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Am not sure if this page exists on the nlab, but I think a nice presentation of this proof technique, would be useful in the development of “new ways of doing mathematics”. I would also like to know of other proof techniques in category theory that are useful but happen to be unpopular, maybe a page can be created to handle such? I would appreciate an example that illustrates transformation of a proof in theory A to a proof in theory B(a concept we may have taken for granted? I don’t know). You can view Olivia Caramello’s general idea
Her 2017 book is mentioned at classifying topos, you could start a new page from there perhaps.
Re #1:
Just to remark that, sociologically, there may be a causation between proclaiming “new ways of doing mathematics” in boldface and these ending up feeling “unpopular” (if they really are unpopular – maybe they are widely appreciated but in less boldface).
I think the idea of making (more) systematic use of Morita equivalences of sites is evidently right and good, and if one finds a class of open problems to which this hammer applies, then it’s bound to be popular.
But the hammer itself is not that new. The comparison lemma is one of the classical results of topos theory, after all, and is all about having different sites for the same topos.
If anyone has the inspiration and energy to write about this, I might suggest a good page title could be Morita equivalence of sites, and that would naturally make an appearance in evident remarks both at site and at category of sheaves; before it gets into the more sophisticated perspective through classifying toposes.
What may be disappointing when one learns a trick is that the idea to apply it is not sufficient to easily find a new, good, needed and doable application. For example, the transfer principle for nonstandard models in model theory (subject sometimes known under nonstandard analysis). It sounds great, but could we easily solve new important problems using it ? Finding applications is much of an art, after few easiest applications are exhausted.
One case of my related interest is in so called colocalization of coGrothendieck Abelian categories. coGrothendieck categories are dual to Grothendieck Abelian categories where exact localizations are in great parallel with open subtoposes; the additive analogue of Grothendieck topologies by Gabriel provide localilization in precise analogy with sheafification, as a square of certain endofunctor H which is an analogue of +-functor, and constructed in the same way. Now if you formally dualize then the localizations will have left adjoint rather than right adjoint, this is known as colocalization (resulting in coreflective rather than reflective subcategories). However, there is a concrete realization of such coGrothendieck categories as a kind of subcategories of modules over certain class of complete topological rings, which is somewhat parallel to Grothendieck categories, but the topological situation has its subtleties and specifics which are not predictable even from the point of view of a rather concrete (rather than only formal) duality invented in this context (Gabriel-Grothendieck-Roos duality) which has a concrete description itself (and relation to some other dualities in different fields, especially in model theory). I think that the point of view on colocalization which is analogous to sheaf theory (additive-like topologies for complete topological rings or somethink like that) is still missing. I hope I can understand something along this way, and make the subject a little easier to comprehend.
Re #4: I think an algebraic approach can help shed some light on the topic, one can regard coGrothendieck Abelian categories as coreflective subcategories of the category of modules over a unital ring by dualizing Gabriel-popescu theorem. It is thus EM category over a comonad on the category of modules over a unital ring. Seems like am tethered to the algebraic approach, but it has huge scope as I have been emphasizing. I think the analogies you mention have a precise formulation since sheafification is reflective localization nlab. Am not familiar with transfer principle but I think constructive proof techniques will not disappoint: particularly am interested in inductive proofs as (iterated)reflective localization.
Re #3:
The fact that a topos admits (infinitely) many different presentations (in terms of sites, theories, etc.) is certainly something well-known since the birth of topos theory; in fact, Grothendieck himself compared the existence of different sites of definition for one topos to the existence of different presentations for a group by generators and relations. The main innovation brought by the ’bridge’ technique consists in the systematic use of topos-theoretic invariants as means for generating connections between different (Morita-equivalent) theories or sites through the ‘computations’ of such invariants in terms of the different presentations of the relevant toposes.
As a matter of fact, the view of toposes as ‘bridges’ has represented a deep paradigmatic shift both with respect to the ‘elementary topos’ tradition which has dominated the categorical community throughout the past forty years, and with respect to the algebraic geometry tradition promoted by the former pupils and colleagues of Grothendieck, which has essentially ‘neglected’ the general theory of toposes in favour of sites. These aspects are thoroughly discussed in this paper, which also identifies the divergence from these two established traditions as the main reason for the methodology “toposes as bridges” not yet being so popular as one would naturally expect considering its technical power. Indeed, as A. Joyal rightly put it, this methodology is a “vast extension of Felix Klein’s Erlangen Program” (note how this observation perfectly fits with Grothendieck’s analogy with groups recalled above). An overview of notable ‘bridges’ obtained so far in different areas of mathematics is provided here.
It is also worth noticing that the notion of ‘bridge’ is a conceptual architecture which makes sense well beyond topos theory (see this paper for some examples of ‘bridges’ outside mathematics).
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