Dear all, Alyssa Renata (Imperial College London) will give a talk for a second edition of this month's PCT seminar (https://pctseminar.github.io/) on Friday August 28 at 10am JST/11am AED (1am UTC). The zoom link to join the seminar is: https://kyoto-u-edu.zoom.us/j/82401794000?pwd=VjVmzVjmlgxhkdHES4xYeQBTpMzQkS... Title: The Game Spectrum of a Monad? Abstract: In recent work joint with Richard Garner and my supervisor Nicolas Wu, inspired by the computational effects interpretation of monads and algebraic theories due to Moggi, Plotkin and Power, we constructed an adjunction between the category of monads on set and the category of small categories and retrofunctors/cofunctors. Under the computational effects interpretation, the monad encodes computations which may interact with or affect some environment. The left adjoint to this construction assigns to each monad what is called its **behaviour category**, which may be seen as a model of the environment itself: the objects are the possible states, and the morphisms transitions between states. This adjunction ought to be seen as a kind of duality, with the behaviour category acting as the spectrum of a monad. The first half of the talk will be dedicated to reviewing this construction and its surrounding adjunction. The second half of the talk is about my recent attempt with Richard at rectifying a defect of the theory: many monads have trivial behaviour category! This is undesirable because of an analogy we want to develop with schemes: the monads replace the (commutative) rings, while the behaviour category should replace the affine schemes. But of course, an affine scheme/spectrum should faithfully encode its corresponding ring/monad, and this is evidently not so for the behaviour category. Our proposal then is to replace the behaviour category by a particular 2-player game (between the program and the environment), from which the data of the behaviour category can be extracted by inspecting its (winning) strategies. The point is that the behaviour category is trivial when there are no winning strategies, but this need not imply triviality of the game itself. To be more precise, we will investigate the relationship with the template games of Paul-Andre Mellies, which he used to build higher-dimensional models of linear logic. We hope to see you there! Talks after posted on youtube afterwards: https://www.youtube.com/@PCTSeminar Best wishes, Soichiro Fujii, Zeinab Galal, JS PL