Topological Galois theory.

*(English)*Zbl 1401.18007The article extends Grothendieck’s generalisation of Galois theory: it does so by studying the representation of some atomic two-valued toposes as toposes of continuous actions over a topological group. In this way, equivalences of Galois type are established in new contexts, such as graphs and finite groups.

The long, complex, and very deep paper starts by analysing Grothendieck’s theory of Galois categories along the lines which will be developed in the rest of the article: in particular, it discusses how the scope of Grothendieck’s theory is narrower than the introduced approach. Section 2 is devoted to construct the necessary relationships between topological groups and the associated categories of continuous actions on discrete sets. Specifically, the notion of algebraic base is introduced and used to obtain a representation theorem for the topos of continuous actions over a topological group in terms of an algebraic base. In turn, the topological groups that are isomorphic to the group of automorphisms of the canonical point of the associated topos of continuous actions, are characterised in terms of algebraic bases. These results provide the fundamental set of tools to develop the subsequent parts.

Section 3 states and proves the main representation theorem in the article, which provides sufficient conditions to say when a topos of sheaves equipped with the atomic topology is equivalent to the topos of continuous actions over the group of automorphisms on a suitable point, endowed with a suitable topology. Here, the word “suitable” indicates where the sufficient conditions act.

The equivalence of toposes in the main representation theorem is induced by a functor \(F: \mathbb{C}^{op} \to \mathbf{Cont}_t(\mathrm{Aut}_{\mathbb{C}}(u))\). Section 4 analyses under which conditions \(F\) is full and faithful. This analysis yields an elementary process to complete a category \(\mathbb{C}\), which satisfies the conditions to make \(F\) full and faithful, so that \(F\) becomes an equivalence. In turn, the analysis requires to discuss the relationship between regular and strict monomorphisms, and what can be said about the relationship between \(\mathbb{C}\) and \(\mathrm{Aut}_{\mathbb{C}}(u)\) when \(F\) is not full and faithful.

Section 5 illustrates many examples of concrete Galois theories obtained applying the results in Sections 3 and 4: in particular, Galois theories are derived from graphs, Boolean algebras, finite groups, and linear orders. Also, the classical Galois theory and Grothendieck’s one are shown to be instances of the general framework, among other results.

Section 6 summarises the main methods to synthesise Galois theories applying the techniques and results in Section 3 and 4. The Conclusions illustrate some promising areas in which these results can be applied or could be extended to.

The paper is long, very deep, really significant, and just scratches the surface of possible applications, although those proposed in Section 5 are important by themselves and illuminating about the level of generality of the framework.

Despite the effort the author put in making the work accessible, which allows the casual reader to get an idea of the contribution, the reader who wants to fully understand the depth of this important work has to master topos theory, has to have a good knowledge of Grothedieck’s Galois theory, and should have at least some acquaintance with the topose-as-bridges paradigm, see, e.g., [O. Caramello, Ann. Pure Appl. Logic 167, No. 9, 820–849 (2016; Zbl 1343.18005)].

The long, complex, and very deep paper starts by analysing Grothendieck’s theory of Galois categories along the lines which will be developed in the rest of the article: in particular, it discusses how the scope of Grothendieck’s theory is narrower than the introduced approach. Section 2 is devoted to construct the necessary relationships between topological groups and the associated categories of continuous actions on discrete sets. Specifically, the notion of algebraic base is introduced and used to obtain a representation theorem for the topos of continuous actions over a topological group in terms of an algebraic base. In turn, the topological groups that are isomorphic to the group of automorphisms of the canonical point of the associated topos of continuous actions, are characterised in terms of algebraic bases. These results provide the fundamental set of tools to develop the subsequent parts.

Section 3 states and proves the main representation theorem in the article, which provides sufficient conditions to say when a topos of sheaves equipped with the atomic topology is equivalent to the topos of continuous actions over the group of automorphisms on a suitable point, endowed with a suitable topology. Here, the word “suitable” indicates where the sufficient conditions act.

The equivalence of toposes in the main representation theorem is induced by a functor \(F: \mathbb{C}^{op} \to \mathbf{Cont}_t(\mathrm{Aut}_{\mathbb{C}}(u))\). Section 4 analyses under which conditions \(F\) is full and faithful. This analysis yields an elementary process to complete a category \(\mathbb{C}\), which satisfies the conditions to make \(F\) full and faithful, so that \(F\) becomes an equivalence. In turn, the analysis requires to discuss the relationship between regular and strict monomorphisms, and what can be said about the relationship between \(\mathbb{C}\) and \(\mathrm{Aut}_{\mathbb{C}}(u)\) when \(F\) is not full and faithful.

Section 5 illustrates many examples of concrete Galois theories obtained applying the results in Sections 3 and 4: in particular, Galois theories are derived from graphs, Boolean algebras, finite groups, and linear orders. Also, the classical Galois theory and Grothendieck’s one are shown to be instances of the general framework, among other results.

Section 6 summarises the main methods to synthesise Galois theories applying the techniques and results in Section 3 and 4. The Conclusions illustrate some promising areas in which these results can be applied or could be extended to.

The paper is long, very deep, really significant, and just scratches the surface of possible applications, although those proposed in Section 5 are important by themselves and illuminating about the level of generality of the framework.

Despite the effort the author put in making the work accessible, which allows the casual reader to get an idea of the contribution, the reader who wants to fully understand the depth of this important work has to master topos theory, has to have a good knowledge of Grothedieck’s Galois theory, and should have at least some acquaintance with the topose-as-bridges paradigm, see, e.g., [O. Caramello, Ann. Pure Appl. Logic 167, No. 9, 820–849 (2016; Zbl 1343.18005)].

Reviewer: Marco Benini (Buccinasco)

##### Keywords:

Grothendieck topos; Galois theory; Galois category; atomic topos; automorphism group; ultrahomogeneous structure; atomic and complete theory
Full Text:
DOI

##### References:

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