Robert Atkey - Containers

MSP Strathclyde: the sound gets better after 5 minutes or so, ... Containers are described like polynomial functors and you can get a category out of them in about 24 minutes and 41 seconds. By 37 minutes 42 seconds you can represent constructive propositions and some sort of smooth manifolds. By 50 minutes and 6 seconds you get container morphisms and you can make a goal-directed theorem prover thanks to Gordon, Milner and Wadsworth. 

 

There's more:

 

And there are five lectures so far on Tangent Categories:

 

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