An introduction to Sweedler Theory
Ulrik Sørgaard Djupvik
Tallinn University of Technology
Abstract:
This talk will be divided into two parts.
In the first part, I will cover the main results from Sweedler theory in the classical setting of vector spaces over a field. In particular, I will first describe the finite dual and show how this construction may be used to conduct the cofree coalgebra construction. Thereafter, I will define measuring maps using the structure of the convolution algebra; while also providing a few examples of such maps. Combining these two constructions, I will describe some of the Sweedler operations, which will allow us to define what is meant by the term "Sweedler theory".
In the second part, I will show how this theory naturally lives in the framework of
V-graded categories. In particular I will describe how three of the Sweedler operations may be described as (co)represented functors. I will then describe how measuring maps may be defined in the context of duoidal categories, which witnesses monoids being graded by comonoids. Lastly, I will discuss how this phenomenon can be lifted to the context of modules and comodules, inspiring the use of
V-graded categories in a double categorical framework.