Abstract
Erratum, 11 July 2022: This is an updated version of the original paper \cite{Moore1} in which the notion of reparametrization category was incorrectly axiomatized. Details on the changes to the original paper are provided in the Appendix.A reparametrization category is a small topologically enriched semimonoidal category such that the semimonoidal structure induces a structure of a 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 biclosed 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.
Highlights
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 as a tensor product of two maps
A d-space is just a topological space equipped with a distinguished set of continuous maps playing the role of the execution paths of a concurrent process
Flows are difficult to use because they are roughly speaking d-spaces without an underlying topological space. They just have an underlying homotopy type as defined in [12]. This fact makes the category of flows quite unwieldy, it has mathematical features not shared by the other topological models of concurrency, like the possibility of defining functorial homology theories for branching and merging areas of execution paths [9, 11]
Summary
The category Top is either the category of ∆-generated spaces or the full subcategory of ∆Hausdorff ∆-generated spaces (cf. [19, Section 2 and Appendix B]). The inclusion functor from the full subcategory of ∆-generated spaces to the category of general topological spaces together with the continuous maps has a right adjoint called the ∆kelleyfication functor. The latter functor does not change the underlying set. Note that the paper [15], which is used several times in this work, is written in the category of ∆-generated spaces; it is still valid in the category of ∆-Hausdorff ∆-generated spaces This point is left as an exercise for the idle mathematician: see [19, Section 2 and Appendix B] for any help about the topology of ∆-generated spaces.
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