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Homotopy Type Theory by Vladimir Voevodsky is another possibility. This is an attempt to link Topology and Computer Science


Back in the day there was Feynman's Lectures on Computation. Hint: pdf can be found by searching


See also nLab


one should never forget Jacob Lurie's "Higher Topos Theory" which is 1000 pages just like that


Actually I recommend against readin it as it only covers 2 of the 4 topics you discuss (Topology and Logic). However it certainly has applications to the other two.

The infinity groupoid models of type theory have already revolutionized our understanding of equality in type theory. So far, the 21st century has been an incredible time for logicians.

There are also older, and very different topological models for typed lambda calculi (see e.g. http://www.cs.bham.ac.uk/~mhe/papers/entcs87.pdf). These motivate things like Escardo's "seemingly impossible functional programs" (http://math.andrej.com/2007/09/28/seemingly-impossible-funct...) and, along different lines, Abstract Stone Duality (http://www.paultaylor.eu/ASD/).

Lurie's "On Infinity Topoi"[0] preceded the book and is, I quote[1],

> 50 pages of pure cake, beautifully and informally written. The book is 1000 pages long. There is some new cake there, but not 20 times as much.

[0] https://arxiv.org/abs/math/0306109

[1] https://mathematicswithoutapologies.wordpress.com/2015/05/13...

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