ABSTRACT

A model category is a category possessing all small limits and colimits, together with three distinguished classes of morphisms, called weak equivalences, fibrations, and cofibrations, satisfying several axioms. For example, the category of topological spaces has a model structure Top in which the weak equivalences are the weak homotopy equivalences, all objects are fibrant, and the cofibrant objects are retracts of CW-complexes. The structure of a model category enables one to have a well-defined homotopy category without running into set-theoretic obstacles. A bisimplicial space is a double Segal space if and only if it is a Segal object in Segal spaces. A 2-category can be defined to be a category enriched in categories. There is a model structure CSS on the category of bisimplical spaces in which the fibrant objects are precisely the 2-fold complete Segal spaces.