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We are interested in homotopy theory in a variety of categories, and we want to compare these. There is an efficient machinery due to Quillen, which encodes this structure. We have used this language in our discussion of simplicial sets. In addition to weak equivalences (which is all that is needed to form the homotopy category) we have fibrations and cofibrations satisfying certain axioms. This structure ensures that the homotopy category actually exists, but more importantly it encodes the deeper homotopical structures, making a large class of arguments formal. It also makes comparison between different homotopical structures more transparent.
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