Co-smash products in semi-abelian algebra

Van der Linden, Tim [UCL]

We give an overview of the use of co-smash products and the induced (possibly higher-order) Higgins commutators in semi-abelian algebra. The idea is to express other commutators (the Huq commutator, the Smith commutator, the weighted commutator) in terms of the Higgins commutator, and use these expressions to characterise categorical-algebraic properties. We focus on (i) exactness conditions of the change-of-base functors in the fibration of points and (ii) characterising internal categories and (higher-dimensional) central extensions.