Grothendieck's theory of fibred categories for monoids

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.