ENRICHED INDEXED CATEGORIES Contents 1. Introduction

ENRICHED INDEXED CATEGORIES 631G × K f×1 K H × KG∆G × G1 G ×gG × L1 H ×gf×1 L H × L∆G × G ∆×1G1 G ×∆ G × G × GG (1 G,f) G × Hff×1 HH∆H H × HFigure 1: Homotopy pullback squareswith a G-action. With fiberwise cartesian monoidal structures (and G-actions induced bythe diagonal), this yields a Grp(S)-indexed cartesian monoidal category Act(S).Since GS is monadic over S with monad (G × −), Act(S) has any fiberwise limitsthat S has, and any fiberwise colimits that S has and that are preserved by × in eachvariable. And if S is cartesian closed, then Act(S) is closed, with the exponentials in GSbeing those of S with a conjugation action.Now if S has coequalizers preserved by × in each variable, then the restriction functorsf ∗ : G → H have left adjoints. Namely, for a group homomorphism f : G → H andX ∈ GS, we define f ! X to be the coequalizer of the two mapsH × G × X ⇒ H × Xinduced by the action of G on X, and by f followed by the multiplication of H. Similarly,if S is cartesian closed and has equalizers, then f ∗ has a right adjoint, with f ∗ X definedto be the equalizer of the analogous pair of mapsX H ⇒ X G×H .Unfortunately, these adjoints do not satisfy the Beck-Chevalley condition for all pullbacksquares in S. However, we do have the Beck-Chevalley condition for three importantclasses of pullback squares, shown in Figure 1. (For example, this follows from [Joh02,B2.5.11].) Note that these squares are all pullbacks in any category with finite products,whether or not it has all pullbacks.Only once or twice in this paper will we use the Beck-Chevalley condition for fullygeneral pullback squares; in most cases we will only need it for these particular ones,along with their transposes, and their cartesian products with fixed objects. In [PS12],we said that an indexed category had indexed homotopy coproducts if its restrictionfunctors f ∗ have left adjoints satisfying the Beck-Chevalley condition for these pullbacksquares. But since Act(S) is the only example we will discuss in this paper 2 which fails2 There are other examples discussed in [PS12], in which the fiber categories V X are “homotopycategories”. However, these examples are usually not very interesting to enrich in, both because theytend to lack fiberwise limits and colimits, and because the resulting enriched categories would be only“up to unspecified homotopy”, not up to coherent homotopy.