Enriched algebras, spectra and homotopy limits

Abstract

O. Introduction. The purpose of this paper is the same as that of [5]; to show how certain properties of homotopy limits are consequences of what either are or should be standard facts about categories enriched in a closed category. The property to be explained here is as follows: in [16], Thomason shows that the degreewise homotopy limit of a diagram of pointed simplicial spectra is a pointed simplicial spectrum. He calls this the homotopy limit in the category of such spectra. It is eminantly reasonable to suppose that, in fact, it is the homotopy limit in this category, but two things have to be proved, i) . The category Spec K, of pointed simplicial spectra is a complete simplicial category, since only such categories have homotopy limits, ii) . The component projections pr n : Spec K, ÷ K,, for each degree n, have simplicial!y enriched left adjoints, and hence preserve homotopy limits.