Homotopy theory of cofibration categories

Abstract

We construct a fibration category of cofibration categories which constitutes a convenient framework for the homotopy theory of cofibration categories.

Frames in cofibration categories

“…The definition readily dualizes to yield fibration categories which are models of finitely complete homotopy theories or κ-complete homotopy theories depending on the choice of axioms. The main result of [21] establishes the homotopy theory of cofibration categories in the form of a fibration category. We recall the prerequisite definitions before stating the theorem.…”

“…Our results are based on the techniques of [21] and we start by summarizing the contents of this paper. The central notion is that of cofibration categories which are slightly modified duals of Brown's categories of fibrant objects [3].…”

“…The central notion is that of cofibration categories which are slightly modified duals of Brown's categories of fibrant objects [3]. Definition 1.1 [21,Definition 1.1] A cofibration category is a category C equipped with two subcategories: the subcategory of weak equivalences (denoted by ∼ →) and the subcategory of cofibrations (denoted by ) such that the following axioms are satisfied (here, an acyclic cofibration is a morphism that is both a weak equivalence and a cofibration).…”

“…This paper is the second in the series of three that summarize the results of the author's thesis [19] (see also [20] for a slightly edited version). The first one [21] constructs a fibration category of cofibration categories which provides a convenient framework for the homotopy theory of cofibration categories. In particular, the notion of a fibration of cofibration categories introduced there is a crucial tool in the present paper.…”

“…1 where we prove a number of preliminary results about diagrams in cofibration categories and fibrations between fibration categories. This section builds directly on [21]. In Sect.…”