Slice Category

Definition

Given a category $C$ and an object $A$ in $C$ , the slice category $C / A$ (also called the over category) has:

objects: pairs $(X, f)$ where $X$ is an object of $C$ and $f : X \to A$ is a morphism in $C$
morphisms: a morphism from $(X, f)$ to $(Y, g)$ is a morphism $h : X \to Y$ in $C$ such that the following triangle commutes:

g \circ h = f

or equivalently:

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
X \arrow[rr, "h"] \arrow[dr, "f"'] & & Y \arrow[dl, "g"] \\
& A &
\end{tikzcd}
\end{document}

Composition and identities are inherited from $C$ .

The coslice category (or under category) $A / C$ is defined dually, with objects $(X, f)$ where $f : A \to X$ .

The slice category $C / A$ has a forgetful functor to $C$ that sends $(X, f)$ to $X$
If $C$ has pullbacks, then $C / A$ has all limits that $C$ has
The terminal object in $C / A$ is $(id_{A} : A \to A)$
Slice categories play a central role in the theory of fibrations and toposes

In $Set$ , the slice category $Set / A$ has objects that are functions $f : X \to A$ , which can be viewed as $A$ -indexed families of sets
In a poset viewed as a category, $P / a$ consists of all elements $x \leq a$
The slice category $Cat / C$ consists of functors with codomain $C$