Homotopical interpretation of globular complex by multipointed d-space

TL;DR

This paper establishes a homotopical framework for globular complexes, modeling higher dimensional automata, and connects it with flows and directed algebraic topology through a combinatorial model category.

Contribution

It constructs a model category where globular complexes are the cellular objects, linking them to flows and directed topology via Quillen adjunctions.

Findings

Globular complexes form the cellular objects of a new model category.

The homotopy category of this model is equivalent to that of flows.

The homotopy type functor of flows is a derived functor in this framework.

Abstract

Globular complexes were introduced by E. Goubault and the author in arXiv:math/0107060 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 introduced in arXiv:math/0308054. 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 constructed in arXiv:math/0308063 can be interpreted up to equivalences of categories as the total left derived functor of a left Quillen adjoint.