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Theories, Sites, Toposes: Relating and studying mathematical theories through topos-theoretic 'bridges'
According to Grothendieck, the notion of topos is "the bed or deep river where come to be married geometry and algebra, topology and arithmetic, mathematical logic and category theory, the world of the continuous and that of discontinuous or discrete structures." It is what he had "conceived of most broad to perceive with finesse, by the same language rich of geometric resonances, an "essence" which is common to situations most distant from each other, coming from one region or another of the vast universe of mathematical things."
The aim of this book is to present a theory and a number of techniques which allow to give substance to Grothendieck's vision by building on the notion of classifying topos educed by categorical logicians. Mathematical theories (formalized within first-order logic) give rise to geometric objects called sites; the passage from sites to their associated toposes embodies the passage from the logical presentation of theories to their mathematical content, i.e. from syntax to semantics.
The essential ambiguity given by the fact that any topos is associated in general with an infinite number of theories or different sites allows to study the relations between different theories, and hence the theories themselves, by using toposes as 'bridges' between these different presentations. The expression or calculation of invariants of toposes in terms of the theories associated with them or their sites of definition generates a great number of results and notions varying according to the different types of presentation, giving rise to a veritable mathematical morphogenesis.
The aim of this book is to present a theory and a number of techniques which allow to give substance to Grothendieck's vision by building on the notion of classifying topos educed by categorical logicians. Mathematical theories (formalized within first-order logic) give rise to geometric objects called sites; the passage from sites to their associated toposes embodies the passage from the logical presentation of theories to their mathematical content, i.e. from syntax to semantics.
The essential ambiguity given by the fact that any topos is associated in general with an infinite number of theories or different sites allows to study the relations between different theories, and hence the theories themselves, by using toposes as 'bridges' between these different presentations. The expression or calculation of invariants of toposes in terms of the theories associated with them or their sites of definition generates a great number of results and notions varying according to the different types of presentation, giving rise to a veritable mathematical morphogenesis.
- GenresMathematics
382 pages, Hardcover
Published February 21, 2018
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June 22, 2019
The mathematical promise of using (Grothendieck) toposes as 'bridges' to connect different mathematical settings stems from several observations:
1) Every Grothendieck topos is a classifying topos of a first-order geometric theory
2) Every geometric theory is classified by a Grothendieck topos (indeed, we can canonically construct a topos via the category of sheaves of on the syntactic category)
3) There are first-order geometric theories which are not bi-interpretable (i.e. their syntactic categories are not equivalent), but can nonetheless be related on the level of toposes if they are Morita equivalent (i.e. they are classified by the same topos, up to equivalence)
In this monograph, Caramello lays the foundation of this technique, particularly with a view towards: (i) understanding the interaction between the logical and geometric/topological aspects of a topos; and (ii) understanding presheaf-type toposes, and how they relate to the rest of Groth. toposes.
The first 4 chapters are devoted to introducing the preliminary background to topos theory and developing aspects the bridge technique, and are fairly readable. I found the background chapter rather terse and difficult to digest, given that it was my first proper point of contact with topos theory, but the second time reading was easier.
Chapters 5-8 were very technical, and I confess I skimmed a lot of it - I basically decided I didn't really need to know the nitty-gritty of how these results were proved, I just needed to know the results themselves. Anyway, these chapters are motivated by the fact that every Grothendieck topos can be understood as a subtopos of a presheaf topos - Caramello unpacks what this means on the logical side of things (among other things) in painstaking detail.
Chapters 9-10 were more pleasant to read because they discussed applications of the bridge technique, and they serve (if nothing else) as a quick primer on the work that Olivia has been doing in applying this bridge technique to other areas of mathematics. Slightly disappointed that I didn't see the stuff about Nori motives or "independence of l", but this is treated in her Habilitation, available on her website, so I'll read about those applications there.
1) Every Grothendieck topos is a classifying topos of a first-order geometric theory
2) Every geometric theory is classified by a Grothendieck topos (indeed, we can canonically construct a topos via the category of sheaves of on the syntactic category)
3) There are first-order geometric theories which are not bi-interpretable (i.e. their syntactic categories are not equivalent), but can nonetheless be related on the level of toposes if they are Morita equivalent (i.e. they are classified by the same topos, up to equivalence)
In this monograph, Caramello lays the foundation of this technique, particularly with a view towards: (i) understanding the interaction between the logical and geometric/topological aspects of a topos; and (ii) understanding presheaf-type toposes, and how they relate to the rest of Groth. toposes.
The first 4 chapters are devoted to introducing the preliminary background to topos theory and developing aspects the bridge technique, and are fairly readable. I found the background chapter rather terse and difficult to digest, given that it was my first proper point of contact with topos theory, but the second time reading was easier.
Chapters 5-8 were very technical, and I confess I skimmed a lot of it - I basically decided I didn't really need to know the nitty-gritty of how these results were proved, I just needed to know the results themselves. Anyway, these chapters are motivated by the fact that every Grothendieck topos can be understood as a subtopos of a presheaf topos - Caramello unpacks what this means on the logical side of things (among other things) in painstaking detail.
Chapters 9-10 were more pleasant to read because they discussed applications of the bridge technique, and they serve (if nothing else) as a quick primer on the work that Olivia has been doing in applying this bridge technique to other areas of mathematics. Slightly disappointed that I didn't see the stuff about Nori motives or "independence of l", but this is treated in her Habilitation, available on her website, so I'll read about those applications there.
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