Jérémy Dubut's defense

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I defended my thesis on Monday, 11th September, 2017.




Studying a system that evolves with time through its geometry is the main purpose of directed algebraic topology. This topic emerged in computer science, more particularly in true concurrency, where Pratt introduced the higher dimensional automata (HDA) in 1991 (actually, the idea of geometry of concurrency can be tracked down Dijkstra in 1965). Those automata are geometric by nature: every set of n processes executing independent actions can be modeled by a n-cube, and such an automaton then gives rise to a topological space, obtained by glueing such cubes together. This space naturally has a specific direction of time coming from the execution flow. It then seems natural to use tools from algebraic topology to study those spaces: paths model executions, homotopies of paths, that is continuous deformations of paths, model equivalence of executions modulo scheduling of independent actions, and so on, but all those notions must preserve the direction. This brings many complications and the theory must be done again. In this thesis, we develop homotopy and homology theories for those spaces with a direction. First, my directed homotopy theory is based on deformation retracts, that is continuous deformation of a big space on a smaller space, following directed paths that are inessential, meaning that they do not change the homotopy type of spaces of executions. This theory is related to categories of components and higher categories. Secondly, my directed homology theory follows the idea that we must look at the spaces of executions and those evolves with time. This evolution of time is handled by defining such homology as a diagram of spaces of executions and comparing such diagrams using a notion of bisimulation. This homology theory has many nice properties: it is computable on simple spaces, it is an invariant of our homotopy theory, it is invariant under simple action refinements and it has a theory of exactness.

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