Cohomology and Monoidal Groupoids

In this chapter, we analyze the structure of monoidal abelian groupoids by developing a 3-dimensional Schreier-Grothendieck theory of factor sets that leads to their classification via of 3rd \(\mathsf {D}\) -cohomology classes of monoids. We also consider the case of monoidal abelian groupoids endowed with a coherent monoid \(\Gamma \) -action by monoidal endofunctors. The particular case of categorical groups was dealt with by Grothendieck and Sinh in 1975. A fundamental result about them was their classification by Eilenberg-Mac Lane 3rd-cohomology classes of groups. The appearance of monoidal groupoids in several branches of mathematics makes them interesting in their own right. Here, we focus on the special case of monoidal abelian groupoids (whose automorphism groups are all abelian).