ENRICHED INDEXED CATEGORIES Contents 1. Introduction

ENRICHED INDEXED CATEGORIES 629it has indexed (co)products if and only if V has both indexed (co)products and smallfiberwise ones.Noting that Fam(⋆) ∼ = Set, we see that by applying this construction to Const(⋆, V)we reproduce our original example Fam(V) from Example 2.2.2.31. Example. If V is an S-indexed monoidal category and F : S ′ → S is any functor,then there is an S ′ -indexed monoidal category F ∗ V defined by (F ∗ V ) X = V F (X) : thefiberwise monoidal structure of V passes immediately to F ∗ V . (If F does not preservefinite products, then the external product of F ∗ V may differ from that of V .)2.32. Example. The category of topological spaces has pullbacks, but is not locallycartesian closed, so its self-indexing does not have indexed products and is not closed. Itssubcategories of “compactly generated spaces” and “k-spaces” are cartesian closed, butstill not locally cartesian closed. However, the references given in [MS06, §1.3] show thatif we take S to be the category of compactly generated spaces, and for X ∈ S we take K Xto be the category of k-spaces over X, then we do obtain an indexed cosmos K . Sincenot every k-space is compactly generated, this indexed cosmos K is larger than Self (S)(which is not an indexed cosmos).2.33. Example. If S is a category with finite limits and finite colimits which are preservedby pullback, then there is an S-indexed monoidal category Self ∗ (S) whose fiberSelf ∗ (S) X is the category of sectioned objects over X. For such A and B, the fiberwisesmash product is the following pushout.A ⊔ X BA × X BX A ∧ X B.This defines a monoidal structure with the unit object X → X ⊔ X → X, which hasindexed coproducts preserved by ∧. If S is locally cartesian closed, it is an indexedcosmos.More generally, a similar construction can be applied to any V with fiberwise finitelimits and colimits, with the fiberwise colimits preserved by ⊗. For instance, startingfrom Example 2.32 we obtain an indexed cosmos K ∗ of sectioned topological spaces.2.34. Example. Let S be locally cartesian closed with countable colimits, and let Ab(S) Xbe the category of abelian group objects in S/X. The countable colimits in S enable us todefine free abelian group objects. Thus by [Joh02, D5.3.2], Ab(S) has indexed productsand coproducts and fiberwise finite limits and colimits. The countable colimits also enableus to define a tensor product, making Ab(S) into an indexed cosmos.More generally, if V is an S-indexed cartesian cosmos with countable fiberwise colimits,we can define an S-indexed cosmos Ab(V ) whose fiber over X is the category ofabelian groups in V X . For example, from Example 2.32 we obtain an indexed cosmos oftopological abelian groups.