Yang-Baxter elements and a new proof of homological stability for the mapping class group of surfaces

A Yang-Baxter element in a monoidal category gives a weak form of braiding. We explain how such an element allows to define a semi-simplicial set whose connectivity rules homological stability for certain automorphism groups in the category, and how this can be used in the category of bimarked surfaces to give a quite direct proof of slope 2/3 stability of the homology of the mapping class groups of surfaces. This is joint work with Oscar Harr and Max Vistrup.