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An Extension of the Galois Theory of Grothendieck
71 pages, Paperback
First published December 31, 1984
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July 2, 2019
The type-setting of this monograph makes it really hard to read sometimes - not clear when proofs end, and another section begins. I am also told that the exposition on descent theory for toposes is done better in Johnstone's Elephant. We shall see. I found the terminology used here somewhat confusing (what we normally call a frame, the authors call a locale, i.e. a poset admitting arbitrary suprema + finite infima satisfying a distributive law; and what we call frame, they call a "Space"), and some of the reasoning was a bit too quick for me in some areas.
Anyway, one theme I found interesting was how toposes provide a setting for understanding how algebra and geometry (or "spatial") concepts interact. The monograph begins with the study of locales in an arbitrary topos, whose structure has clear analogies with the commutative ring. This leads the authors to develop an algebraic theory of descent in terms of A-modules, which they later generalise to the context of toposes. In order to make this generalisation work, there were a few intervening chapters that introduced and developed the concepts of "openness" (e.g. open subspaces, open mappings etc.) and "spatial reflections", i.e. concepts which emphasised the more spatial aspects of toposes, which were later used to give necessary and sufficient criteria for when we have an effective descent morphism.
I was particularly intrigued by the material on open geometric morphisms - both the geometric and logical interpretations of it. In particular, we have the following proposition:
TFAE:
1) A morphism p: E--> S is open
2) For any open space X of E, p!X is an open space of S.
3) p*:S-->E preserves universal quantification.
2) gives us a more geometric/topological interpretation of what an open geometric morphism is, which is not altogether surprising.
The logical characterisation of open morphisms, however, surprised me. Inverse image functors of geometric morphisms do not, in general, preserve universal quantification which is something we have to be careful about if we want to stay within constructive mathematics. Thus, the result that something spatial could have some genuinely non-trivial logical implications was something I found interesting.
I don't have a particularly clear-headed understanding of how all this machinery gives us an extension of Grothendieck's Galois Theory (the monograph just says a few lines about this in the introduction), and I need to get my hands dirty when it comes to understanding how to use descent methods effectively, but it's a start.
Anyway, one theme I found interesting was how toposes provide a setting for understanding how algebra and geometry (or "spatial") concepts interact. The monograph begins with the study of locales in an arbitrary topos, whose structure has clear analogies with the commutative ring. This leads the authors to develop an algebraic theory of descent in terms of A-modules, which they later generalise to the context of toposes. In order to make this generalisation work, there were a few intervening chapters that introduced and developed the concepts of "openness" (e.g. open subspaces, open mappings etc.) and "spatial reflections", i.e. concepts which emphasised the more spatial aspects of toposes, which were later used to give necessary and sufficient criteria for when we have an effective descent morphism.
I was particularly intrigued by the material on open geometric morphisms - both the geometric and logical interpretations of it. In particular, we have the following proposition:
TFAE:
1) A morphism p: E--> S is open
2) For any open space X of E, p!X is an open space of S.
3) p*:S-->E preserves universal quantification.
2) gives us a more geometric/topological interpretation of what an open geometric morphism is, which is not altogether surprising.
The logical characterisation of open morphisms, however, surprised me. Inverse image functors of geometric morphisms do not, in general, preserve universal quantification which is something we have to be careful about if we want to stay within constructive mathematics. Thus, the result that something spatial could have some genuinely non-trivial logical implications was something I found interesting.
I don't have a particularly clear-headed understanding of how all this machinery gives us an extension of Grothendieck's Galois Theory (the monograph just says a few lines about this in the introduction), and I need to get my hands dirty when it comes to understanding how to use descent methods effectively, but it's a start.
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