Clifford semigroups and the monoidal Grothendieck construction
arXiv:2607.12944
The paper shows that the known correspondence between Clifford semigroups and functors from a semilattice to groups arises from the monoidal Grothendieck construction, uses this to describe the whole category of Clifford semigroups and build factorisation systems, and extends the approach to relate inverse semirings with lax monoidal functors into abelian groups.
Abstract
Clifford semigroups are known to correspond to functors from a semilattice into the category of groups. We show that this correspondence is an instance of the monoidal Grothendieck construction. Moreover, applying the Grothendieck construction to the functor sending a semilattice L to the functor category [L, Grp] yields the category of all Clifford semigroups. We use this to construct a number of factorisation systems on the category of Clifford monoids. Finally, we prove a general result on taking monoids in a monoidal fibration and apply it to give a correspondence between inverse semirings and lax monoidal functors from idempotent semirings into the category of abelian groups.
30 pages