- Simon HENRY
College de France
3 Rue d'Ulm
Courriel: henry at phare dot normalesup dot org
I am currently working in the College de France in Paris.
I am intrested in relations between non commutative geometry and topos theory. I've done my thesis on this subject under the superision of Alain Connes.
Roughly, topos theory and C*-algebra theory are in some sense two generalisations of ordinary topology, and my goal is to understand if and how they are related. I am studying more precisely continuous fields of Hilbert spaces over toposes, i.e. Hilbert spaces defined in the internal logique of a topos. The algebra of endomorphisms of such a Hilbert space is a C* algebra closely related to the geometry of the topos. The study of the properties of such continuous fields allow to understand several relations between topos theory and operator algebras.
The introduction of my thesis (linked above) contains a more detailled explanation of theses ideas.
More recently, I have been interested in higher category theory, and more precesely on whether it is possible to develope a formalism for higher category that works well within intuitionist mathematics, with the immediate application to topos theory and Higher topos theory in mind. For this reason I have been interested in globular model for Higher category and homotopy type theory.
I am also active on mathoverflow.
PapersHere is a list of my main (pre)publications, with for each an informal abstract trying to present the main ideas of the paper, its place in my research program and the relations between them.
Measure theory over boolean toposes (arXiv:1411.1605 )
We construct a topos theoretical analogue of the modular time evolution of von Neuman algebras.
Each Von Neuman algebra is canonically endowed with an outer action of the group of real numbers called its modular time evolution. It is non trivial exactly when the algebra is of type III (more precesely when it has a non tirivial type III component). We give a similar construction which endows every integrable boolean locally separated topos with a canonical principal bundle for the group of real numbers which can be used to describe the modular time evolution of certain algebras of endomorphisms of Hilbert bundle of the topos. This canonical bundle over a topos corresponds to the bundle of "measures", and the fact that it is a principal bundle is a form of Radon-Nikodym theorem. The central notion of this paper are the "invariant measures" over a boolean topos, and this "modular" principal bundle we constructed classifies such invariant measures: local sections of the bundle corresponds to local invariant measures and globale sections correspond to invariant measures, hence (when thought of as a cohomology class) this bunle is the obstruction to the existence of such an invariant measure.
Localic metric spaces and the localic Gelfand duality (arXiv:1411.0898 )Published in Advances in mathematics 294 (2016).
Locale are the object of study of "pointfree topology", they are analogous to topological spaces with exceptions that they can fail to have points. A locale which has enough point is the same as a sober topological space, and (assuming the axiom of choice) every locally compact locale has enough point. In classical mathematics the difference between locale and topological space is relatively small, but in the absence of the axiom of choice this difference become a lot more important, and in both vases locales tend to be better behaved than topological spaces.
In the paper, we develope the theory of metric locales (that is locale endowed with a distance function) in a constructive framework and extend the already known theory of localic completion. We then use this to develope a theory of localic Banach space and localic C*-algebra to prove a conjecture of C.J.Mulvey and B.Banachewski that there is a constructive Gelfand duality relating localic commutative unital C* algebra and compact regular locales (while their version of the Gelfand duality was relating ordinary C*-algebras with compact completely regular locales).
The intuitive idea behind this results is relatively simple: classically, for comact Hausdorff locales, complete regularity follow from the Urysohn lemma, which relies on the axiom of (dependant) choice. But constructively, complete regularity is not automatic and is only about the existence of points for the "locale of continuous functions on the space", hence if we are willing to give up this existence of points in the C* algebras we consider it should be possible to relax complete regularity into just the Hausdorff property. And this is what we proove in the present paper.
This article is higly inspired from the idea of geometric logic, in fact our main result is that the localic C*-algebras form in some sense the stack-completion of the pre-stack of ordinary C*-algebras for the topology of open surections of toposes. This allows to deduce the localic Gelfand duality from its usual (constructive) counterpart by using descent theory.
The fact that localic C*-algebras form a stack for the topology of effective descent morphism of locales (inclusing open and proper surjections) allow to define them on an arbitrary localic groupoid (which are more general than toposes) in a way which is invariant under equivalence of localic groupoids.
My thesis contains, as its chapter 3, an extended version of this paper that includes a spatiality theorem inspired by a similar result of Dal Soglio-Hérault and Douady, proving that over toposes satisfying a condition analogous to paracompactness and assuming the axiom of dependant choices in the base univers there is no difference between localic Banach spaces and ordinary Banach spaces in the topos. Over a topological space, internal ordinary Banach spaces correspond to semi-continuous fiels of Banach spaces whch have sufficently many local sections while internal localic Banach spaces correspond to the more general notion of semi-continous fields of Banach spaces, and in this case this is exactly the results proved by Dal Soglio-Hérault and Douady.
Constructive Gelfand duality for non-unital commutative C*-algebras (arXiv:1412.2009)
We prove that the constructive form of the Gelfand duality can be extended to non-unital C*-algebras by giving constructive version of both the one point compactification of a locally compact locale and the unitarization of a C*-algebra. We also take this opportunity to prove constructively some classically well known results about the gelfand duality, like the fact that ideal of a commutative C*-algebra corresponds bijectively to the open subspace of its spectrum and that proper maps between spectrums correspond to non-degenerate morphisms between the algebras.
We also prove that all these results hold as well for the localic C* algebras introduced in my other paper mentioned just above. Finally we prove a purely constructive result that says that the norm of a commutative C*-algebra takes values in the continuous real numbers if and only if the canonical map from the spectrum of this algebra to the base topos is an open map. This last result already appears independently in a paper of T.Coquand for unital ordinary algebra. When interpreted internally in the topos of sheaves over a topological space it means that a semi-continuous field of commutative C* algebra over a space X is a continuous field if and only if the proper map Y -> X associated to the field by the Gelfand duality is also an open map.
We consider this paper as of lesser importance and its goal was mostly to prove some results that play a key role in the next paper (and that will probably be usefull for some other work as well).
Toward a non-commutative Gelfand duality: Boolean locally separated toposes and Monoidal monotone complete C*-categories (arXiv:1501.07045)
In this paper we prove that a boolean locally separated topos (that is, essentially those toposes to which our construction of the modular time evolution done in " Measure theory over boolean toposes" apply) can be reconstructed out of their category of Hilbert bundles as long as this one is consiered as a symmetric monoidal C*-category.
More precisely, we prove that any boolean locally separated topos is the classyfing topos for the theory of normal, non-degenerate, monoidal symmetric *-representation of its category of Hilbert bundles. There is two versions of the theorem one involving the full category of Hilbert bundles and the other slightly easier involving a full subcategory of so called "square integrable" Hilbert bundles.
At the end of this paper we explain how we think this result is a special case of a much more general duality between certain localic groupoids and certain monoidal symmetric C* categories whose precise statement is still conjectural.
This paper as been released as a preliminary draft, and will probably stay definitely in that form: indeed some results we obtained shortly after (in "On toposes generated by cardinal finite objects", "Complete C*-categories and a topos theoretic Green-Julg theorem") produces exactly the tools that were needed to extend the results of this paper to more general (non-boolean) toposes as well, making most of the paper sligtly obsolete. Hence we will probably re-write it completely at some point to includes this more general result.
A Geometric Bohr topos (arXiv:1502.01896)
This paper is a short note that we wrote to illustrate the ideas of "Localic Metric spaces and the localic Gelfand duality" and more generally of geometric logic, and to apply them to the construction of the Bohr topos in the topos theoretic approach to physics.
The Bohr topos is one of the central construction of the topos theoretic approches to theoretical physique. It associates to every C* algebra A a space S(A) (its Bohr topos) which has points corresponding to the commutative sub-C*-algebras of A and which is endowed with a bundle of commutative C*-algebras. From my understanding this construction is suppose to embodies the "classical information" that can be read in a given quantum system
It has been pointed out that this construction suffer several problems including the fact that it is not "geometric" (i.e. stable under base change along geometric morphisms), and when applied to examples it seems to be completely ignoring some important topological information of A.
In this short note, we propose a modified version of this construction that relly on our work done in "Localic Metric spaces and the localic Gelfand duality". Indeed from the properties of the Bohr topos it seems that what was intended was somehow to construct "the space of all commutative sub-algebras of A". The descent argument used in our previous work shows that it is impossible to have such a space in a "pullback stable" way: there cannot be a classifying topos for commutative sub-algebras of a given Banach algebra or C*-algebra because those do not satisfy descent, but it become possible if one try instead to construct a classifying space for localic commutative sub-algebras instead, and that is exactly what we do in the present note.
Another "improvement" of the Bohr toposes with very similar properties has been proposed by G.Renault in his thesis, but only for finite dimension algebras. His construction endows the Bohr topos with a nicer topology by consdering it as a patching of various sub-spaces of Grassmannian manifolds (the subalgebras of fixed dimension). We have NOT checked whether or not our construction and his are equivalent for finite dimensional algebras, but in any case they are very close and have the same points.
On toposes generated by cardinal finite objects (arXiv:1505.04987)
One of the Key result in our work on boolean toposes ("Measure theory over boolean toposes" and "Toward a non-commutative Gelfand duality:...") is that separated boolean toposes are generated by objects which are internally finite (for boolean toposes Kuratowski finite and cardinal finite are equivalent). In order to generalize some of the methods used to handle boolean toposes we needed a similar result for non boolean toposes and in a constructive context. In this paper we prove that constructively a hyperconnected separate locally decidable topos is generated by objects that are internally cardinal finite. The proof is quite difficult because we want this result to be constructive, if we are working over a boolean base then one can apply an argument very similar to the one for boolean toposes and prove this in a few line.
But having this result constructively is extremely important: if one start from a topos which is separated and locally decidable then internally in its localic reflection it can be seens as a hyperconnected separated and locally decidable topos, hence one can apply our first resul internally and we obtains a generating familly of object that satisfies a kind of finitness condition.
Using this technique we prove in this paper that a topos is generated by cardinal finite object if and only if it is separated, locally decidable and its localic reflection is zero dimensional (generated by complemented sub-object).
This finitness result is a very important step toward extending the results of "Toward a non-commutative Gelfand duality:..." to some non-boolean toposes, as well as in the results of my paper "Complete C*-categories and a topos theoretic Green-Julg theorem" mentioned just below.
Complete C*-categories and a topos theoretic Green-Julg theorem (arXiv:1512.03290)
This a long paper that have several interconnected goals. First, it is apparently one of the first paper to deal with non-commutative C* algebras in constructive mathematics hence we had to prove a lot of basic results about those, for example we proved constructively the quite non trivial result that the various definitions of positivity are all equivalent in a (non-commutative) C* algebra. We introduce the notion of spectrum, Hilbert modules, hereditary sub-algebras and so on in constructive mathematics.
One of the main goals of this paper is introduce the concept of "complete C* category". For W* categories (the several object generalization of Von Neumann algebras) there is a very well behaved notion of completness (at least in non-constructive mathematics): a W* category is complete if it has arbitrary orthogonal sums and spliting of symetric projections. One can them prove that a functor preserve those "limits" (i.e. orthogonal sums and splitting of symetric projections) if and only if it is normal, that the category of W*-modules over a Von Neumann algebra is the free completion of the algebra while presrving its normal structure, that there is a good notion of generators and comparison theory in W*-categories, and that complete W*-category with a generator are exactly category of W* modules over a von Neumann algebra (those results are proved in the paper "W*-categories" by Ghey, Lima and Roberts). Unfortunately this notion of completeness for W* categories is very far from being ordinary categorical completeness.
There is a lot of example of C* category that we want to think of as "complete": complete W* categories of couse, but also category of Hilbert modules over a C* algebra, a small C* category, or a pro-C*-alebra, and of couse the category of Hilbert spaces over a topos, which is our main concern.
We develope a notion of complete C*-categories and "continuous functors" between them that includes all those examples (and some others) and allow to prove some results similar to those for complete W* categories (in particular for comparison theory and generators). We also give an abstract characterization of category of Hilbert modules as the complete C* categories having enough "absolutely compact morphisms".
Finally, we apply this characterization of C* categories of Hilbert modules to the C* category of Hilbert spaces over a topos and combining this to the results of "On toposes generated by cardinal finite objects" we prove our topos theoretic Green-Julg theorem: If T is a separated, locally decidable topos whose localic reflection is locally compact then there is a C* algebra C(T) attached to this topos such that the category of Hilbert space over T is equivalent to the category of Hilbert C(T) modules. There is also a constructive version of this result, as well as a version "with coeficients".
Work in progress
- There is a way to use the topos theoretic Green-Julg theorem to attach reduced and maximal C* algebras to toposes that only satisfies its hypothesis locally. While separation is a very restrictive condition (for example, we expect that one can only attach a type I C*-algebra to a topos that satisfies the hypothesis of the Green-Julg theorem), locale separation is a condition which is a lot more reasonable and I will write a paper explaining how a lots of construction of C* algebras from classical data can be presented in terms of the C* algebra of such a topos. (this will includes for example étale groupoids, including dynamical system and foliations, inverse semi-group, graph, generalizations of graph like P-graph, topological graph or Ultra-graph...)
- The next step would to relate the (co)homological properties of the topos to the (co)homological properties of the attached C* algebra. For examples for foliations the cohomology is the same as the cyclic cohomology of the subalgebra of smooth funtions, the Baum-Connes conjectures and its generalization can be seen as relations between the K-theory of the reduced algebra and a kind of K-theory of the topos, and there is a lots of examples of computations of the K-theory of graphs algebras from property of the underlying graphs, as well as example of computation of K-theory for other kind of C* algebras that also comes from toposes.
- I have recently observed that some toposes have a "Enveloping Von Neumann algebra" (for toposes having a reduced and maximal algebra, it is the enveloping von Neumann algebra of the maximal C* algebra ). It is defined in the following way, for any topos T, we have defined in "Measure theory over boolean toposes" a notion of "interable generalized measure class" over T. To any such measure class there is a von Neumann algebra attached, in a lots of situation there is a maximal such measure class that defines the enveloping von Neumann algebra. W*-Modules over this algebra correspond to a notion of ``integtable'' fields of Hilbert spaces over the topos. In a lot of examples (including all etale groupoids) this algebra have a natural description in terms of a site of definition for T, more precisely it seems that representations of this algebra can be decribed in terms of a notion of "representation of the site" or "co-Hilbert spaces" over T. I'm trying to find a general result in this spirit. From a very optimistic stand point one can hope that it is also possible to define co-Hilbert C modules for an arbitrary C* algebra C and to use this to give a universal property of the maximal C* algebra of a topos when it exists.