ENRICHED INDEXED CATEGORIES Contents 1. Introduction

ENRICHED INDEXED CATEGORIES 6232.3. Example. Let S be a category with finite limits, and let V X = S/X, with thecartesian product (which is pullback in S). This is a cartesian monoidal fibration calledthe self-indexing of S; we write it as Self (S). Its total category ∫ Self (S) is thecategory S 2 of arrows in S, its external product is just the cartesian product in S.See [Shu08] and [PS12] for further study of indexed monoidal categories; the latterincludes an informal string diagram calculus.Now, as is the case with classical enriched category theory, we frequently need completenessconditions on V . By a fiberwise limit or colimit, we mean a limit or colimit ina fiber category V X which is preserved by all functors f ∗ . If κ is a regular cardinal, wesay that V is fiberwise κ-complete if it has all fiberwise limits of cardinality < κ, andwe say V is fiberwise complete if it has all small fiberwise limits. Of course we havesimilar notions of fiberwise cocompleteness.The other important sort of (co)limit for indexed categories is the following.2.4. Definition. V has S-indexed coproducts if(i) each functor f ∗ : V Y → V X has a left adjoint f ! , and(ii) for any pullback squarehkin S, the induced Beck-Chevalley transformation k ! h ∗ → g ∗ f ! is an isomorphism.Dually, V has S-indexed products if each f ∗ has a right adjoint f ∗ satisfying an analogouscondition.It is well-known that the adjoints f ! exist if and only if the fibration ∫ V → S is alsoan opfibration.2.5. Example. Fam(V) has any fiberwise limits and colimits that V has, and has Setindexed(co)products iff V has (co)products.2.6. Example. If S has finite limits, then Self (S) has fiberwise finite limits. It isfiberwise complete if S is complete, and has any fiberwise colimits that S has. It alwayshas S-indexed coproducts, and has S-indexed products if and only if S is locally cartesianclosed.We say that V is κ-complete if it is fiberwise κ-complete and has indexed products,and similarly it is κ-cocomplete if it is fiberwise κ-cocomplete and has indexedcoproducts.Now in the case when V has indexed coproducts, there is a third variant of themonoidal structure. For A ∈ V X×Y and B ∈ V Y ×Z , we defineA ⊗ [Y ] B = π Y ! ∆ ∗ Y (A ⊗ B),gf