Homotopy in semi-abelian categories: an overview

In the first part of the talk we give an overview of some major differences between abelian and semi-abelian categories from a homotopical-algebraic point of view. Many constructions which are common in the abelian context become impossible to carry out once the hom-sets lose their additive structure. One way to deal with this problem is to use simplicial techniques [2, 4], but perhaps this is not the only solution. In the second part we focus on joint work-in-progress with Mathieu Duckerts-Antoine towards homotopy of maps between objects of a given semi-abelian category—a non-additive version of the homotopy theory introduced in [1, 3]. References [1] B. Eckmann, Homotopie et dualité, Colloq. Topologie Algébrique, Louvain (1956), 41–53. [2] T. Everaert and T. Van der Linden, Baer invariants in semi-abelian categories II: Homology, Theory Appl. Categ. 12 (2004), no. 4, 195–224. [3] P. J. Hilton, Homotopy theory of modules and duality, Proc. Mexico Sympos. (1958), 273–281. [4] T. Van der Linden, Simplicial homotopy in semi-abelian categories, J. K-Theory 4 (2009), no. 2, 379–390