Monoids, Monoidal Grothendieck Construction and Clifford Semigroups

The Grothendieck construction exhibits one of the most fundamental equivalences in category theory. It gives the correspondence between Grothendieck fibrations and indexed categories, allowing one to freely move between the two worlds and enjoy the perks of both. In the last years, the theory of the Grothendieck construction has been actively expanded and generalized to broader settings, as well as successfully applied to both category theory itself and most of the areas of mathematics. Notably, Joe Moeller and Christina Vasilakopoulou extended the Grothendieck construction to the monoidal setting, motivated by abundant examples in algebra, dynamical systems, graphs and networks. They established an equivalence between monoidal fibrations and lax monoidal pseudofunctors into $\mathrm{Cat}$.

In this talk, we present new categorical results on the monoidal Grothendieck construction as well their applications to the theory of Clifford semigroups and inverse semirings. Clifford semigroups are semigroups equipped with a weak form of inverses. In the commutative case, they are also known as inverse semigroups. They encompass a much broader set of examples, including partial bijections and semilattices. Inverse semirings are then semirings whose additive monoid is an inverse semigroup. A remarkable example is given by bounded polynomials.

We present what happens when taking monoids in the monoidal Grothendieck construction. We prove that monoids in the monoidal total category given by the monoidal Grothendieck construction precisely correspond to the Grothendieck construction of a pseudofunctor that takes monoids in the fibres. The monoids in the fibres are here considered with respect to a structure of monoidal category that is induced from the starting monoidal indexed category.

We then apply this result and the monoidal Grothendieck construction to categorically capture Clifford semigroups and inverse semirings. We show that Clifford semigroups equivalently correspond to discrete fibrations in groups over a meet semilattice. We then prove that, exactly as discrete fibrations can be collected into a Grothendieck fibration over $\mathrm{Cat}$, the category of Clifford semigroups can be obtained as a monoidal Grothendieck construction over the category of semilattices. This extends the Structure Theorem for Clifford semigroups, giving rise to new applications in semigroup theory. In particular, we obtain results on factorization systems for Clifford semigroups.

Finally, we apply our result on monoids in the monoidal Grothendieck construction to the monoidal total category of inverse semigroups, in the commutative case. We obtain that inverse semirings equivalently correspond to taking monoids in the fibres, which are functor categories into the category $\mathrm{Ab}$ of abelian groups. We prove that, surprisingly, the induced monoidal structure on the fibres is precisely that of the Day convolution.