Kathryn Hess - Comonads to Calculus



55:12 Wouldn't it be great if you could cook up a comonad that gives a tower of categories that sum to differential calculus on a real vector space? See below. 

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What is a comonad on a category? It's just a monad on the opposite category: 

... "Set theory is linear coalgebra on vector spaces": 

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But they have to make everything much more complicated:

We discuss differentiable smoothness structures on 𝑹⁴ from three different categorical perspectives. The first one relies on considering open atlases on 𝑹⁴ with certain (not all) of its local charts residing in a smooth topos. Thus exotic smooth functions on 𝑹⁴ are finelly approached without any use of Casson handles and handle decompositions. The second approach takes into account entire space of all smoothness structures on 𝑹⁴ . Forcing extensions naturally order the structures and show a way towards new smooth invariants of exotic 𝑅⁴'s. The third approach shows how logical structure of quantum mechanics enforces exotic smoothness at large cosmological scales.

For a gentle introduction to these issues, see Frederic Schuller's lectures, in particular this one on Differentomorphism classes of real n-spaces. 

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