Homotopy Type Theory
hom functor

Idea

The hom-sets of a precategory are functorial in nature.

Definition

For any precategory 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')

See also

Category theory functor Yoneda lemma precategory

References

HoTT book

category: category theory