Monads are used in functional programming as a means of modeling and encapsulating computational effects at an appropriate level of abstraction. In previous work, we have introduced monad-based dynamic logic and have used it for the specification and verification of various monadic functional-imperative algorithms over different monads. The semantics of this logic was originally defined over pa...

The Eilenberg-Moore constructions and a Beck-type theorem for pairs of monads are described. More specifically, a notion of a Morita context comprising of two monads, two bialgebra functors and two connecting maps is introduced. It is shown that in many cases equivalences between categories of algebras are induced by such Morita contexts. The Eilenberg-Moore category of representations of a Mor...

Consider categories with involutions which fix objects, functors which preserve involution, and natural transformations. In this setting certain natural adjunctions become universal and, thereby, become constructible from abstract data. Although the formal theory of monads fails to apply and the Eilenberg-Moore category fails to fit, both are successfully adapted to this setting, which is a 2-c...

Talk 1: Big Goal of Alg Top, operads and model categories, fix notation for model categories, remarks about how difficult it is to verify model category axioms. Motivation from equivariant spectra, and discussion of Kervaire. Monoidal model categories, define the inherited model structure on the category of algebras over an operad. Basic facts about Bousfield localization. Preservation theorem ...

Containers are a neat representation of a wide class of set functors. We have previously [1] introduced directed containers as a concise representation of comonad structures on such functors. Here we examine interpreting the opposite categories of containers and directed containers. We arrive at a new view of a di↵erent (considerably narrower) class of set functors and monads on them, which we ...

We consider representations of quivers in arbitrary categories and twisted representations of quivers in arbitrary tensor categories. We show that if A is an abelian category, then the category of representations of a quiver in A is also abelian, and that the category of twisted linear representations of a quiver is equivalent to the category of linear (untwisted) representations of a different...