Relative categories: Another model for the homotopy theory of homotopy theories

Abstract

We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, "functors between two homotopy theories form a homotopy theory", or more precisely that the category of such models has a well-behaved internal hom-object.

How to glue derived categories

“…Then to finish the proof, it suffices to show that λ * also commutes with U † . Since for any map m : 2 , this trivially follows from (7.7).…”

“…At present, there are several constructions of ∞-categories in the literature (e.g. quasicategories of Joyal as developed by Lurie [33,34], complete Segal spaces of Rezk [41], and an attempt at a more invariant treatment by Barwick and Kan [2]). All of them are quite heavy and/or inexplicit.…”

“…Following [2], we start with the notion of a relative category. Definition 2.1 A relative category C, W is a category C equipped with a class of morphisms W that contains all identity maps.…”

“…Extend f to a cofiber square (2.4) in the sense of Definition 2.8, say with X = Y . Then f represents S(h(X )) in Ho(C [2] , W [2] ), and being cocartesian, (2.4) induces an isomorphism (h(X )) ∼ = h(Y ). Since X = Y is an acyclic object, and C ⊂ C is a pointed model embedding, X = Y lies in C. Then by Definition 2.12 (ii), Y ∈ C, and by Definition 2.12 (i), (2.4) is homotopy cartesian, thus induces an isomorphism h(…”

“…related by a map (4.1) form a precofibration over the category [2] = {0, 1, 2}, not a prefibration, and the category of sections of this precobiration has nothing to do with the iterated comma-category R( q).…”