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I’ve added references at cohesive topos and Birkhoff’s theorem to Lawvere’s recent paper Birkhoff’s Theorem from a geometric perspective: A simple example, which appears in an obscure Iranian journal.
Hm, I had forgotten about the entry on the “other” Birkhoff theorem (the Birkhoff-von Neumann theorem on doubly stochastic matrices), where Ian Durham and quantum channels participated. From the looks of it, some clean-up and expansion would be in order.
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