Determining Closed Model Category Structures

Closed model categories are a general framework introduced by Quillen [15] in which one can do homotopy theory. An alternative framework has been developed by Baues [2]. Putting a closed model structure on a category not only allows one to use techniques of homotopy theory to study it but also allows one to better understand the category through the concepts and constructions that come with the structure. The usefulness of closed model categories in homotopy theory has been firmly established. It seems likely that they could also be useful in other areas of mathematics. Our main theorem 7.1 and its variant 7.2 for the first time give general conditions for the existence of a closed model category structure on a category. All previous proofs of closed model category structures have depended on either ad hoc arguments or the existence of a closed model category structure on some other category. Our conditions have been designed to make giving a category a closed model structure a simple task. Only a few hypotheses need be verified. Hopefully our construction will prove useful as a tool for other researchers. As an illustrative example we give a model category structure for CDGA(R). In the next few paragraphs we give an overview of the contents and main arguments of the paper. For convenience we refer the reader to the relevant definitions within the paper. A closed model category (Definition 3.4) is a category with three distinguished classes of morphisms: cofibrations; fibrations; weak equivalences. We use the word acyclic as the adjective corresponding to the noun weak equivalence. Each map is required to have two