Richard Southwell Being Norman Wildberger

41:31 That point of perspectivity is invisible from the perspective of the camera!  

See Coxeter's Projective Geometry on the Internet Archive. 

Subscribe to Richard Southwell.

This is a placeholder for Norman's upcoming video on polynomial functors in terms of slice categories and adjunctions or something, ... 

Well, Norman didn't show up, so I found this talk by Simon Willerton on the Categorical notion behind the Legendre-Fenchel Transform.

1:06:16 Interesting question about the Cauchy completion of the rationals. It reminded me of the theorem of Kronecker for some reason.

See Lawvere's 1984 paper State Categories, Closed Categories and the Existence of Semi-continuous Entropy Functions.

I think that if you just consider a finite complete lattice then you will get most of this structure (of the Legendre-Fenchel transform) in a topological space. Then maybe you can extend it to an infinite lattice using some model in projective geometry. So one side of the Fenchel duality you have the infs/sups and on the other side you have the points/lines. Then to get the classical Legendre-Fenchel theorem you would need a model that was a vector space and its dual and that's where the metric would come from. I am not sure this is how could do Category theory, but it's how I would try to program a module to implement this in Standard ML. It's interesting to me that the soundness of higher-order functors in Standard ML was proved using the Grothendieck construction. I have a feeling that, because of this, any category you construct using that module system must have some correspondence, somehow, to the theory that models the soundness of the type system: it must be in some sense an instance of that construction, mustn't it? Well, it would be a nice property for a programming language to have, I think. See Harper, Mitchell and Moggi Higher-order modules and the phase distinction and System Fω And ML Module Semantics, also Simon Peyton-Jones on impredicativity in this post of mine.

Subscribe to Applied Category Theory @ UCR.

In this famous quote of Grothendieck's he uses three pairs as his example of how the theory connects streams of mathematics, continuous and discrete:

 

See The Category Enriched over the Category of Finite Sets, The Finitely Triangulated Manifold and the Magnitude of a Finite Category.

 

How's that for perspectivity Norman? See Thierry Coquand on Computational Interpretation of Topos Theory.

Subscribe to L’Università degli Studi dell’Insubria.