These lecture notes were written for a short course to be delivered in March 2017 at the Atlantic Algebra Centre of the Memorial University of Newfoundland, Canada. Folklore says that (Hopf) bialgebras are distinguished algebras whose representation category admits a (closed) monoidal structure. Here we discuss generalizations of (Hopf) bialgebras based on this principle. • The first lecture is...

The context of this article is the program to develop monoidal bicategories with a feedback operation as an algebra of processes with applications to concurrency theory The objective here is to study reachability minimization and minimal realization in these bicategories In this set ting the automata are cells in contrast with previous studies where they appeared as objects As a consequence we ...

We address the question of how elegantly to combine a number of different structures, such as finite product structure, monoidal structure, and colimiting structure, on a category. Extending work of Marmolejo and Lack, we develop the definition of a pseudo-distributive law between pseudo-monads, and we show how the definition and the main theorems about it may be used to model several such stru...

We show that the basic categorical concept of an s-algebra as derived from the theory of Segal’s Γ-sets provides a unified description of several constructions attempting to model an algebraic geometry over the absolute point. It merges, in particular, the approaches using monöıds, semirings and hyperrings as well as the development by means of monads and generalized rings in Arakelov geometry....

The concept of refinement in type theory is a way of reconciling the “intrinsic” and the “extrinsic” meanings of types. We begin with a rigorous analysis of this concept, settling on the simple conclusion that the type-theoretic notion of “type refinement system” may be identified with the category-theoretic notion of “functor”. We then use this correspondence to give an equivalent type-theoret...

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 ...

We introduce a generalisation of monads, called relative monads, allowing for underlying functors between different categories. Examples include finite-dimensional vector spaces, untyped and typed λ-calculus syntax and indexed containers. We show that the Kleisli and Eilenberg-Moore constructions carry over to relative monads and are related to relative adjunctions. Under reasonable assumptions...

We introduce update monads as a generalization of state monads. Update monads are thecompatible compositions of reader and writer monads given by a set and a monoid. Distributivelaws between such monads are given by actions of the monoid on the set.We also discuss a dependently typed generalization of update monads. Unlike simple updatemonads, they cannot be factored into a ...