arXiv subjects

Simon Henry

Publications and source records attributed to Simon Henry.

30 records · Page 2Linked to original sources

arXiv2018

Regular polygraphs and the Simpson conjecture

We prove Carlos Simpson's "semi-strictification" (or "weak unit") conjecture in the case of infinity-groupoids. More precisely, we introduce two precise versions of the conjecture, the "general" and the "regular" conjecture, involving two different notions of "non-unital categories". The "general" version involve infinity-categories where absolutely all composition operations (horizontal, vertical and whiskering) are defined and compatible, the "regular" version involve infinity-categories where all the composition operations corresponding to "regular" pasting diagram are defined and compatible. In both case we construct (weak) model structures on these categories such that fibrant objects have weak units and weak inverse. We prove the regular version of the conjecture using the original strategy of Kapranov and Voevodsky, together with our previous work on polygraphs. The general version cannot be proved by these methods and is still open. In order to do this we also study some subtle property of the combinatorics of polygraphs, and we construct a new counting function for polygraphs, inspired by previous work of Makkai.

math.CT↗

arXiv2017

The localic Istropy group of a topos

It has been shown by J.Funk, P.Hofstra and B.Steinberg that any Grothendieck topos T is endowed with a canonical group object, called its isotropy group, which acts functorially on every object of T. We show that this group is in fact the group of points of a localic group object, called the localic isotropy group, which also acts on every object, and in fact also on every internal locales and on every T-topos. This new localic isotropy group has better functoriality and stability property than the original version and shed some lights on the phenomenon of higher isotropy observed for the ordinary isotropy group. We prove in particular using a localic version of the isotropy quotient that any geometric morphism can be factored uniquely as a connected atomic geometric morphism followed by a so called "essentially anisotropic" geometric morphism, and that connected atomic morphism are exactly the quotient by an open isotropy action.

math.CT↗

arXiv2016

The convolution algebra of an absolutely locally compact topos

We introduce a class of toposes called "absolutely locally compact" toposes and of "admissible" sheaf of rings over such toposes. To any such ringed topos $(\mathcal{T},A)$ we attach an involutive convolution algebra $\mathcal{C}_c(\mathcal{T},A)$ which is well defined up to Morita equivalence and characterized by the fact that the category of non-degenerate modules over $\mathcal{C}_c(\mathcal{T},A)$ is equivalent to the category of sheaf of $A$-module over $\mathcal{T}$. In the case where $A$ is the sheaf of real or complex Dedekind numbers, we construct several norms on this involutive algebra that allows to complete it in various Banach and $C^*$-algebras: $L^1(\mathcal{T},A)$, $C^*_{red}(\mathcal{T},A)$ and $C^*_{max}(\mathcal{T},A)$. We also give some examples where this construction corresponds to well known constructions of involutive algebras, like groupoids convolution algebra and Leavitt path algebras.

math.CT↗

arXiv2016

Algebraic models of homotopy types and the homotopy hypothesis

We introduce and study a notion of cylinder coherator similar to the notion of Grothendieck coherator which define more flexible notion of weak infinity groupoids. We show that each such cylinder coherator produces a combinatorial semi-model category of weak infinity groupoids, whose objects are all fibrant and which is in a precise sense "freely generated by an object". We show that all those semi model categories are Quillen equivalent together and Quillen to the model category of spaces. A general procedure is given to produce such coherator, and several explicit examples are presented: one which is simplicial in nature and allows the comparison to the model category for spaces. A second example can be describe as the category of globular sets endowed with "all the operations that can be defined within a weak type theory". This second notion seem to provide a definition of weak infinity groupoids which can be defined internally within type theory and which is classically equivalent to homotopy types. Finally, the category of Grothendieck infinity groupoids for a fixed Grothendieck coherator would be an example of this formalism under a seemingly simple conjecture whose validity is shown to imply Grothendieck homotopy hypothesis. This conjecture seem to sum up what needs to be proved at a technical level to ensure that the theory of Grothendieck weak infinity groupoid is well behaved.

math.CT↗

arXiv2016

On toposes generated by cardinal finite objects

We give a characterizations of toposes which admit a generating family of objects which are internally cardinal finite (i.e. Kuratowski finite and decidable) in terms of "topological" conditions. The central result is that, constructively, a hyperconnected separated locally decidable topos admit a generating family of cardinal finite objects. The main theorem is then a generalization obtained as an application of this result internally in the localic reflection of an arbitrary topos: a topos is generated by cardinal finite objects if and only if it is separated, locally decidable, and its localic reflection is zero dimensional.

math.CT↗

arXiv2015

Complete C*-categories and a topos theoretic Green-Julg theorem

We investigate what would be a correct definition of categorical completeness for C*-categories and propose several variants of such a definition that make the category of Hilbert modules over a C*-algebra a free (co)completion. We extend results about generators and comparison theory known for W*-categories with direct sums and splitting of symmetric projections to our "complete C*-categories" and we give an abstract characterization of categories of Hilbert modules over a C*-algebra or a C*-category as "complete C*-category having enough absolutely compact morphisms (and a generator)". We then apply this to study the category of Hilbert spaces over a topos showing that this is an example of a complete C*-category. We prove a topos theoretic Green-Julg theorem: The category of Hilbert spaces over a topos which is locally decidable, separated and whose localic reflection is locally compact and completely regular is a category of Hilbert modules over a C*-algebras attached to the topos. All the results in this paper are proved constructively and hence can be applied themselves internally to a topos. Moreover we give constructive proof of some known classical results about C*-algebras and Hilbert modules.

math.CT↗

arXiv2015

A Geometric Bohr topos

In this short note, we construct a variant of the Bohr topos of a C*-algebra which takes into account the topology of the algebra in a finer way and such that this construction is stable under pullback along geometric morphisms. This generalizes a construction for finite dimensional algebras of G.Raynaud. Our idea is to construct the Bohr topos of a C*-algebra A as the sublocale of the lower power locale of the localic completion of A which classifies the commutative localic sub-C*-algebras of A.

math.CT↗

arXiv2015

Constructive Gelfand duality for non-unital commutative C*-algebras

We prove constructive versions of various usual results related to the Gelfand duality. Namely, that the constructive Gelfand duality extend to a duality between commutative nonunital C*-algebras and locally compact completely regular locales, that ideals of a commutative C*-algebras are in order preserving bijection with the open sublocales of its spectrum, and a purely constructive result saying that a commutative C*-algebra has a continuous norm if and only its spectrum is open. We also extend all these results to the case of localic C*-algebras. In order to do so we develop the notion of one point compactification of a locally compact regular locale and of unitarization of a C*-algebra in a constructive framework.

math.CT↗

arXiv2015

Toward a non-commutative Gelfand duality: Boolean locally separated toposes and Monoidal monotone complete $C^{*}$-categories

** Draft Version ** To any boolean topos one can associate its category of internal Hilbert spaces, and if the topos is locally separated one can consider a full subcategory of square integrable Hilbert spaces. In both case it is a symmetric monoidal monotone complete $C^{*}$-category. We will prove that any boolean locally separated topos can be reconstructed as the classifying topos of "non-degenerate" monoidal normal $*$-representations of both its category of internal Hilbert spaces and its category of square integrable Hilbert spaces. This suggest a possible extension of the usual Gelfand duality between a class of toposes (or more generally localic stacks or localic groupoids) and a class of symmetric monoidal $C^{*}$-categories yet to be discovered.

math.CT↗

arXiv2014

Measure theory over boolean toposes

In this paper we develop a notion of measure theory over boolean toposes which is analogous to noncommutative measure theory, i.e. to the theory of von Neumann algebras. This is part of a larger project to study relations between topos theory and noncommutative geometry. The main result is a topos theoretic version of the modular time evolution of von Neumann algebra which take the form of a canonical R+*-principal bundle over any integrable locally separated boolean topos.

math.CT↗

arXiv2013

Toposes, quantales and C* algebras in the atomic case

We start by reviewing the relation between toposes and Grothendieck quantales. We improve results of previous work on this relation by giving both a characterisation of the map from the tensor product of two internal sup-lattices to another sup-lattice and a description of the category of internal locales of a topos in terms of the associated Grothendieck quantale. We then construct a convolution product, corresponding to internal composition of matrices, on the set of positive lower semi-continuous functions on the underlying locale of the quantale attached to a topos. In good cases, this convolution product does restrict into a well defined convolution product on a subset of the set of continuous functions and defines a convolution C* algebra attached to the quantale. In the last part of this article we investigate in details these attached C* algebras in the special case of an atomic topos. In this situation the related Grothendieck quantale corresponds to a hypergroupoid. Relatively simple finiteness conditions on this hypergroupoid appear in order to obtain an interesting C* algebra. This algebra corresponds to a hypergroupoid algebra which comes endowed with an arithmetic sub-algebra and a time evolution. We conclude by showing that the existence of a hypergroupoid satisfying all the requirements attached to a specified atomic topos is equivalent to the fact that the topos is locally decidable and locally separated. Also in this situation the time evolution only depends on the topos and is described by a (canonical) principal Q+* bundle on the topos. The BC-system and more generally the double cosets algebras are special cases of this situation.

math.CT↗

arXiv2013

Symmetrization of monoïds as hypergroups

We adapt the construction of the Grothendieck group associated to a commutative monoïd to handle idempotent monoïds. Our construction works for a restricted class of commutative monoïds, it agrees with the Grothendieck group construction in many cases and yields a hypergroup which solves the universal problem for morphisms to hypergroups. It gives the expected non-trivial hypergroup construction in the case of idempotent monoïds.

math.AG↗