nLab
overcategory

Definition

The slice category or over category $C/c$ of a category $C$ over an object $c\in C$ has

• objects that are all arrows $f\in C$ such that $\mathrm{cod}(f)=c$, and
• morphisms $g:X\to X\prime \in C$ from $f:X\to c$ to $f\prime :X\prime \to c$ such that $f\prime \circ g=f$.

$$C/c=\left\{\begin{array}{ccccc}X& & \stackrel{g}{\to }& & X\prime \\ & {}_f\searrow & & {\swarrow }_{f\prime }\\ & & c\end{array}\right\}$$C/c = \left\lbrace \array{ X &&\stackrel{g}{\to}&& X' \\ & {}_f \searrow && \swarrow_{f'} \\ && c } \right\rbrace

The slice category is a special case of a comma category.

There is a forgetful functor $U_c:C/c\to C$ which maps an object $f:X\to c$ to its domain $X$ and a morphism $g:X\to X\prime \in C$ (from $f:X\to c$ to $f\prime :X\prime \to c$ such that $f\prime \circ g=f$) to the morphism $g:X\to X\prime$.

The dual notion is an under category.

Examples

• If $C=P$ is a poset and $p\in P$, then the slice category $P/p$ is the down set $\downarrow (p)$ of elements $q\in P$ with $q\le p$.

• If $1$ is a terminal object in $C$, then $C/1$ is isomorphic to $C$.

Properties

The assignment of overcategories $C/c$ to objects $c\in C$ extends to a functor

$$C/(-):C\to \mathrm{Cat},$$C/(-) : C \to Cat \,,

Under the Grothendieck construction this functor corresponds the the codomain fibration

$$\mathrm{cod}:[I,C]\to C$$cod : [I,C] \to C

from the arrow category of $C$.

Generalizations

• The notion of over category applicable to (∞,1)-categories is discussed at over quasi-category.

• Similarly, there is a notion of model structure on an over category.