category theory

# Contents

## Definition

For $C$ a locally small category, its hom-functor is the functor

$\mathrm{hom}:{C}^{\mathrm{op}}×C\to \mathrm{Set}$hom : C^{op} \times C \to Set

from the product category of the category $C$ with its opposite category to the category Set of sets, which sends

• an object $\left(c,c\prime \right)\in {C}^{\mathrm{op}}×C$, i.e. a pair of objects in $C$, to the hom-set ${\mathrm{Hom}}_{C}\left(c,c\prime \right)$ in $C$, the set of morphisms $q:c\to c\prime$ in $C$;

• a morphism $\left(c,c\prime \right)\stackrel{}{\to }\left(d,d\prime \right)$, i.e. a pair of morphisms

$\begin{array}{cc}c& c\prime \\ {↑}^{f}& {↓}^{g}\\ d& d\prime \end{array}$\array{ c & c' \\ \uparrow^{\mathrlap{f}} & \downarrow^{\mathrlap{g}} \\ d & d' }

in $C$ to the function ${\mathrm{Hom}}_{C}\left(c,c\prime \right)\to {\mathrm{Hom}}_{C}\left(d,d\prime \right)$ that sends

$\left(q:c\stackrel{}{\to }c\prime \right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↦\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\left(g\circ q\circ f\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}\begin{array}{ccc}c& \stackrel{q}{\to }& c\prime \\ {↑}^{f}& & {↓}^{g}\\ d& & d\prime \end{array}\right)\phantom{\rule{thinmathspace}{0ex}}.$(q : c \stackrel{}{\to} c') \;\;\; \mapsto \;\;\, \left( g \circ q \circ f \; : \; \array{ c &\stackrel{q}{\to}& c' \\ \uparrow^{\mathrlap{f}} && \downarrow^{\mathrlap{g}} \\ d && d' } \right) \,.

More generally, for $V$ a closed symmetric monoidal category and $C$ a $V$-enriched category, its hom-functor is the functor

$C\left(-,-\right):{C}^{\mathrm{op}}×C\to V$C(-,-) : C^{op} \times C \to V

that sends objects $c,c\prime \in C$ to the hom-object $C\left(c,c\prime \right)\in V$.

Some categories $C$ are equipped with an operation that behaves like a hom-functor, but takes values in $C$ itself

$\left[-,-\right]:{C}^{\mathrm{op}}×C\to C\phantom{\rule{thinmathspace}{0ex}}.$[-,-] : C^{op} \times C \to C \,.

Such an operation is called an internal hom functor, and categories carrying this are called closed categories.

## Properties

### Representable functors

Given a hom-functor $\mathrm{hom}:{C}^{\mathrm{op}}×C\to \mathrm{Set}$, for any object $c\in C$ one obtains a functor

${h}^{c}:C\to \mathrm{Set}$h^c: C \to Set

given by ${h}^{c}≔\mathrm{hom}\left(c,-\right)$ and a functor

${h}_{c}:{C}^{\mathrm{op}}\to \mathrm{Set}$h_c : C^{op} \to Set

given by ${h}_{c}≔\mathrm{hom}\left(-,c\right)$, i.e. by fixing one of the arguments of $\mathrm{hom}:{C}^{\mathrm{op}}×C\to \mathrm{Set}$ to be $c$.

Formally this is

$\mathrm{hom}\left(c,-\right):C\stackrel{\simeq }{\to }*×C\stackrel{\left(c,\mathrm{Id}\right)}{\to }{C}^{\mathrm{op}}×C\stackrel{\mathrm{hom}\left(-,-\right)}{\to }\mathrm{Set}$hom(c,-) : C \stackrel{\simeq}{\to} * \times C \stackrel{(c,Id)}{\to} C^{op} \times C \stackrel{hom(-,-)}{\to} Set

and

$\mathrm{hom}\left(-,c\right):C\stackrel{\simeq }{\to }{C}^{\mathrm{op}}×*\stackrel{\left(\mathrm{Id},c\right)}{\to }{C}^{\mathrm{op}}×C\stackrel{\mathrm{hom}\left(-,-\right)}{\to }\mathrm{Set}\phantom{\rule{thinmathspace}{0ex}}.$hom(-,c) : C \stackrel{\simeq}{\to} C^{op} \times * \stackrel{(Id,c)}{\to} C^{op} \times C \stackrel{hom(-,-)}{\to} Set \,.

Functors of the form ${C}^{\mathrm{op}}\to \mathrm{Set}$ are called presheaves on $C$, and functors equivalent to $\mathrm{hom}\left(-,c\right)$ are called representable functors or representable presheaves on $C$.

Functors of the form $C\to \mathrm{Set}$ are called copresheaves on $C$, and functors equivalent to $\mathrm{hom}\left(c,-\right)$ are called corepresentable functors or representable copresheaves on $C$.

### Preservation of limits

The hom-functor preserves all limits in both arguments separately. This means:

• for fixed object $c\in C$ the functor $\mathrm{hom}\left(c,-\right):C\to \mathrm{Set}$ sends limit diagrams in $C$ to limit diagrams in $\mathrm{Set}$;

• for fixed object $c\prime \in C$ the functor $\mathrm{hom}\left(-,c\prime \right):{C}^{\mathrm{op}}\to \mathrm{Set}$ sends limit diagrams in ${C}^{\mathrm{op}}$ – which are colimit diagrams in $C$! – to limit diagrams in $\mathrm{Set}$.

For instance for

$\begin{array}{ccc}y{×}_{x}z& \to & y\\ ↓& & ↓\\ z& \to & x\end{array}$\array{ y \times_x z &\to& y \\ \downarrow && \downarrow \\ z &\to& x }

a pullback diagram in $C$ and for $c\in C$ any object, the induced diagram

$\begin{array}{cccc}{\mathrm{Hom}}_{C}\left(c,y\right){×}_{{\mathrm{Hom}}_{C}\left(c,x\right)}{\mathrm{Hom}}_{C}\left(c,z\right)\simeq & {\mathrm{Hom}}_{C}\left(c,y{×}_{x}z\right)& \to & {\mathrm{Hom}}_{C}\left(c,y\right)\\ & ↓& & ↓\\ & {\mathrm{Hom}}_{C}\left(c,z\right)& \to & {\mathrm{Hom}}_{C}\left(c,x\right)\end{array}$\array{ Hom_C(c,y) \times_{Hom_C(c,x)} Hom_C(c,z)\simeq & Hom_C(c,y \times_x z) &\to& Hom_C(c,y) \\ & \downarrow && \downarrow \\ & Hom_C(c,z) &\to& Hom_C(c,x) }

in Set is again a pullback diagram. A moment of reflection shows that this statement is equivalent to the very definition of limit.

### Relation to profunctors

The hom-functor $\mathrm{hom}:{C}^{\mathrm{op}}×C\to \mathrm{Set}$ is also the identity profunctor ${1}_{C}:C⇸C$.

One way to see this is to notice that its adjunct

$C\to \left[{C}^{\mathrm{op}},\mathrm{Set}\right]$C \to [C^{op}, Set]

under the internal hom adjunction in the 1-category Cat is the functor

$C\stackrel{\mathrm{id}}{\to }C\stackrel{j}{\to }\left[{C}^{\mathrm{op}},\mathrm{Set}\right]\phantom{\rule{thinmathspace}{0ex}},$C \stackrel{id}{\to} C \stackrel{j}{\to} [C^{op}, Set] \,,

where $j$ is the Yoneda embedding. Profunctors $F:{C}^{\mathrm{op}}×C\to \mathrm{Set}$ whose hom-adjunct is of the form $C\stackrel{F}{\to }C\stackrel{j}{\to }\left[{C}^{\mathrm{op}},\mathrm{Set}\right]$ for $F$ an ordinary functor are those in the inclusion of these ordinary functors into profunctors. So the hom-functor is the image of the identity functor under this inclusion.

## Examples

homotopycohomologyhomology
$\left[{S}^{n},-\right]$$\left[-,A\right]$$\left(-\right)\otimes A$
category theorycovariant homcontravariant homtensor product
homological algebraExtExtTor
enriched category theoryendendcoend
homotopy theoryderived hom space $ℝ\mathrm{Hom}\left({S}^{n},-\right)$cocycles $ℝ\mathrm{Hom}\left(-,A\right)$derived tensor product $\left(-\right){\otimes }^{𝕃}A$

Revised on May 7, 2013 10:00:56 by Anonymous Dragon? (201.124.238.159)