ENRICHED INDEXED CATEGORIES Contents 1. Introduction

ENRICHED INDEXED CATEGORIES 655Proof. If A is an indexed V -category with transition functors f ∗ : A X → A Y , then itis easy to check that for any x ∈ A X , the object f ∗ x ∈ A Y is a restriction of x along fin ΘA . Thus ΘA , and anything isomorphic to it, is a V -fibration.Conversely, given a V -fibration B, we complete the above construction of an indexedV -category ΛB as follows. We choose, for every x and f, a restriction f ∗ x, and definethe functor f ∗ : (f ∗ ) • ΛB Y → ΛB X to take x to f ∗ x. The definition of restriction ensuresthat this can be extended to a fully faithful V Y -functor, and the essential uniqueness ofrestrictions ensures that they are coherent. Finally, it is straightforward to check thatΘΛB ∼ = B in V -CAT .As remarked above, in order to fully characterize the image of Θ, we will also need tolimit the functors we consider.6.6. Definition. A V -functor f : A → B between large V -categories is called indexedif ɛf x is an identity for all x.If we now let V -FIB denote the sub-2-category of V -CAT consisting of the V -fibrations, the indexed V -functors between them, and all the V -natural transformationsbetween those, then it is easy to extend Λ to a 2-functor V -FIB → V -Cat. Our doubleuse of the word ‘indexed’ is unproblematic because of the following result.6.7. Theorem. The 2-functors Θ and Λ are inverse 2-equivalences between V -Cat andV -FIB.Proof. Left to the reader.Since 2-equivalences preserve all 2-categorical structure, we can use indexed V -categoriesand V -fibrations interchangeably, just as we do for ordinary fibrations and pseudofunctors,and we will rarely distinguish notationally between them.6.8. Remark. This 2-equivalence also extends to profunctors. We could define a virtualequipment of indexed V -profunctors and show it is equivalent to the restriction of V -PROFto the V -fibrations and indexed functors. However, for our purposes it will suffice to notethat for V -fibrations A and B, we have an equivalence of categoriesV -Prof(A , B) ≃ V -PROF (A , B)connecting indexed V -profunctors, as in Definition 4.9, to V -profunctors as considered in§5. This equivalence is constructed just as for the hom-objects of categories, by restrictingalong diagonals and projections.In contrast to the classical case, however, it turns out that by including the nonindexedV -functors, we can put back in the large V -categories that aren’t V -fibrationsand still maintain a biequivalence. We first observe the following.6.9. Proposition. If B is a V -fibration, then any V -functor F : A → B is naturallyisomorphic to an indexed one.