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My work is about the relations between homotopy theory, in particular categorical homotopy theory, and concurrency in computer science. I have introduced several geometric models of concurrency, some of them are variants of models studied by other authors. These variants must be introduced for mathematical reasons, to allow some homotopical structures to be revealed: I do not work with Grandis' $d$-spaces, but rather with multipointed d-spaces, I do not work with Cattani-Sassone's higher dimensional transition systems, but rather with a topological locally presentable variant, and I do not work with enriched categories, but with a variant called a flow. Of course all semantics do factorize by these variants. And my goal is to understand their homotopical properties and how they are related to one another and to non-geometric models of concurrency like process algebras.