From simplicial homotopy to crossed module homotopy in modified categories of interest

Limits in modified categories of interest

‎We firstly prove the completeness of the category of crossed modules in a modified category of interest‎. ‎Afterwards‎, ‎we define pullback crossed modules and pullback cat objects that are both obtained by pullback diagrams with extra structures on certain arrows‎. ‎These constructions unify many corresponding results for the cases of groups‎, ‎commutative algebras and can also be adapted to ...

Stable Homotopy Categories

Introduction. The Freudenthal suspension theorem implies that the set of homotopy classes of continuous maps from one finite complex to another is eventually invariant under iterated suspension of the complexes. In this "stable range" the set of homotopy classes is also well behaved in other ways. For example, it has in a natural way the structure of an abelian group. These stable groups of hom...

Sheafifiable Homotopy Model Categories

If a Quillen model category can be specified using a certain logical syntax (intuitively, “is algebraic/combinatorial enough”), so that it can be defined in any category of sheaves, then the satisfaction of Quillen’s axioms over any site is a purely formal consequence of their being satisfied over the category of sets. Such data give rise to a functor from the category of topoi and geometric mo...

Homotopy colimits in model categories

Homotopy Theory of Model Categories

In this paper some questions in the homotopy theory of model categories are answered. The important results of this paper are Theorems B, C and D which state that pushouts, sequential direct limits, and realizations of simplicial objects respect weak equivalences, provided sufficient cofibrancy is present. Section 1 presents a model category structure on the simplicial objects over a model cate...