nLab hom-functor




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

hom:C op×CSet 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:ccq : 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:cc)(gqf: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×CV C(-,-) : C^{op} \times C \to V

that sends objects c,cCc,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×CC. [-,-] : 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×ASethom_A : A^{op} \times A \to \mathit{Set}

It takes a pair (a,b):(A op) 0×A 0A 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)(gfgf):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')


Representable functors

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

h c:CSet h^c: C \to Set

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

h c:C opSet h_c : C^{op} \to Set

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

Formally this is

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


hom(,c):CC op×*(Id,c)C op×Chom(,)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 opSetC^{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 CSetC \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 cCc \in C the functor hom(c,):CSethom(c,-) : C \to Set sends limit diagrams in CC to limit diagrams in SetSet;

  • for fixed object cCc' \in C the functor hom(,c):C opSethom(-,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 cCc \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×CSethom : C^{op}\times C\to Set is also the identity profunctor 1 C:CC1_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

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

where jj is the Yoneda embedding. Profunctors F:C op×CSet\mathbf{F} : C^{op} \times C \to Set whose hom-adjunct is of the form CFCj[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.


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

Last revised on June 7, 2022 at 15:38:27. See the history of this page for a list of all contributions to it.