ENRICHED INDEXED CATEGORIES Contents 1. Introduction

ENRICHED INDEXED CATEGORIES 633as in [DL07], we define an object of V (S/X)op to be small if it is a small (V-weighted)colimit of such representables. Representable objects are closed under restriction, sincef ∗ (F g ) ∼ = F f ∗ g; hence so are small objects. Similarly, representable objects are closedunder tensor products, since F g ⊗ X F h∼ = Fg×X h; hence so are small objects.Now f ∗ : V (S/Y )op → V (S/X)op has a partial left adjoint f ! defined on small objects,which takes them to small objects: we define f ! (F g ) = F fg and extend cocontinuously.Similarly, all homs V ? (A, B) exist when A is small: we define V X (F g , B)(W h −→ X) =B(g × X h) and extend cocontinuously, and construct the other homs from this as usual.3. Small V -categoriesLet S be a category with finite products and V an S-indexed monoidal category, withcorresponding fibration ∫ V → S. In this section we describe a notion of “small V -category” which directly generalizes internal categories and small enriched categories.(In §5 we will see that there is also another, less elementary, notion of “smallness” forV -categories.)We will use the following notation:A φ B❴ ❴ X fYto indicate that φ : A → B is a morphism in ∫ V lying over f : X → Y in S. Of course, togive such a φ is equivalent to giving a morphism A → f ∗ B in V X , but using morphismsin ∫ V often makes commutative diagrams less busy (since there are fewer f ∗ ’s to notate).3.1. Definition. A small V -category A consists of:(i) An object ɛA ∈ S.(ii) An object A ∈ V ɛA×ɛA .(iii) A morphism in ∫ V :(iv) A morphism in ∫ V :idsI ɛA A❴❴ɛA ∆ɛA × ɛAcompA ⊗ ɛA A A❴❴ɛA × ɛA × ɛA π2ɛA × ɛA