Andrea Cappelletti will give a talk on
Extensive categories of comonoids in pointed categories with smash product
Abstract:
To quote a well-known slogan in the category theory community: “extensive categories are categories in which sums exist and are well-behaved”. In this spirit, Carboni and Janelidze proved that for any good extensive category, the category of pointed objects inherits a canonical monoidal structure via the smash product (which is a categorical generalization of the classical smash product of topological spaces). Moreover, they established that the original extensive category can be fully recovered as the category of comonoids equipped with diagonal comultiplication.
This raises a natural inverse question: given a pointed category equipped with a well-behaved smash product, is the category of diagonal comonoids extensive?
The talk addresses the problem by investigating the structural properties of this category. Alongside our main result, we will show how the proposed framework allows for the recovery of several well-known algebraic constructions.