We prove that the homotopy theory of cofibration categories is equivalent to the homotopy theory of cocomplete quasicategories. This is achieved by presenting both homotopy theories as fibration categories and constructing an explicit equivalence between them.
We construct a fibration category of cofibration categories which constitutes a convenient framework for the homotopy theory of cofibration categories.
Abstract. We show that the quasicategory of frames of a cofibration category, introduced by the second-named author, is equivalent to its simplicial localization.
We present two new proofs of Simon Henry’s result that the category of simplicial sets admits a constructive counterpart of the classical Kan–Quillen model structure. Our proofs are entirely self-contained and avoid complex combinatorial arguments on anodyne extensions. We also give new constructive proofs of the left and right properness of the model structure.
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