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Joint Los Angeles Topology Seminar

Monday, November 7, 2022
4:30pm to 5:30pm
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On the general notion of homotopy-invariant properties

UCLA, Room 4645 in the Geology Building

When considering topological spaces with algebraic structures, there are certain properties which are invariant under homotopy equivalence, such as homotopy-associativity, and others that are not, such as strict associativity. A natural question is: which properties, in general, are homotopy invariant? As this involves a general notion of "property", it is a question of mathematical logic, and in particular suggests that we need a system of logical notation which is somehow well-adapted to the homotopical context. Such a system was introduced by Voevodsky under the name Homotopy Type Theory. I will discuss a sort of toy version of this, which is the case of "first-order homotopical logic", in which we can very thoroughly work out this question of homotopy-invariance. The proof of the resulting homotopy-invariance theorem involves some interesting ("fibrational") structures coming from categorical logic.

For more information, please contact Math Department by phone at 626-395-4335 or by email at [email protected].