ENRICHED INDEXED CATEGORIES Contents 1. Introduction

ENRICHED INDEXED CATEGORIES 6576.12. Example. If A is an S-internal category regarded as a small Self (S)-category,then ΓA is the locally internal category classically associated to A.6.13. Remark. On the other hand, if we write V -CAT ind for the sub-2-category ofV -CAT containing all the V -categories but only the indexed V -functors, then the nonfullinclusion V -CAT ind ↩→ V -CAT is not a biequivalence. This is relevant because ifwe were to restrict ourselves to data contained in the bicategory constructed from V(rather than the whole equipment), then the indexed V -functors would be the only typeof morphism available. For this reason, the authors of [BCSW83, BW87, Bet89, Bet00]had to impose extra conditions at least as strong as being a V -fibration in order to obtainan equivalence with indexed V -categories (in the case V = Self (S), which is the onlyone they considered).Most V -categories which arise “in nature” are either small or are V -fibrations. Wecan regard the other large V -categories as a technical tool which makes it easier to relatethese two most important types. (As we will see in §8, set-sized V -categories are alsoconvenient to use as diagram shapes.)6.14. Remark. We can also define an anologue of the general hom-functors from theend of §2 for any V -fibration A ; we setA Y,[W ] (B, C) = π W ∗ ∆ ∗ Y ×W A (B, C)∼ = πW ∗ A X×Y ×Z×W (π ∗ XB, π ∗ ZC).When we consider tensors, cotensors, and monoidal structures for V -categories, we willalso find analogues for V -fibrations of the various types of monoidal structure on V .7. Change of cosmos and underlying indexed categoriesIf V is an S-indexed monoidal category and W is a T-indexed one, then by a laxmonoidal morphism Φ : V → W we mean a commutative square∫V Φ ∫WSΦsuch that Φ : S → T preserves finite products (hence is strong cartesian monoidal),Φ : ∫ V → ∫ W is lax monoidal and preserves cartesian arrows, and the square commutesin the 2-category of lax monoidal functors. If Φ : S → T is an identity, as is often thecase, we say that Φ is a morphism over S.In this situation, we have induced operations Φ • from small, large, and indexed V -categories to the corresponding sort of W -categories, and similarly for functors, transformations,profunctors, multimorphisms, and so on, which we call change of cosmos. T