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In this paper we show that model categories of a very broad class can be replaced up to Quillen equivalence by simplicial model categories. 1.1. Summary. Model categories were introduced by Quillen (Q) as a very general setting in which one can 'do homotopy theory'. Among them is a distinguished class called the simplicial model categories, which are essentially characterized by the following property: certain formulae which in an arbitrary model category hold only up to homotopy, in simplicial model categories they actually hold on the nose. Actually, such formulae can be made to hold on the nose in an arbitrary model category, but only at the expense of carefully making choices; what's useful about simplicial model categories is that they come with a canonical set of choices already made. This turns out to be quite a convenience, and simplicial model categories are much easier to work with than arbitrary ones. It is certainly not true that all model categories of interest are simplicial, but many of them happen to be. It is natural to wonder if there is a signicant dierence between simplicial and non-simplicial model categories. The obvious question is, what do we mean by 'signicant'? Now Quillen also introduced a notion of equivalence between model categories, which involves having a pair of adjoint functors satisfying certain conditions. Two model categories which are 'Quillen equivalent' represent the same underlying ho- motopy theory. The purpose of this note is to show that a very large class of model categories are Quillen equivalent to simplicial ones. In other words, for this class of model categories there is no signicant dierence between simplicial and non-simplicial, at least as far as the underlying homotopy theory is concerned. More explicitly, if M is a model category, then one can look at the category of simplicial objects sM . This is also a model category in a standard way|it has a so-called Reedy model structure, which is usually not simplicial. There is also an obvious pair of adjoint functors c : M sM : ev0: Here c (the left-adjoint) associates to X 2 M the discrete simplicial object con- sisting of X in every dimension, and ev0 sends Z 2 sM to its zeroth object Z0. Under broad conditions on the model category M , we describe a way of making

Replacing model categories with simplicial ones