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Pré-Publication, Document De Travail Année : 2022

Directed degeneracy maps for precubical sets

Résumé

Transverse (symmetric precubical) sets were introduced to make the construction of the parallel product with synchronization for process algebras functorial. It is proved that one can do directed homotopy on transverse sets in the following sense. A q-realization functor from transverse sets to flows is introduced using a q-cofibrant replacement functor of flows. By topologizing the cotransverse maps, the cotransverse topological cube is constructed. It can be regarded both as a cotransverse topological space and as a cotransverse Lawvere metric space. A natural realization functor from transverse sets to flows is introduced using Raussen's notion of natural $d$-path extended to transverse sets thanks to their structure of Lawvere metric space. It is proved that these two realization functors are homotopy equivalent by using the fact that the small category defining transverse sets is c-Reedy in Shulman's sense. This generalizes to transverse sets results previously obtained for precubical sets. Note that a precubical set and the free transverse set generated by it have isomorphic realizations as flows.
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Dates et versions

hal-03771404 , version 1 (07-09-2022)
hal-03771404 , version 2 (31-10-2022)
hal-03771404 , version 3 (20-03-2024)

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Philippe Gaucher. Directed degeneracy maps for precubical sets. 2022. ⟨hal-03771404v1⟩
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