Contents

Definition

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

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

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

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

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

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

  $$(q:c\stackrel{}{\to }c\prime )\mapsto (g\circ q\circ f:\begin{array}{ccc}c& \stackrel{q}{\to }& c\prime \\ {\uparrow }^f& & {\downarrow }^g\\ d& & d\prime \end{array}).$$(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

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

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

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

Properties

Representable functors

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

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

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

Formally this is

and

Functors of the form $C^{\mathrm{op}}\to \mathrm{Set}$ are called presheaves on $C$, and functors equivalent to $\mathrm{hom}(-,c)$ 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 $\mathrm{hom}(c,-)$ 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 $\mathrm{hom}(c,-):C\to \mathrm{Set}$ sends limit diagrams in $C$ to limit diagrams in $\mathrm{Set}$;

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

For instance for

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

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 $\mathrm{hom}:C^{\mathrm{op}}\times C\to \mathrm{Set}$ is also the identity profunctor $1_C:C⇸C$.

One way to see this is to notice that its adjunct

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

where $j$ is the Yoneda embedding. Profunctors $F:C^{\mathrm{op}}\times C\to \mathrm{Set}$ whose hom-adjunct is of the form $C\stackrel{F}{\to }C\stackrel{j}{\to }[C^{\mathrm{op}},\mathrm{Set}]$ 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

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Related concepts

	homotopy	cohomology	homology
	$[S^n,-]$	$[-,A]$	$(-)\otimes A$
category theory	covariant hom	contravariant hom	tensor product
homological algebra	Ext	Ext	Tor
enriched category theory	end	end	coend
homotopy theory	derived hom space $ℝ\mathrm{Hom}(S^n,-)$	cocycles $ℝ\mathrm{Hom}(-,A)$	derived tensor product $(-){\otimes }^{𝕃}A$

• internal hom, enriched hom, derived hom space, Ext