Clifford semigroups and the monoidal Grothendieck construction

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.