Goubault, Eric and Mimram, Samuel - Directed Homotopy in Non-Positively Curved Spaces

lmcs:5731 - Logical Methods in Computer Science, July 13, 2020, Volume 16, Issue 3 - https://doi.org/10.23638/LMCS-16(3:4)2020
Directed Homotopy in Non-Positively Curved Spaces

Authors: Goubault, Eric and Mimram, Samuel

A semantics of concurrent programs can be given using precubical sets, in order to study (higher) commutations between the actions, thus encoding the "geometry" of the space of possible executions of the program. Here, we study the particular case of programs using only mutexes, which are the most widely used synchronization primitive. We show that in this case, the resulting programs have non-positive curvature, a notion that we introduce and study here for precubical sets, and can be thought of as an algebraic analogue of the well-known one for metric spaces. Using this it, as well as categorical rewriting techniques, we are then able to show that directed and non-directed homotopy coincide for directed paths in these precubical sets. Finally, we study the geometric realization of precubical sets in metric spaces, to show that our conditions on precubical sets actually coincide with those for metric spaces. Since the category of metric spaces is not cocomplete, we are lead to work with generalized metric spaces and study some of their properties.

Volume: Volume 16, Issue 3
Published on: July 13, 2020
Submitted on: August 30, 2019
Keywords: Computer Science - Logic in Computer Science,Computer Science - Distributed, Parallel, and Cluster Computing,Mathematics - Algebraic Topology,55M99, 68Q10,G.0,F.3.2


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