C\xybox{ \xymatrix{F(a) \ar[d]_f \ar[r]^x & F(c) \ar[d]^g \ G(b) \ar[r]^y & G(d)} } \end{xy}$
  is commutative: .