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Tim Hosgood (Mar 17 2021 at 19:21):Wednesday, 22:00 UTC
Tim Hosgood (Mar 17 2021 at 19:21):Abstract:
A set-theoretic polynomial functor is naturally a combinatorial object, which might be imagined as an abacus: a collection of columns, each equipped with a set of beads. Moving quickly but with little outside knowledge beyond the Yoneda lemma, I will use these abacuses to very concretely tell the story of P, the framed bicategory of polynomial comonads and bicomodules. In particular, I'll discuss both theory and applications of P.
For theory I'll begin with an overview of some pleasing properties enjoyed by the category of polynomials. Then I'll explain Ahman-Uustalu's result that—up to isomorphism—polynomial comonads are categories and Garner's result that the bicomodules between them are parametric right adjoints between the associated copresheaf categories. In particular every copresheaf topos is found as a hom-category in P. I'll also discuss the cofree comonad construction.
For applications, I'll recall the notion of Moore machines and show that they are objects in a copresheaf topos C-Set for a particular category C; I'll explain how the story generalizes to arbitrary C. I'll also explain how databases and data migration functors live naturally in this setting. Time permitting I may also discuss how deep learning and cellular automata show up in P.