hom-functor in nLab

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Definition

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

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

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

• an object (c,c′)∈C op×C(c, c') \in C^{op} \times C, i.e. a pair of objects in CC, to the hom-set Hom C(c,c′)Hom_C(c,c') in Set, the set of morphisms q:c→c′q : c \to c' in CC;

• a morphism (c,c′)→(d,d′)(c,c') \stackrel{}{\to} (d,d'), i.e. a pair of morphisms

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

  in CC to the function Hom C(c,c′)→Hom C(d,d′)Hom_C(c,c') \to Hom_C(d,d') that sends

  (q:c→c′)↦(g∘q∘f:c →q c′ ↑ f ↓ g d d′). (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 VV a closed symmetric monoidal category and CC a VV-enriched category, its enriched hom-functor is the enriched functor

C(−,−):C op×C→V C(-,-) : C^{op} \times C \to V

that sends objects c,c′∈Cc,c' \in C to the hom-object C(c,c′)∈VC(c,c') \in V.

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

[−,−]:C op×C→C. [-,-] : C^{op} \times C \to C \,.

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

In homotopy type theory

Note: the HoTT book calls a category a “precategory” and a univalent category a “category”, but here we shall refer to the standard terminology of “category” and “univalent category” respectively.

For any category AA, we have a hom-functor

hom A:A op×A→Sethom_A : A^{op} \times A \to \mathit{Set}

It takes a pair (a,b):(A op) 0×A 0≡A 0×A 0(a,b):(A^{op})_0 \times A_0 \equiv A_0\times A_0 to the set hom A(a,b)hom_A(a,b). For a morphism (f,f′):hom A op×A((a,b),(a′,b′))(f,f') : hom_{A^{op} \times A}((a,b),(a',b')), by definition we have f:hom A(a′,a)f:hom_A(a',a) and f′:hom A(b,b′)f': hom_A(b,b'), so we can define

(hom A) (a,b),(a′,b′)(f,f′)≡(g↦f′gf):hom A(a,b)→hom A(a′,b′)(hom_A)_{(a,b),(a',b')}(f,f') \equiv (g\mapsto f' g f) : hom_A(a,b) \to hom_A(a',b')

Properties

Representable functors

Given a hom-functor hom:C op×C→Sethom:C^{op}\times C\to Set, for any object c∈Cc \in C one obtains a functor

h c:C→Set h^c: C \to Set

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

h c:C op→Set h_c : C^{op} \to Set

given by h c≔hom(−,c)h_c\coloneqq hom(-,c), i.e. by fixing one of the arguments of hom:C op×C→Sethom: C^{op} \times C \to Set to be cc.

Formally this is

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

and

hom(−,c):C→≃C op×*→(Id,c)C op×C→hom(−,−)Set. 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→SetC^{op} \to Set are called presheaves on CC, and functors naturally isomorphic to hom(−,c)hom(-,c) are called representable functors or representable presheaves on CC.

Functors of the form C→SetC \to Set are called copresheaves on CC, and functors naturally isomorphic to hom(c,−)hom(c,-) are called corepresentable functors or representable copresheaves on CC.

Preservation of limits

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

• for fixed object c∈Cc \in C the functor hom(c,−):C→Sethom(c,-) : C \to Set sends limit diagrams in CC to limit diagrams in SetSet;

• for fixed object c′∈Cc' \in C the functor hom(−,c′):C op→Sethom(-,c') : C^{op} \to Set sends limit diagrams in C opC^{op} – which are colimit diagrams in CC! – to limit diagrams in SetSet.

For instance for

y× xz → y ↓ ↓ z → x \array{ y \times_x z &\to& y \\ \downarrow && \downarrow \\ z &\to& x }

a pullback diagram in CC and for c∈Cc \in C any object, the induced diagram

Hom C(c,y)× Hom C(c,x)Hom C(c,z)≃ Hom C(c,y× xz) → Hom C(c,y) ↓ ↓ Hom C(c,z) → Hom C(c,x) \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×C→Sethom : C^{op}\times C\to Set is also the identity profunctor 1 C:C⇸C1_C: C ⇸ C.

One way to see this is to notice that its adjunct

C→[C op,Set] C \to [C^{op}, Set]

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

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

where jj is the Yoneda embedding. Profunctors F:C op×C→Set\mathbf{F} : C^{op} \times C \to Set whose hom-adjunct is of the form C→FC→j[C op,Set]C \stackrel{F}{\to} C \stackrel{j}{\to} [C^{op}, Set] for FF 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

…

• hom-set, hom-object

• hom-functor

• enriched hom-functor, internal hom-functor

• derived hom-functor

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