27th of May, 2021. Part of the Topos Institute Colloquium.
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Abstract: The general higher-categorical semantics of homotopy type theory involves (∞,1)-toposes and Quillen model categories. However, for many applications it suffices to consider (2,1)-toposes, which are reasonably concrete categorical objects built out of ordinary groupoids. In this talk I'll describe how to interpret homotopy type theory in (2,1)-toposes, and some of the applications we can get from such an interpretation. I will assume a little exposure to type theory, as in Dan Christensen's talk from April, but no experience with higher toposes or homotopy theory. This talk will also serve as an introduction to some basic notions of Quillen model categories