Sheaf Theory, Probability and Topology

See Popper's Non-dualist Theory of Objective Knowledge and the remark Popper made in 1977 about abstract invariants being causally efficient:

I think that these considerations tell us a lot about natural selection. While Darwin still worried that he could not explain variation, and while he felt uneasy about being forced to look at it as chancelike, we can now see that the chancelike character of mutations, which may go back to quantum indeterminacy, explains how the abstract invariances of the environment, the somewhat abstract selection pressures, can, by selection, have a downward effect on the concrete living organism — an effect that may be amplified by a long sequence of generations linked by heredity.

See Samson Abramsky's talk on Quantum Monads in Martin Elsman - Deriving a Kronecker-Free Functional Quantum Simulator where he explains the sheaf-theory approach to contextuality in Quantum Mechanics:

See The Sheaf-Theoretic Structure Of Non-Locality and Contextuality by Samson Abramsky and Adam Brandenburger. 

Then see Mike Titelbaum and Ted Theodosopoulos - Topological Obstacles to Shared Priors where some significant topological features arise in theories about measure spaces and sigma-algebras:


See the paper Topological Obstructions to Shared Priors by Owen D. Biesel, Colin McSwiggen, Ted Theodosopoulos and Michael G. Titelbaum.

There are some really interesting features of stochasticity in terms of the kind of abstract causation which it can support!

 

10:45 We often seem to employ arguments in reverse order to the intuitions which produced them: 

39:14 Indivisibility turns up all over the place, especially in biology. 

44:37 What role the wave function plays in the theory:

It's the indivisibility condition which ultimately produces the interference phenomena in the representation by a Markov process cf 'revealed preference theory' in economics in the 'topological obstacles' discussion above 50:28:

See also the bit about higher-order probabilities at 25:00 - 29:00 in the 'topological obstacles' discussion above.

See also Jacob Barandes' Talk at ESA Quantum Technologies Conference 2025.

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