ENRICHED INDEXED CATEGORIES Contents 1. Introduction

ENRICHED INDEXED CATEGORIES 6475.9. Example. Any large V -category A has a unit profunctor A : A −→ −↦ A which ismade up of its hom-objects.5.10. Example. If H : A −↦−→ B is a V -profunctor and f : A ′ → A and g : B ′ → B areV -functors, then there is a V -profunctor H(g, f) : A ′ −→ −↦ B ′ defined byH(g, f)(b, a) = (ɛf × ɛg) ∗( H(gb, fa) )In particular, for a V -functor f : A → B, we have the representable V -profunctorsB(1, f) : A −↦−→ B and B(f, 1) : B −↦−→ A .To define the composite of V -profunctors H : A −→ −↦ B and K : B −→ −↦ C in general, weneed V to have fiberwise colimits of the size of the collection of objects of B. Therefore,if this collection is itself large (in the sense of the ambient set theory), we cannot expectsuch composites to exist. For this reason, it is useful to introduce the following notion.5.11. Definition. For V -profunctors H : A −→ −↦ B, K : B −→ −↦ C , and L : A −→ −↦ C , abimorphism φ : H, K → L consists of(i) For each a, b, c, a morphism in ∫ V :(ii) The following diagrams commute:H(b, a) ⊗ ɛb K(c, b) φ abcL(c, a)❴❴ɛa × ɛb × ɛc ɛa × ɛcH(b, a) ⊗ ɛb K(c, b) ⊗ ɛc C (c ′ , c)1⊗actH(b, a) ⊗ ɛb K(c ′ , b)φ⊗1L(c, a) ⊗ ɛc C (c ′ , c)actφ L(c ′ , a)(5.12)A (a, a ′ ) ⊗ ɛa H(b, a) ⊗ ɛb K(c, b)act⊗1H(b, a ′ ) ⊗ ɛb K(c, b)1⊗φA (a, a ′ ) ⊗ ɛa L(c, a)actφ L(c, a ′ )(5.13)H(b, a) ⊗ ɛb B(b ′ , b) ⊗ ɛb ′ K(c, b ′ )1⊗actH(b, a) ⊗ ɛb K(c, b)act⊗1H(b ′ , a) ⊗ ɛb ′ K(c, b ′ )φφ L(c, a)(5.14)