## The topological behaviour category of an algebraic theory

### Richard Garner – 22 June 2022

In this talk, we explain how the topological comodels of any (finitary) algebraic theory T are the continuous left B-sets for a certain topological category B, called the topological behaviour category. We compute these topological behaviour categories explicitly for various theories of interest.
If time permits, we explain how this gives rise to a Galois adjunction between topological categories and algebraic theories, whose fixpoints are, to one side, the source-etale topological categories with Stone space of objects; and, to the other, the algebraic theories with cartesian closed category of models.

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