Six model categories for directed homotopy

Modelling fundamental 2-categories for directed homotopy (*)

Directed Algebraic Topology is a recent field, deeply linked with ordinary and higher dimensional Category Theory. A 'directed space', e.g. an ordered topological space, has directed homotopies (generally non reversible) and fundamental n-categories (replacing the fundamental ngroupoids of the classical case). Finding a simple model of the latter is a non-trivial problem, whose solution gives r...

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...

Model Structures for Homotopy of Internal Categories

The aim of this paper is to describe Quillen model category structures on the category CatC of internal categories and functors in a given finitely complete category C. Several non-equivalent notions of internal equivalence exist; to capture these notions, the model structures are defined relative to a given Grothendieck topology on C. Under mild conditions on C, the regular epimorphism topolog...

Homotopy colimits in model categories

On Equivariant Homotopy Theory for Model Categories

Two approaches to equivariant homotopy theory in a topological or ordinary Quillen model category are studied and compared. For the topological model category of spaces, we recover that the categories of topological presheaves indexed by the orbit category of a fixed topological group G and the category of G-spaces form Quillen equivalent model categories.