The Fundamental Groupoid and the Geometry of Monoids

This thesis is divided in two equal parts. We start the first part by studying the Kato-spectrum of a commutative monoid M, denoted by KSpec(M). We show that the functor M → KSpec(M) is representable and discuss a few consequences of this fact. In particular, when M is additionally finitely generated, we give an efficient way of calculating it explicitly. We then move on to study the cohomology theory of monoid schemes in general and apply it to vector- and particularly, line bundles. The isomorphism class of the latter is the Picard group. We show that under some assumptions on our monoid scheme X, if k is an integral domain (resp. PID), then the induced map Pic(X) → Pic(Xk) from X to its realisation is a monomorphism (resp. isomorphism). We then focus on the Pic functor and show that it respects finite products. Finally, we generalise several important constructions and notions such as cancellative monoids, smoothness and Cartier divisors, and prove important results for them. The main results of the second part can be summed up in fewer words. We prove that for good topological spaces X the assignment U → II₁(U) is the terminal object of the 2-category of costacks. Here U is an open subset of X and II₁(U) denotes the fundamental groupoid of U. This result translates to the étale fundamental groupoid as well, though the proof there is completely different and involves studying and generalising Galois categories.