Sunday, September 29, 2013

On Topos Theory

I hesitate to write about Topos Theory since I am not an expert in this area. However it seems to me a subject of great importance which is being presented in a distorted fashion of late.

Grothendieck introduced toposes as a notion both of space and of categories of space, a notion which permits spatial intuition to be applied to widely different parts of mathematics. It is a huge conceptual advance on Hausdorff's notion of topological space.

Lawvere and Tierney extracted elementary axioms of toposes which also served as an algebra of higher-order predicate logic, as boolean algebras are an algebra of propositional logic through Lawvere's discovery that both higher-order and quantifiers arise as adjoints.

Grothendieck's work lead, for example, to Deligne's proof of the Weil conjectures used in such concrete results as Zhang's theorem on gaps in the primes.

Lawvere and Tierney's work lead to a revolution in logic, but also to the possibility of a further development of Grothendieck's notion of space for teaching and progress in geometry.

I mention now three  developments which seem to miss the point of the great  conceptual development introduced by Grothendieck.

Such projects as polymath8, attempting to extract the best possible constants in Zhang's Theorem,  instead of taking advantage of the conceptual progress of topos theory, seek to eliminate it from applications by special arguments.

Another recent claim is that by concentrating on the axiomatic method in developing topos theory there have been a very limited number of applications in concrete mathematics, and that it would be better to use toposes as a tool to study relations between sites.

A third current opinion strongly supported is that topos theory should be supplanted by infinity-topos theory.

(This is how I understand the situation. I have not worked directly on applications of topos theory, though I have used various of the exactness properties in my research into the foundations of computer science. I have done a little work on the abstract theory of toposes, the introduction of lex total categories with Ross Street - see the earlier post about the Isle of Thorns 1976 - and the characterization of relations in an elementary topos with Aurelio Carboni.)

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