category theory

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Definition

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

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

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

• an object $(c, c') \in C^{op} \times C$, 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

$\array{ c & c' \\ \downarrow^{\mathrlap{f^{op}}} & \downarrow^{\mathrlap{g}} \\ d & d' }$

in $C$ to the function $Hom_C(c,c') \to Hom_C(d,d')$ that sends

$(q : c \stackrel{}{\to} c') \;\;\; \mapsto \;\;\, \left( g \circ q \circ f \; : \; \array{ c &\stackrel{q}{\to}& c' \\ \uparrow^{\mathrlap{f}} && \downarrow^{\mathrlap{g}} \\ d && d' } \right) \,.$

Note: when the symbol $\circ$ is used, it denotes traditional right-to-left order of composition. For those who prefer the left-to-right order, the symbol $;$ may be used in place of $\circ$. Further discussion of this should go to the nForum page here.

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

$C(-,-) : C^{op} \times C \to V$

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

$[-,-] : C^{op} \times C \to C \,.$

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

Properties

Representable functors

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

$h^c: C \to Set$

given by $h^c\coloneqq hom(c,-)$ and a functor

$h_c : C^{op} \to Set$

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

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

and

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

Functors of the form $C^{op} \to Set$ are called presheaves on $C$, and functors naturally isomorphic 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 naturally isomorphic to $hom(c,-)$ 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 $hom(c,-) : C \to Set$ 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

$\array{ y \times_x z &\to& y \\ \downarrow && \downarrow \\ z &\to& x }$

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

$\array{ Hom_C(c,y) \times_{Hom_C(c,x)} Hom_C(c,z)\simeq & Hom_C(c,y \times_x z) &\to& Hom_C(c,y) \\ & \downarrow && \downarrow \\ & Hom_C(c,z) &\to& Hom_C(c,x) }$

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

One way to see this is to notice that its adjunct

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

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

$C \stackrel{id}{\to} C \stackrel{j}{\to} [C^{op}, Set] \,,$

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.

Examples

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

Revised on August 16, 2016 07:51:28 by Todd Trimble (67.81.95.215)