ENRICHED INDEXED CATEGORIES Contents 1. Introduction

ENRICHED INDEXED CATEGORIES 6434.7. Example. If S has finite products and V is an ordinary monoidal category as inExample 2.42, then an indexed Psh(S, V)-category A consists of, in particular, for eachX ∈ S, a V (S/X)op -enriched category A X . However, a V (S/X)op -enriched category isequivalently a functor A X : (S/X) op → V-CAT whose image consists of functors thatare the identity on objects.Moreover, by definition of the functor f ∗ : V (S/X)op → V (S/Y )op and by full-faithfulnessof f ∗ : (f ∗ ) • A Y → A X , for any objects a, b of A Y the hom-object A Y (f)(a, b) ∈ V mustbe isomorphic to A X (1 X )(f ∗ a, f ∗ b), and similarly for all the category structure. Thus,the V-category A Y (f) is completely determined by the V-category A X (1 X ) and thefunction f ∗ : ob(A Y (1 Y )) → ob(A X (1 X )).Furthermore, the action on hom-objects of the functors in the image of the functorA X : (S/X) op → V-CAT assemble exactly into an extension of this function on objectsto a V-functor A Y (1 Y ) → A X (1 X ). Adding in the pseudofunctoriality constraints,we see that an indexed Psh(S, V)-category is equivalently an ordinary pseudofunctorS op → V-CAT . In fact, the 2-category Psh(S, V)-Cat is 2-equivalent to the 2-category[S op , V-CAT ] of pseudofunctors, pseudonatural transformations, and modifications.In particular, an indexed Psh(S, Set)-category is merely an ordinary S-indexed category(with locally small fibers). It is known (e.g. [Joh02, B2.2.2]) that locally internalcategories can be identified with indexed categories that are “locally small” in an indexedsense. This corresponds to identifying indexed Self (S)-categories with indexedPsh(S, Set)-categories whose hom-presheaves A X (a, b) ∈ Set (S/X)op are all representable.(The connection with the usual definition of “locally small indexed category” will be moreevident in §5.)4.8. Remark. In [GG76] indexed V -categories are called V -enriched fibrations. Wehave chosen a different terminology because indexed V -categories are more analogous topseudofunctors than to fibrations. In §6 we will see a more ‘fibrational’ approach.It is straightforward to define indexed V -profunctors as well.4.9. Definition. For indexed V -categories A and B, an indexed V -profunctor consistsof a V X -enriched profunctor H X : A X −→ −↦ B X for each X, together with isomorphisms(f ∗ ) • H Y ∼ = H X satisfying evident coherence axioms. We obtain a categoryV -Prof(A , B) of indexed V -profunctors.However, rather than develop the theory of indexed V -categories any further here, wewill instead move on to a more general notion which includes both small V -categoriesand indexed V -categories as special cases.5. Large V -categoriesContinuing with our minimal assumptions that S has finite products and V is an S-indexed monoidal category, we will now define a different sort of “large V -category”which more obviously includes the small ones from §3. The relationship of these large