Homotopy theory of Moore flows (I)

Abstract

A reparametrization category is a small topologically enriched symmetric semimonoidal category such that the semimonoidal structure induces a structure of a commutative semigroup on objects, such that all spaces of maps are contractible and such that each map can be decomposed (not necessarily in a unique way) as a tensor product of two maps. A Moore flow is a small semicategory enriched over the closed semimonoidal category of enriched presheaves over a reparametrization category. We construct the q-model category of Moore flows. It is proved that it is Quillen equivalent to the q-model category of flows. This result is the first step to establish a zig-zag of Quillen equivalences between the q-model structure of multipointed d-spaces and the q-model structure of flows.

Homotopy theory of Moore flows (II)

“…It is not in [14]. The proof is given in this section and not in Section 7 to recall [14,Corollary 5.13] which also helps to understand the geometric contents of the tensor product of G-spaces. (…”

“…Presentation. This paper is the companion paper of [14]. The purpose of these two papers is to exhibit, by means of the q-model category of Moore flows (cf.…”

“…The Quillen equivalence between flows and Moore flows is proved in [14,Theorem 10.9]. The Quillen equivalence between multipointed d-spaces and Moore flows is proved in Theorem 8.1.…”

“…It illustrates the interest of this structure already for calculating spaces of execution paths of cellular multipointed d-spaces. The interest of this structure is beyond directed homotopy theory, as remarked in the introduction of [14] where possible connections with type theory are briefly discussed. The potential of this semimonoidal structure is visible in the proofs of Proposition 6.3, Theorem 7.2, Theorem 7.3 and Corollary 7.…”

“…The enriched category G is an example of a reparametrization category in the sense of [14,Definition 4.3]…”