Another Excellent Talk on HoTT by Steve Awodey

First watch this talk by Andrej Bauer on Type Universes


If you're not afraid of going insane, read Per Martin-Löf's On The Meanings of the Logical Constants and the Justifications of the Logical Laws. (1983). See also Wikipedia's Intuitionistic type theory - Martin-Löf type theories. If that does drive you nuts, then you have nothing to lose so you can just go ahead and read Girard et al's Proofs and Types.

Here's Steve's talk:

At 12:00 he mentions that the idea of infinity groupoids came from topology and that they turned up in MLTT as identity types, but they represent the abstract stucture of a space, if you look at its paths and path between paths and so on, ad infinitum. At 15:42 on the different picture of the Universe of all types as points on a two-dimensional plane with the homotopy type on one axis and the cardinality on the other: once you take into account the homotopy types you are able to reason effectively about the Universes. There is an analogy here in physical theories too. See Foundations of Physics and Mathematics.

His slides are here: https://awodey.github.io/talks/AwodeyHIM2024.pdf.

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And I just found this talk by Steve on Category Theory Foundations. He says Category Theory is the abstract algebra of functions. I knew that, but I didn't know that I knew it until just now! See  Emily Riehl Doing Stratospheric Category Theory.

Part II


Part III:


 Part IV:

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A few days ago I also found this more philosophical take of his on Mac Lane and Carnap's Logical Syntax of Language given on 14 October 2005.

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