ENRICHED INDEXED CATEGORIES Contents 1. Introduction

ENRICHED INDEXED CATEGORIES 6353.8. Definition. If V has indexed coproducts preserved by ⊗, then for any objectX ∈ S, there is a small V -category δX with ɛ(δX) = X and δX = (∆ X ) ! I X . We call itthe discrete V -category on X.3.9. Definition. Let A and B be small V -categories. A V -functor f : A → B consistsof:(i) A morphism ɛf : ɛA → ɛB in S.(ii) A morphism in ∫ V :A❴ɛA × ɛAfB❴ɛf×ɛf ɛB × ɛB.(iii) The following diagrams commute:idsI ɛA AI ɛBids BfandA ⊗ ɛA A comp Af⊗fB ⊗ ɛB B comp B.f3.10. Examples. Evidently a Fam(V)-functor is a V-enriched functor, and a Self (S)-functor is an S-internal functor. The other specific examples are similar.3.11. Remark. If δX is the discrete V -category on X ∈ S as in Definition 3.8, then aV -functor f : δX → A is uniquely determined by its underlying morphism ɛf : X → ɛAin S (the unit axiom forces its action on homs to be induced by ids : I ɛA → A).If V lacks indexed coproducts, in which case δX may not exist as a V -category,it is nevertheless often convenient to abuse language and allow the phrase “V -functorδX → A” to refer simply a morphism X → ɛA in S.3.12. Definition. Let f, g : A → B be V -functors. A V -natural transformationα: f → g consists of:(i) A morphismαI ɛA B❴❴ɛA (ɛg,ɛf)ɛB × ɛB.(3.13)