Frames in cofibration categories

Abstract

We introduce the quasicategory of frames of a cofibration category, i.e. a new model of the (∞, 1)-category associated with a cofibration category.

Homotopy theory of cocomplete quasicategories

“…N f C . Then [23,Lemma 4.8] implies that ˆ. To see that it is a weak equivalence, it is enough to verify the approximation properties of Proposition 1.8.…”

“…In [23] The results of the last section heavily depend on the methods of [24; 23] which in turn involve a lot of notation useful in expressing properties of N f C in terms of various diagrams in C . In this section, we recall some of that notation and prove a few auxiliary lemmas.…”

“…Another lemma that we will need says that up to equivalence all frames are Reedy cofibrant replacements of constant diagrams. Perhaps the most useful result of [23] characterizes universal cones K B ! N f C in terms of the corresponding diagram D.K B / !…”

“…The main result of the first one [24] was existence of a fibration category of cofibration categories. In the second one [23] we introduced the quasicategory of frames which is a new construction of the .1; 1/-category associated to a cofibration category. In the present paper we construct a fibration category of cocomplete quasicategories and prove that the quasicategory of frames functor is an equivalence of fibration categories.…”

Homotopy theory of cofibration categories

“…(See also [Szu14b] for a slightly edited version.) The second paper [Szu16] introduces a new construction of a cocomplete quasicategory associated with a cofibration category called the quasicategory of frames. The third one [Szu] shows that it is possible to construct a cofibration category from a cocomplete quasicategory in a way that establishes an equivalence between the homotopy theory of cofibration categories and the homotopy theory of cocomplete quasicategories.…”