nLab
under category

Definition

Given a category $C$ and an object $c\in C$, the under category (also called coslice category) $c\downarrow C$ (also written $c/C$ and sometimes, confusingly, $c\backslash C$) is the category whose

• objects are morphisms in $C$ starting at $c$; $c\to d$

• morphisms are commuting triangles $\begin{array}{ccc}& & c\\ & \swarrow & & \searrow \\ d_1& & \to & & d_2\end{array}.$

The under category $c\downarrow C$ is a kind of comma category; it is the strict pullback

$$\begin{array}{ccc}c\downarrow C& \to & \mathrm{pt}\\ \downarrow & & {\downarrow }^{\mathrm{pt}\mapsto c}\\ [I,C]& \stackrel{d_0}{\to }& C\end{array}$$\array{ c\downarrow C &\to& pt \\ \downarrow && \;\;\downarrow^{pt \mapsto c} \\ [I,C] &\stackrel{d_0}{\to}& C }

in Cat, where

• $I$ is the interval category $\{0\to 1\}$;

• $[I,C]$ is the internal hom in Cat, which here is the arrow category $\mathrm{Arr}(C)$;

• the functor $d_0$ is evaluation at the left end of the interval;

• $\mathrm{pt}$, the point, is the terminal category, the 0th oriental, the 0-globe;

• the right vertical morphism maps the single object of the point to the object $c$.

The left vertical morphism $c\downarrow C\to C$ is the forgetful morphism which forgets the tip of the triangles mentioned above.

The dual notion is an over category.

Examples

• ${\mathrm{Set}}_{*}$, the category of pointed sets, is the undercategory $\mathrm{pt}\downarrow \mathrm{Set}$, where $\mathrm{pt}\simeq \{•\}$ is the singleton set.

• The category of commutative algebras over a field $F$ is the category $F\downarrow$Ring of commutative rings under $F$.

Related concepts

• over-category

  • slice category

  • under category

  • over topos

• over (∞,1)-category,

  • model structure on an over category

  • over-(∞,1)-topos