Learning from Grothendieck
I received the annual IHES newsletter this week. It featured an extract of Barry Mazur's foreword to Paumier and Blieck's A History of the IHES. Mazur mentions Grothendieck's ability to make observations of the form '"X" (i.e., whatever he was talking about) "n'est rien d'autre qu'un Y", thereby creating a(n) "Y" that critically changes our point of view (in language and concept) towards "X".'
This is a spirit that has been adopted throughout category theory, allowing one to understand new concepts as generalisations or special cases of existing ones to connect everything together, while increasing the number of perspectives that one has on any given object of their interests. Indeed, there is a special case of this kind of observation that is particularly relevant to us in topos theory: "X is just a Y in the topos E".
Recall that one of the ways of understanding (and working with) toposes is as universes of sets, since we can interpret the constructive axioms of set theory in any topos. As such, we can do mathematics in any topos, and directly lift constructive arguments valid in the "usual" category of sets to any topos. In particular, that means that when faced with a complicated object X, if we can push the structure of X into the topos, we can understand X as a much simpler object in that topos. As an elementary example, take Ivan Tomašić's paper A topos-theoretic view of difference algebra, where he exploits the observation that "a difference Y [where Y is a set, category, ring...] is just a Y in the topos of -sets". In this case, the 'difference' structure, which is an operator commuting in a suitable way with all of the algebraic structure of interest, has been exported to the topos, allowing us to reason with these algebraic objects in a familiar way in calculations, with the unique caveat of having to translate (or externalise) the result at the end of the calculation.
A more classical example is that when we encounter a geometric structure having an underlying topological space X, we can re-express that geometric structure in terms of a structure sheaf on X, which typically is "just" a (local) ring in the topos of sheaves on X. The fact that we have a lot of techniques for working with rings means that we can directly exploit this, which is how we arrive at a many general homology and cohomology theories, albeit with a lot of work involving abelian categories in the interim.
All in all, I think this approach has a lot of potential to be exploited further. We still have a lot to learn from Grothendieck!
I'm definitely enamoured with this point of view. And I also think is the instance of a much more common problem solving technique. I've tried to explain this in a 'seminar' I've given for a course, this January, whose very biref notes I attach here.
In summary: 'X is just a Y in the topos E' is an instance of offsetting part of the complexity of a description to the environment in which we carry out such description. E.g. describing the structure sheaf of a scheme X in Set can be simplified if we carry out the same description in ShX, which is arguably a more convoluted universe. Yet when reasoning about such descriptions, the environment is not involved and thus we earned something.
This problem solving technique is widespread in calculus, where it's usually called 'change of variables', or in physics where it's called 'change of coordinates'.
Here are the notes I was referring to:
This general methodology is commonly known under the name of relativization technique, and can indeed be very powerful, in particular within topos theory. Of course, its usefulness and applicability greatly depends on the effectiveness and flexibility of the formalism that one has at one's disposal for performing such relativizations, that is, of the techniques for describing 'internalizations', 'externalizations', the behaviour of properties and constructions with respect to change of base etc.
There is actually a lot of work to do in this direction within topos theory, as the existing formalisms based on internal categories and internal sites are very rigid (for instance, they do not accommodate stacks, which are the 'right' notion of category internal to a topos of sheaves, nor they naturally integrate arguments using generalized elements). This makes passages from one base topos to another generally non-straightforward from the point of view of presentations of the toposes over them. To address this problem, with @RZanfa we have been developing a theoretical framework for relative topos theory based on stacks and indexed categories, which should be much more "user-friendly" and amenable to computations (we plan to make a first draft of our work available soon).
@matteocapucci Concerning 'change of variables' or 'change of coordinates', I tend to view this more as an instance of the 'bridge' technique (which consists in expressing 'invariants' in terms of different presentations of a given object) than of the relativization method, even though of course the two techniques are not at all unrelated; indeed, relativization generally leads to different presentations for the objects under consideration and hence is a natural source of 'bridges'.
It made me try to rephrase it as I enjoy to do most as soon as I can, i.e. in terms of metamathelatical, syntactic interpretations between each and every synthetic formalizations of the concepts involved, namely here in the case of the example:
"a difference Y [where Y is a set, category, ring...] is just a Y in the topos of -sets"
, the Axiomatic Theories of Topos TT, of Ys TY, of Difference Ys TDY, and of course the background theory you were tacitly entirely talking in, say by chance ZFC (for instance).
Here's what I mean: this "X n'est rien d'autre qu'un Y" genius conceptual intuition formally amounts in the above synthetic framework to say that well, it is often fruitful in order to study some interpretations (here of TDY into ZFC) to consider a subsystem of the source (here TY subsystem of TDY) that we understand way better the interpretations and have many more tools to study, and to factorize the initial interpretations as composed of much simpler interpretations of the subsystem into an intermediate theory (here the categorical semantics into TT) and of an already well-known and studied interpretation of this intermediary into the target of the initial one (here the interpretation "Topos of -Sets" of TT into ZFC)!
I'm very sorry not to be able to post the diagrammatic square, but it's a real draft^^ It's actually not a mere square by the way, since the situation makes a detour through the theory of topos equipped with a Y object TTYO and its composite set-theoretic semantics as the "Topos of -Sets equipped with a difference Y".
So here is, whether it's enlightening or not for one, what actually happens formally, at the most metamathematical and synthetic level (I presume), when conceptually this relativization technique occurs.
In accordance with my paper :"Post cartesian rationality", and precisely with this resume :
May I suggest that :
- an object of Lawvere's topos form would be approached following process (a),
- when Grothendieck's one, as "le lit du discret et du continu", whould be discussed follwing (b)
- an a bridge, as proposed by Olivia Caramello, would be a matter of (c) level ?
@grothenditque There is no need to relegate ZFC to the meta-theory: a topos comes with internal logic built in, and so rather than going to the elementary theory of toposes and having to build from there, we work over a familiar topos of sets which validate ZFC by assumption. So I was saying "a difference Y in Sets" corresponds to "a Y in the category of N-sets". This can be extended by replacing Sets with any (elementary) topos having a natural number object (since we are just using the fact that such an object has a canonical monoid, and hence category, structure), but we cannot do this in an arbitrary elementary topos.
@a-c-o-simon your link goes not to a "paper" but to the blog post which you have already shared in several places on this forum. It has no specific references and no detailed explanations of any of the concepts involved beyond a translation of some buzzword philosophical ideas into your own symbols. This is a forum on topos theory, where the common language is expected to be category theory, or at least mathematics. You cannot expect others to be able to decipher or have meaningful discussions on the basis of hieroglyphics you invented to synthesise your own ideas. Moreover, the distinction you apparently imagine between Lawvere's toposes and Grothendieck's has very little to do with discreteness, and both have the same notion of equivalences (which is to say categorical equivalence), so are amenable to Caramello's bridge technique. So even ignoring the table, the distinction of levels you propose is nonsensical.
@morgan-rogers Oh I never denied some changes or generalizations were possible, I was just unfolding what you had actually written, not thinking more in that (different) direction. But ok yes, there is no problem with replacing in my words ZFC by TTNNO, the theory of topos having a natural number object , and then:
the interpretation "Topos of -Sets" of TT into ZFC
by: 'the interpretation "Subtopos of -Objects" of TT into TTNNO', and:
its composite set-theoretic semantics as the "Topos of -Sets equipped with a difference Y"
by: 'its composite interpretation as the "Subtopos of -Objects, but equipped with a difference Y Object" '.
And besides already being a good idea as most of generalizations that work, that is also one as it makes even clearer the fact that the actual duality here is between the 'difference' and the '-' aspects, sets having no role in here. Thanks