Category theory

Our research focuses on categorical algebra: more specifically we develop some new aspects of category theory also useful in

- non-abelian homological algebra;
- higher dimensional homotopical algebra;
- universal algebra and topological algebra;
- descent theory and Galois theory;
- Hopf algebra theory and quandle theory.

We develop (co)homology in non-abelian algebraic categories in several different ways: via exactness properties of higher categories; via higher-dimensional centrality, defined in terms of categorical Galois theory in semi-abelian categories and in related contexts; via higher-order and abstract versions of Schreier–Mac Lane extension theory, which leads to the concept of butterflies and an approach to cohomology of monoids; via the study of protoadditive functors and of non-abelian torsion theories. We are interested in descent and Galois theory in general, exploring its applications in commutator theory and in universal algebra. We also study categorical properties of specific algebraic structures such as cocommutative Hopf algebras and quandles.

We investigate homotopical algebra in its interactions with homology, via Quillen model structures or via higher categories of fractions, and their relationship with categorical-algebraic conditions.