Let $\mathcal{C}$ be a category and two subcategories $D_1$ and $D_2$ of $\mathcal{C}$ . We form a category $(D_1,D_2)$ as follows:
- The objects of $(D_1,D_2)$ are of the form $(a,b,f)$ , where $a$ is an object in $D_1$ , $b$ is an object in $D_2$ , and $f$ is a morphism (in $\mathcal{C}$ ) from $a$ to $b$ :
- The morphisms of $(D_1,D_2)$ 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
is a commutative diagram: $yf = gx$ .
It is easy to check that $(D_1,D_2)$ is indeed a category. For example, given object $(a,b,f)$ , $(1_a,1_b)$ is the corresponding identity morphism. Furthermore, if we have the following two commutative diagrams
we may combine them and form the following commutative diagram
which shows that $h(ix)=(jy)f$ , so that $(ix,jy)\in (D_1,D_2)$ is the composition of $(x,y)$ and $(i,j)$ .
Definition. $(D_1,D_2)$ , so constructed, is called a comma category (the comma between $D_1$ and $D_2$ ), or a slice category.
Examples. In the following examples, identity morphisms and compositions of morphisms are implicitly assumed to be included in the subcategories.
- $D_1$ and $D_2$ each consists of a single object of $\mathcal{C}$ , say $a,b$ , then $(D_1,D_2)$ 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 $(1_a,1_b)$ . $\hom(a,b)$ is a discrete category.
- $D_1$ consists of an object $a$ 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 $a$ 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 $\mathcal{C}$ .
The diagrams $D_1,D_2$ , can be joined to the original category $\mathcal{C}$ via the inclusion functors $I_1,I_2$ :
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 $F,G$ 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
written $(F,G)$ or $(F\downarrow G)$ , consists of the following:
- objects have the form $(a,b,f)$ , where
- $a$ is an object of $\mathcal{A}$ ,
- $b$ is an object of $\mathcal{B}$ , and
- $f:F(a)\to G(b)$ is a morphism in $\mathcal{C}$ ;
- morphisms from $(a,b,f)$ to $(c,d,g)$ have the form $(x,y)$ , where
- $x:a\to c$ is a morphism of $\mathcal{A}$ ,
- $y:b\to d$ is a morphism of $\mathcal{B}$ , such that
- the following diagram
is commutative: $F(y)f=gF(x)$ .
- 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 $(a,b,f)$ is provided by the morphism $(1_a,1_b)$ .
Remark. If $\mathcal{A}$ and $\mathcal{B}$ happen to be subcategories of $\mathcal{C}$ and $F,G$ are the inclusion functors, then we may write $(F,G)$ as $(\mathcal{A},\mathcal{B})$ or $(\mathcal{A}\downarrow \mathcal{B})$ .
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