Clifford semigroups and the monoidal Grothendieck construction

Joint work with Elena Caviglia, Peter F. Faul and Graham Manuell.

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 Cat(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.

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