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 $g:X\to X\prime \in C/c$ (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

$$C/(-):C\to \mathrm{Cat},$$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 $C$. (Note that unless $C$ has pullbacks, this functor is not actually a fibration, though it is always an opfibration.)

Adjunctions on overcategories

Proposition

Let

$$(L⊣R):D\stackrel{\stackrel{L}{\leftarrow }}{\underset{R}{\to }}C$$(L \dashv R) : D \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} C

be a pair of adjoint functors, where the category $C$ has all pullbacks.

Then for every object $X\in C$ there is induced a pair of adjoint functors between the slice categories

$$(L/X⊣R/X):D/(LX)\stackrel{\stackrel{L/X}{\leftarrow }}{\underset{R/X}{\to }}C/X$$(L/X \dashv R/X) : D/(L X) \stackrel{\overset{L/X}{\leftarrow}}{\underset{R/X}{\to}} C/X

where

• $L/X$ is the evident induced functor;

• $R/X$ is the composite

  $$R/X:D/LX\stackrel{R}{\to }C/(RLX)\stackrel{i_X^{*}}{\to }C/X$$R/X : D/{L X} \stackrel{R}{\to} C/{(R L X)} \stackrel{i_{X}^*}{\to} C/X

  of the evident functor induced by $R$ with the pullback along the $(L⊣R)$-unit at $X$.

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.

Proposition

There is an equivalence of categories

$$e:\mathrm{PSh}(C/c)\stackrel{\simeq }{\to }\mathrm{PSh}(C)/Y(c).$$e : PSh(C/c) \stackrel{\simeq}{\to} PSh(C)/Y(c) \,.

Proof

The functor $e$ takes $F\in \mathrm{PSh}(C/c)$ to the presheaf $F\prime :d\mapsto {\bigsqcup }_{f\in C(d,c)}F(f)$ which is equipped with the natural transformation $\eta :F\prime \to Y(c)$ with component map ${\eta }_d{\bigsqcup }_{f\in C(d,c)}F(f)\to C(d,c)$.

A weak inverse of $e$ is given by the functor

$$\overline{e}:\mathrm{PSh}(C)/Y(c)\to \mathrm{PSh}(C/c)$$\bar e : PSh(C)/Y(c) \to PSh(C/c)

which sends $\eta :F\prime \to Y(C))$ to $F\in \mathrm{PSh}(C/c)$ given by

$$F:(f:d\to c)\mapsto F\prime (d){\mid }_c,$$F : (f : d \to c) \mapsto F'(d)|_c \,,

where $F\prime (d){\mid }_c$ is the pullback

$$\begin{array}{ccc}F\prime (d){\mid }_c& \to & F\prime (d)\\ \downarrow & & {\downarrow }^{{\eta }_d}\\ \mathrm{pt}& \stackrel{f}{\to }& C(d,c)\end{array}.$$\array{ F'(d)|_c &\to& F'(d) \\ \downarrow && \downarrow^{\eta_d} \\ pt &\stackrel{f}{\to}& C(d,c) } \,.

Example

Suppose the presheaf $F\in \mathrm{PSh}(C/c)$ does not actually depend on the morphsims to $C$, i.e. suppose that it factors through the forgetful functor from the over category to $C$:

$$F:(C/c{)}^{\mathrm{op}}\to C^{\mathrm{op}}\to \mathrm{Set}.$$F : (C/c)^{op} \to C^{op} \to Set \,.

Then $F\prime (d)={\bigsqcup }_{f\in C(d,c)}F(f)={\bigsqcup }_{f\in C(d,c)}F(d)\simeq C(d,c)\times F(d)$ and hence $F\prime =Y(c)\times F$ with respect to the closed monoidal structure on presheaves.

See also functors and comma categories.

For the analog statement in (∞,1)-category theory see (∞,1)-category of (∞,1)-presheaves -- Interaction with overcategories

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

Limits and colimits

Proposition

A limit in an under category is computed as a limit in the underlying category.

Precisely: let $C$ be a category, $t\in C$ an object, and $t/C$ the corresponding under category, and $p:t/C\to C$ the obvious projection.

Let $F:D\to t/C$ be any functor. Then, if it exists, the limit over $p\circ F$ in $C$ is the image under $p$ of the limit over $F$:

$$p(\lim F)\simeq \lim (pF)$$p(\lim F) \simeq \lim (p F)

and $\lim F$ is uniquely characterized by $\lim (pF)$.

Proof

Over a morphism $\gamma :d\to d\prime$ in $D$ the limiting cone over $pF$ (which exists by assumption) looks like

$$\begin{array}{ccc}& & \lim pF\\ & \swarrow & & \searrow \\ pF(d)& & \stackrel{pF(\gamma )}{\to }& & pF(d\prime )\end{array}$$\array{ && \lim p F \\ & \swarrow && \searrow \\ p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d') }

By the universal property of the limit this has a unique lift to a cone in the under category $t/C$ over $F$:

$$\begin{array}{ccc}& & t\\ & \swarrow & \downarrow & \searrow \\ & & \lim pF\\ \downarrow & \swarrow & & \searrow & \downarrow \\ pF(d)& & \stackrel{pF(\gamma )}{\to }& & pF(d\prime )\end{array}$$\array{ && t \\ & \swarrow &\downarrow & \searrow \\ && \lim p F \\ \downarrow & \swarrow && \searrow & \downarrow \\ p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d') }

It therefore remains to show that this is indeed a limiting cone over $F$. Again, this is immediate from the universal property of the limit in $C$. For let $t\to Q$ be another cone over $F$ in $t/C$, then $Q$ is another cone over $pF$ in $C$ and we get in $C$ a universal morphism $Q\to \lim pF$

$$\begin{array}{ccc}& & t\\ & \swarrow & \downarrow & \searrow \\ & & Q\\ \downarrow & \swarrow & \downarrow & \searrow & \downarrow \\ & & \lim pF\\ \downarrow & \swarrow & & \searrow & \downarrow \\ pF(d)& & \stackrel{pF(\gamma )}{\to }& & pF(d\prime )\end{array}$$\array{ && t \\ & \swarrow & \downarrow & \searrow \\ && Q \\ \downarrow & \swarrow &\downarrow & \searrow & \downarrow \\ && \lim p F \\ \downarrow & \swarrow && \searrow & \downarrow \\ p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d') }

A glance at the diagram above shows that the composite $t\to Q\to \lim pF$ constitutes a morphism of cones in $C$ into the limiting cone over $pF$. Hence it must equal our morphism $t\to \lim pF$, by the universal property of $\lim pF$, and hence the above diagram does commute as indicated.

This shows that the morphism $Q\to \lim pF$ which was the unique one giving a cone morphism on $C$ does lift to a cone morphism in $t/C$, which is then necessarily unique, too. This demonstrates the required universal property of $t\to \lim pF$ and thus identifies it with $\lim F$.

• Remark: One often says ”$p$ reflects limits” to express the conclusion of this proposition. A conceptual way to consider this result is by appeal to a more general one: if $U:A\to C$ is monadic (i.e., has a left adjoint $F$ such that the canonical comparison functor $A\to (UF)-\mathrm{Alg}$ is an equivalence), then $U$ both reflects and preserves limits. In the present case, the projection $p:A=t/C\to C$ is monadic, is essentially the category of algebras for the monad $T(-)=t+(-)$, at least if $C$ admits binary coproducts. (Added later: the proof is even simpler: if $U:A\to C$ is the underlying functor for the category of algebras of an endofunctor on $C$ (as opposed to algebras of a monad), then $U$ reflects and preserves limits; then apply this to the endofunctor $T$ above.)

Proposition

For $C$ a category, $X:D\to C$ a diagram, $C/X$ the comma category (over-category if $D$ is the point) and $F:K\to C/X$ a diagram in the comma category, then the limit ${\lim }_{\leftarrow }F$ in $C/X$ coincides with the limit ${\lim }_{\leftarrow }F/X$ in $C$.

For a proof see at (∞,1)-limit here.

Initial and terminal objects

As a special case of the above discussion of limits and colimits in a slice ${𝒞}_{/X}$ we obtain the following statement, which of course is also immediately checked explicitly.

Corollary

• If $𝒞$ has an initial object $\varnothing$, then ${𝒞}_{/X}$ has an initial object, given by $⟨\varnothing \to X⟩$.

• The terminal object ${𝒞}_{/X}$ is ${\mathrm{id}}_X$.

Related concepts

• over-category

  • slice category

  • under category

  • over topos

• over (∞,1)-category,

  • model structure on an over category

  • over-(∞,1)-topos