Contents

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 $f\prime \circ g=f$.

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

Relation to codomain fibration

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

Under the Grothendieck construction this functor corresponds the the codomain fibration

from the arrow category of $C$.

Presheaves on over-categories and over-categories of presheaves

Let $C$ be a category, $c$ an object of $C$ and let $C/c$ be the over category of $C$ over $c$. Write $\mathrm{PSh}(C/c)=[(C/C{)}^{\mathrm{op}},\mathrm{Set}]$ for the category of presheaves on $C/c$ and write $\mathrm{PSh}(C)/Y(y)$ for the over category of presheaves on $C$ over the presheaf $Y(c)$, where $Y:C\to \mathrm{PSh}(c)$ is the Yoneda embedding.

See also functors and comma categories.

For the analog statement in (∞,1)-category theory see

at (∞,1)-category of (∞,1)-presheaves.

In higher category theory

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.