Grothendieck's theory of fibred categories for monoids
Ilia Pirashvili
TL;DR
The paper transfers Grothendieck's fibre-category framework to monoids by treating monoids as one-object categories and showing prefibrations correspond to Schreier extensions. It proves a 2-equivalence between lax/pseudo $N$-monoids and fibrations/prefibrations over $N$ via the Grothendieck construction $\mathsf{Groth}(A,\phi,\gamma)$, yielding a $2$-semi-direct product structure and a short exact sequence $1\to A\to \mathsf{Groth}(A,\phi,\gamma)\to M\to 1$. By introducing cleavages, it constructs lax actions on the kernel $A$ and derives automorphism- and cohomology-based exact sequences, extending known group-based results to the monoid setting. In the commutative-kernel case, the work recovers canonical correspondences $H^2(N,A)\cong \mathsf{MonExt}(N,A)$ and $H^2(N,A^{\times})\cong \mathsf{RegMonExt}(N,A)$, aligning Schreier extension theory with Grothendieck's perspective.
Abstract
Grothendieck's theory of fibred categories establishes an equivalence between fibred categories and pseudo functors. It plays a major role in algebraic geometry and categorical logic. This paper aims to show that fibrations are also very important in monoid theory. Among other things, we generalise Grothendieck's result slightly and show that there exists an equivalence between prefibrations (also known as Schreier extension in the monoidal world) and lax functors. We also construct two exact sequences which involve various automorphism groups arising from a given fibration. This exact sequence was previously only known for group extensions.