Homotopical Interpretation of Globular Complex by Multipointed D-space

Globular complexes were introduced by E. Goubault and the author to model higher dimensional automata. Globular complexes are topological spaces equipped with a globular decomposition which is the directed analogue of the cellular decomposition of a CW-complex. We prove that there exists a combinatorial model category such that the cellular objects are exactly the globular complexes and such that the homotopy category is equivalent to the homotopy category of flows. The underlying category of this model category is a variant of M. Grandis’ notion of d-space over a topological space colimit generated by simplices. This result enables us to understand the relationship between the framework of flows and other works in directed algebraic topology using d-spaces. It also enables us to prove that the underlying homotopy type functor of flows can be interpreted up to equivalences of categories as the total left derived functor of a left Quillen adjoint.