Configuration spaces for concurrent systems of processes admit some structure of time. I will present a cohomology theory for such "directed spaces." On such spaces, ordinary cohomology groups reveal properties invariant under continuous deformations, while directed cohomology monoids detect finer properties invariant under deformations respecting the temporal structure and thus tease out the "qualitative" structure of time. Directed cohomology extends several well-known properties of its classical analogue: our new invariants admit chain-theoretic constructions, equivalent homotopical descriptions, axiomatic characterizations, and multiplicative structure. After presenting the basic theory of directed homotopy and cohomology, I will sketch real and potential applications to concurrent engineering, string rewriting, and informatics. This talk, aimed at a general audience, assumes no prior experience with directed spaces or cohomology theories.
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