Sweedler duals for monads

Abstract: While the linear dual of any coalgebra is an algebra, the converse is not true; however, there is an adjoint to the coalgebra-to-algebra functor, given by the so-called Sweedler dual.

There is a notion of “linear dual” for an endofunctor of Set, given by homming into the identity functor for the Day convolution structure. Again, this sends comonads to monads, but not vice versa; but again, there is an adjoint. This “Sweedler dual” comonad of a monad was introduced by Katsumata, Rivas and Uustalu in 2019.

The purpose of this talk is to give an explicit construction of the Sweedler dual comonad of any monad on Set. The category of coalgebras for the Sweedler dual turns out to be a presheaf category, whose indexing category can be described explicitly in terms of a kind of computational dynamics of the monad. If time permits, we also describe the source-etale topological category which classifies the topological Sweedler dual comonad of a monad on Set; in particular, this recovers all kinds of etale topological groupoids of interest in the study of combinatorial $C^*$-algebras.

algebraic topologycategory theoryquantum algebra

Audience: learners

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