After 50 years, vivid interest in topos theory seems to have returned to one of the most prestigious research institutes, the IHES. Last november, there was the meeting Topos a l’IHES.
At the meeting, Celine Loozen filmed a documentary which is supposed to have as its title “Unifying Worlds”. Its very classy trailer is now on YouTube (via +David Roberts).
How did topos theory, a topic considered by most to be far too abstract to be useful to main stream mathematics, suddenly return in such force?
It always helps when a couple of world-class mathematicians become interest in the topic, for their own particular reasons. Clearly, the topic gathers considerable momentum if these people are all permanent members of the IHES.
A lot of geometric information is contained in the category of all sheaves on the geometric object. Topos theory offers a way to construct ‘geometries’ out of nothing, that is, out of arbitrary categories.
Take your favourite category $\mathbf{C}$, then “presheaves” on $\mathbf{C}$ are defined to be contravariant functors $\mathbf{C} \rightarrow \mathbf{Sets}$. For any Grothendieck topology on $\mathbf{C}$ one can then restrict to the sub-category of “sheaves” for this topology, and that’s your typical topos.
Alain Connes got interested in topos theory because he observed that even for the most trivial of categories, such as the monoid category with just one object and endomorphisms the multiplicative semigroup $\mathbb{N}_{\geq 1}^{\times}$, and taking the coarsest of all Grothendieck topologies, one gets interesting objects of baffling complexity.
One of the ‘invariants’ one can associate to a topos is its collection of “points”. Together with Katia Consani, Connes computed in Geometry of the Arithmetic Site that the collection of points of this simple presheaf topos is exactly the set of adele classes $\mathbb{Q}^{\ast}_+ \backslash \mathbb{A}^f_{\mathbb{Q}} / \hat{\mathbb{Z}}^{\ast}$.
Here’s what Connes himself said about this revelation (followed by an attempted translation):
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(50.36)
And,in this example, we saw the wonderful notion of a topos, developed by Grothendieck.
It was sufficient for me to open SGA4, a book written at the beginning of the 60ties or the late fifties.
It was sufficient for me to open SGA4 to see that all the things that I needed were there, say, how to construct a cohomology on this site, how to develop things, how to see that the category of sheaves of Abelian groups is an Abelian category, having sufficient injective objects, and so on … all those things were there.
This is really remarkable, because what does it mean?
It means that the average mathematician says: “topos = a generalised topological space and I will never need to use such things. Well, there is the etale cohomology and I can use it to make sense of simply connected spaces and, bon, there’s the chrystaline cohomology, which is already a bit more complicated, but I will never need it, so I can safely ignore it.”
And (s)he puts the notion of a topos in a certain category of things which are generalisations of things, developed only to be generalisations…
But in fact, reality is completely different!
In our work with Katia Consani we saw not only that there is this epicyclic topos, but in fact, this epicyclic topos lies over a site, which we call the arithmetic site, which itself is of a delirious simplicity.
It relies only on the natural numbers, viewed multiplicatively.
That is, one takes a small category consisting of just one object, having this monoid as its endomorphisms, and one considers the corresponding topos.
This appears well … infantile, but nevertheless, this object conceils many wonderful things.
And we would have never discovered those things, if we hadn’t had the general notion of what a topos is, of what a point of a topos is, in terms of flat functors, etc. etc.
(52.27)
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Pierre Cartier has a very wide interest in mathematical theories, the wilder the better: Witt rings, motifs, cosmic Galois groups, toposes…
He must have been one of the first people to speak about toposes at the Bourbaki seminar. In february 1978 he gave the talk Logique, categories et faisceaux, d’apres F. Lawvere et M. Tierney (and dedicated to Grothendieck’s 50th birthday).
He also gave the opening lecture of the Topos a l’IHES conference.
In this fragment of an interview with Stephane Dugowson and Anatole Khelif in 2014 he plays down his own role in the development of topos theory, compared to his contributions in other fields, such as motifs.
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(46:24)
Well, I didn’t invest much time in topos theory.
Except, I once gave a talk at the Bourbaki seminar on the use of topos theory in logic, such as the independence of the axiom of choice, that is, on the idea of forcing.
But, it was just this talk, I didn’t do anything original in it.
Then there is nonstandard analysis, where one can formulate certain things in terms of topos theory. When I got interested in nonstandard analysis, I had this possible application of topos theory in mind.
At the moment when you have a nonstandard model of the integers or more generally of set theory, then one has two models of set theory, that is two different toposes, and then one obviously tries to compare them.
In that sense, I was completely aware of the fact that everything I was doing could be expressed in the language of toposes,or at least in the philosophy of toposes.
I haven’t made any important contributions in that theory, for me it merely remained a tool.
(47:49)
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Laurent Lafforgue says he spend hundredths and hundredths of hours talking to Olivia Caramello about topos theory.
She must have been quite convincing. The last couple of years Lafforgue is a fierce advocate of Caramello’s work.
Her basic idea is that the same topos can arise from two very different mathematical settings (that is, two different categories with Grothendieck topologies can have equivalent categories of sheaves).
The hope then is to translate results from one theory to the other, or as she expresses it, toposes can be used as “bridges” between different mathematical topics.
At the moment though, is seems a bit far fetched for this idea to be relevant to the Langlands programme.
Caramello and Lafforgue have just a paper out: Sur la dualit´e des topos et de leurs pr´esentations et ses applications : une introduction.
The paper is based on a lecture Lafforgue gave in April in Nantes. Here’s the video:
In the introduction they write:
“It is our conviction that the theory of toposes and their representations, with its essential and structural ambiguity, is destined to have an impact on mathematics comparable to the impact group theory has had from the moment, some decades after its discovery by Galois, the mathematical community began to understand it.”