ENRICHED INDEXED CATEGORIES Contents 1. Introduction

ENRICHED INDEXED CATEGORIES 621With the basic theory of limits and colimits available, there are of course many differentdirections in which to develop category theory. We choose only two: presheaf categoriesin §10, and monoidal structures in §11.The goal of §10 is to prove that presheaf V -categories are free cocompletions. Thearguments are purely formal and equipment-theoretic. Rather than restrict ourselves topresheaves on small categories, we consider more generally small presheaves in the senseof [DL07], which form free cocompletions of not-necessarily-small categories.Finally, in §11 we study monoidal V -categories, using for the first time in an essentialway the symmetry of V . We define two tensor products of V -categories, one “indexed”and one not, which extend the biequivalence of §6 to a monoidal biequivalence. MonoidalV -categories are then pseudomonoids with respect to either of these tensor products. Wealso define closed monoidal V -categories by way of profunctors, essentially specializingthe general definitions of [DS97, DMS03, Str04] to V -categories. As in the classicalcase, closed monoidal V -categories correspond closely to monoidal adjunctions involvingV . We conclude with the Day convolution monoidal structure for V -presheaf categories,which, combined with the previous theory, yields some of the most important examples,from [MS06] and [Boh12].1.4. Acknowledgments. I would like to thank my thesis advisor, Peter May, for manythings too numerous to mention. I would also like to thank Anna Marie Bohmann andMarta Bunge, for providing the final impetus for publication. Finally, I would like tothank Hurricane Sandy, for providing a week free from other commitments, in which Iwas able to polish my notes into a readable paper; and the Institute for Advanced Study,for providing an electricity generator during that week.2. Indexed monoidal categoriesLet S be a category with finite products. We write ∆ X : X → X × X for the diagonal ofX ∈ S, and for related maps such as X × Y → X × X × Y . Similarly, we write π X forany projection map in which X is projected away, such as X → 1 or X × Y → Y . If thiswould be ambiguous, such as for the product projections X × X → X, we use numericalsubscripts which again denote the copy being projected away; thus π 1 : X × X → X isthe projection onto the second copy.Our enriched indexed categories will be indexed over S. Their enrichment, on the otherhand, will not be over a monoidal category in the classical sense, but over the followingtype of category.2.1. Definition. An S-indexed monoidal category is a pseudofunctor V : S op →MonCat, where MonCat is the 2-category of monoidal categories, strong monoidal functors,and monoidal transformations.As usual, we write the image of X ∈ S as V X , and the image of f : X → Y asf ∗ : V Y → V X . We write the tensor product and unit of V X as ⊗ X and I X . The