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 $(c, c') \in C^{op} \times C$, i.e. a pair of objects in $C$, to the hom-set $Hom_C(c,c')$ in $C$, the set of morphisms $q : c \to c'$ in $C$;
a morphism $(c,c') \stackrel{}{\to} (d,d')$, i.e. a pair of morphisms
in $C$ to the function $Hom_C(c,c') \to Hom_C(d,d')$ that sends
More generally, for $V$ a closed symmetric monoidal category and $C$ a $V$-enriched category, its hom-functor is the functor
that sends objects $c,c' \in C$ to the hom-object $C(c,c') \in V$.
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.
Given a hom-functor $hom:C^{op}\times C\to Set$, for any object $c \in C$ one obtains a functor
given by $h^c\coloneqq hom(c,-)$ and a functor
given by $h_c\coloneqq hom(-,c)$, i.e. by fixing one of the arguments of $hom: C^{op} \times C \to Set$ to be $c$.
Formally this is
and
Functors of the form $C^{op} \to Set$ are called presheaves on $C$, and functors equivalent to $hom(-,c)$ are called representable functors or representable presheaves on $C$.
Functors of the form $C \to Set$ are called copresheaves on $C$, and functors equivalent to $hom(c,-)$ are called corepresentable functors or representable copresheaves on $C$.
The hom-functor preserves all limits in both arguments separately. This means:
for fixed object $c \in C$ the functor $hom(c,-) : C \to Set$ sends limit diagrams in $C$ to limit diagrams in $Set$;
for fixed object $c' \in C$ the functor $hom(-,c') : C^{op} \to Set$ sends limit diagrams in $C^{op}$ – which are colimit diagrams in $C$! – to limit diagrams in $Set$.
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.
The hom-functor $hom : C^{op}\times C\to Set$ 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 $\mathbf{F} : C^{op} \times C \to Set$ whose hom-adjunct is of the form $C \stackrel{F}{\to} C \stackrel{j}{\to} [C^{op}, Set]$ 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.
…
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 $\mathbb{R}Hom(S^n,-)$ | cocycles $\mathbb{R}Hom(-,A)$ | derived tensor product $(-) \otimes^{\mathbb{L}} A$ |