Slice Category

Definition

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

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

$$\begin{array}{r}g\circ h=f\end{array}$$

or equivalently:

Composition and identities are inherited from $\mathcal{C}$.

Dual Notion

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

Properties

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

Examples

• In $\mathbf{Set}$, the slice category $\mathbf{Set}/A$ 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, $\mathcal{P}/a$ consists of all elements $x\le a$
• The slice category $\mathbf{Cat}/\mathcal{C}$ consists of functors with codomain $\mathcal{C}$

Related Concepts

• Comma Category
• Pullback
• Fibration

References

• riehl2016-category-theory