Topological Galois theory

Abstract

We introduce an abstract topos-theoretic framework for building Galois-type theories in a variety of different mathematical contexts; such theories are obtained from representations of certain atomic two-valued toposes as toposes of continuous actions of a topological group. Our framework extends Grothendieck's theory of Galois categories and allows to build Galois-type equivalences in new contexts, such as for example graph theory and finite group theory

Sites whose topoi are the smooth representations of locally prodiscrete monoids

“…In Section 14, we show how our Y -site and a grid give rise to an ultrahomogeneous object used in Caramello's work [3].…”

“…The main theorem in [3] is Theorem 3.5, but we look instead at the author's Proposition 4.3, which is very close to the former theorem, and closer to our setup.…”

“…Proof. We prove that the pair (C 0 , ι 0 ) satisfies property (3). It remains to be proven that the pair (C 0 , ι 0 ) satisfies property (4).…”

“…We remark here that the underlying category C of a site (C, J) that we consider is opposite to that in [3].…”

Universal locally finite maximally homogeneous semigroups and inverse semigroups

“…In the process of proving part (5) of Theorem 6.1, another interesting structural property of T that we shall establish is that it is isomorphic to its opposite; see Theorem 7.5. Parts (1), (3) and (4) are the most straightforward to prove, so we begin with them. (1) is an immediate consequence of the definition of T and Cayley's Theorem for semigroups.…”