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Paolo Perrone (Jul 28 2020 at 13:53):Hi all! This is the official thread for Eugenia Cheng's talk, "Distributive laws for Lawvere theories".
When: Thursday July 30th, 12 noon EDT (Boston time)
Zoom meeting:
https://mit.zoom.us/j/280120646
Meeting ID: 280 120 646
Youtube live stream:
https://youtu.be/YCJAzV1g6Xo
Paolo Perrone (Jul 28 2020 at 13:57):Paper here: https://compositionality-journal.org/papers/compositionality-2-1/
Paolo Perrone (Jul 30 2020 at 15:48):Hello! The talk will start in 12 minutes!
Brian Pinsky (Jul 30 2020 at 16:18):Is 0 a natural number here? I'm not sure how one would implement nullary operations here
Paolo Perrone (Jul 30 2020 at 16:18):They are "constants". Think, the neutral element of a monoid
Jesse Sigal (Jul 30 2020 at 16:33):I have an example related to that great cylindrical diagram about monads and algebras.
One can define a weak distributive law for the nonempty powerset monad and the finite distribution monad. The nonempty powerset monad is not finitary, and so does not correspond to a Lawvere theory. However, when you consider the finite fragment (so do the Kleisli construction mentioned), the weak distributive law for the monads shows up as a "distributive tensor" of the Lawvere theories of nondeterminism and probabilistic nondeterminism, where nondeterminism distributes over probabilistic nondetermism
x +p (y | z) = (x +p y) | (x +p z)
where +p is choice with probability p and | is nondeterministic choice. Call this Lawvere theory combined choice.
Thus, the Lawvere theory for combined choice gives an algebra/model based on the full nonempty powerset monad, and corresponds to the wider circle of models on the bottom of the cylinder.
Jesse Sigal (Jul 30 2020 at 16:38):Sources of the example:
K.-H. Cheung, ‘Distributive Interaction of Algebraic Effects’, p. 200.
D. Varacca and G. Winskel, ‘Distributing probability over non-determinism’, Math. Struct. in Comp. Science, vol. 16, no. 01, p. 87, Feb. 2006, doi: 10.1017/S0960129505005074.
A. Goy and D. Petrişan, ‘Combining probabilistic and non-deterministic choice via weak distributive laws’, in Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science, Saarbrücken, Germany, Jul. 2020, pp. 454–464, doi: 10.1145/3373718.3394795.
Brian Pinsky (Jul 30 2020 at 16:42):The periodic internet disconnecting is actually kind of nice; it gives me opportunities to read all the wikipedia pages I open that I usually have to go back and read afterwards
Nathanael Arkor (Jul 30 2020 at 17:09):There's a syntactic presentation of distributive laws for Lawvere theories in this paper by Piróg and Staton.
Valeria de Paiva (Jul 30 2020 at 17:23):@Paolo can we have the slides, please?
Brendan Fong (Jul 30 2020 at 17:23):We're now at the pub here!
https://gather.town/8u0TXRzLpBsKwHjz/mit-categories-seminar
password: yonedalemma
Paolo Perrone (Jul 31 2020 at 14:05):Video here!
https://youtu.be/t4pwM8h8XyY
