**Venue:** Seminar Room "-1"– Department of Mathematics – Via Sommarive, 14 Povo - Trento**Hour**: 2.30 pm

**Davide Trotta**-*PhD**in Mathematics*

*Abstract:*

During this seminar we will give an introduction to the theory of 2-monads, explaining how it provides an abstract syntax-free approach to universal algebra, and how it allows us to give a precise definition of what it means to be a "property" for a category and what it means to be a "structure".

The main idea is that the notion of algebraic extra structure on a category is somewhat wider than that of "algebra for a 2-monad". In an example so simple as that of finite products, we know precisely in what sense the structure is "unique to within a unique isomorphism"; but it is not so obvious what such uniqueness should mean in the case of a general 2-monad on a 2-category. For example: a category may bear many monoidal structures, but (to within a unique isomorphism) only one structure of “category with finite products”. To capture such distinctions, we consider on a 2-category those 2-monads for which algebra structure is essentially unique if it exists, giving a precise mathematical definition of “essentially unique” and investigating its consequences.

After a general introduction on these topics, we will explain how these results about 2-monad theory can be used to study some completions of primary doctrines in a more general contest, with focus on the existential completion.

**Supervisor: **Roberto Zunino