@TECHREPORT{Reedy74homotopytheory, author = {C. L. Reedy}, title = {Homotopy theory of model categories}, institution = {}, year = {1974} }

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Abstract

In this paper some questions in the homotopy theory of model categories are answered. The important results of this paper are Theorems B, C and D which state that pushouts, sequential direct limits, and realizations of simplicial objects respect weak equivalences, provided sufficient cofibrancy is present. Section 1 presents a model category structure on the simplicial objects over a model category. This is done partly to provide a justification for the term cofibrant, as applied to certain simplicial objects, and also to show that any object can be “approximated ” by a cofibrant object. The lemmas in this section are presented without proof, since the proofs are easy, and of an entirely category theoretic nature. In section 2 it is shown that, in model categories, weak equivalences respect pushouts and sequential direct limits. That weak equivalences respect pushouts is particularly important since this shows that any closed model category is a suitable category for homology theory (see [1]). In section 3 the result that the realization of a weak equivalence of cofibrant objects is a weak equivalence is proven. Certain stronger results in the case of simplicial topological spaces are also mentioned. Section 4 discusses special results about simplicial simplicial sets, including the result that realization is isomorphic to the diagonal, a useful result which is not widely known, and not original with the author.