Categorical Non-commutative Geometry

Category theory has been applied to operator algebraic settings since many years ago, probably starting with the pioneering work of John Roberts in algebraic quantum field theory around 1980.

As far as we know, the study of "categorical non-commutative geometry", in the setting of A.Connes' spectral triples, started around 2002-2003 as a byproduct of our research project "Modular Spectral Triples in Non-commutative Geometry and Physics" (Thai Research Fund Grant: RSA4580030). There, in order to identify a non-commutative configuration space from a non-commutative phase-space obtained by Tomita-Takesaki modular theory, a kind of polarization was necessary and for this purpose we were motivated to introduce a definition of sub-object and study the most elementary notion of morphism of spectral triples that was later published in the research paper A Category of Spectral Triples and Discrete Groups with Length Function (arXiv version).

The research on categorical non-commutative geometry soon started to become one of the main areas of our activity as documented in the survey paper Non-commutative Geometry Categories and Quantum Physics (updated arXiv-version) that can still be considered a fairly good introduction to the subject and in the new review paper Categorical Non-commutative Geometry (arXiv version).

Apart from the study of alternative simple notions of morphism of spectral triples in A Remark on Gel'fand Duality for Spectral Triples (arXiv version), we started a project of "categorification" of A.Connes non-commutative geometry. Around 2006 we introduced the terms "horizontal categorification" and "vertical categorification" in order to distinguish the categorical "many-object version" of usual mathematical concepts from the more demanding "higher-morphism" counterpart and we concentrated in proving a horizontal categorified version of Gel'fand Naimark duality for commutative full C*-categories, a result that has been published in A Horizontal Categorification of Gel'fand Duality (arXiv version). The spectrum of such a C*-category consists of a specific Fell line-bundle that we have called "topological spaceoid".

Spaceoids have been alternatively described in three different equivalent ways in the subsequent work Enriched Fell Bundles and Spaceoids (arXiv version), where we also introduced the notion of weak involutive bicategory and Fell bundles enriched in it.

Recent work in this direction (2017-2018), in collaboration with my PhD student Natee Pitiwan, culminated in a proof of a generalization of Gel'fand-Naumark duality theorem to the case of non-commutative unital C*-algebras, spectrally analyzed in term of spaceoids. This would be of some interests also for its implications on operational foundations of algebraic quantum theory and its relations with classical physics.
The horizontal categorification of Krein-C*-algebras (essentially categories of linear bounded operators between complete semi-definite linear spaces) has also been investigated in Krein C*-categories (arXiv version). Categories of Krein-C*-modules have been investigated for some time (mostly in Master thesis and senior unergraduate projects with some of my ex students) and the resulting notions have been published in Krein C*-modules (arXiv version). A spectral theory for some Krein C*-algebras (using spaceoids) has been also presented in Spectral Theory on Commutative Krein C*-algebras (arXiv version).

A first preliminary objective of this research would be to provide explicit examples of functors from suitable categories of geometrical spaces (such as for example oriented Riemannian or spinorial compact manifolds) to categories of spectral triples (such as the category described by B.Mesland [arXiv:0904.4383v1] or possibly some variants of it). Very elementary and partial results in this direction have been presented in a short survey paper Remarks on Morphisms of Spectral Geometries (arXiv version). For this purpose we plan to spend some time investigating the categorical structures involved in the case of oriented spaces, measure spaces, (Riemannian/ Hermitian) manifolds/bundles equipped with connections or with spinorial bundles, etc. In a related effort (in collaboration with Fabian Germ - at that time preparing a senior undergraduate project) we started to study how to associate spectral geometric data to certain diffeological spaces (Spectral Geometries for Some Diffeologies - talk in Warsaw): the hope is that the wider diffeological environment might help to simplify "spectral reconstruction theorems" both for spaces and their morphisms.

A lot of effort has been dedicated to (and is still focused onto) the study of vertical categorifications of C*-categories:

Other lines of research that we are planning to close in the near future include:

Much more ambitious goals that we would like to address in the longer term are: Other important applications of such mathematical structures in physics are also likely, for example, in the context of loop quantum gravity, we might provide: 
Modular Non-commutative Geometry and Modular Algebraic Quantum Gravity

The attempt to generalize the setting of R.Haag-D.Kastler algebraic quantum field theory, eliminating the assumptions on the existence of a background space-time in view of a theory of quantum gravity, where space-time is recovered a-posteriori by states on algebras of observables has been a main motivation (with actually a quite long list of proposals in the past).
In this wider context, my main research interests, since 1995, have been devoted to the investigation of the possibility to use of Tomita-Takesaki modular theory of Von Neumann algebras in order to associate,
to suitable (KMS) states over C*-algebras, non-commutative spectral geometries essentially similar to A.Connes real spectral triples (although the first ideological motivation in this kind of research in modular spectral geometry came from works by J.-L.Sauvageot on non-commutative geometries associated to semigroups of completely positive contractions).
The main research project was announced in the final section of the review Non-commutative Geometry Categories and Quantum Physics and has been described in some detail in the subsequent paper Modular Theory, Non-commutative Geometry and Quantum Gravity (arXiv version), where some strong relations with the work in modular spectral triples by A.Carey and collaborators have been established.

What has been achieved so far is a definition of modular spectral geometries associated to any pair given by a C*-algebra and a KMS state over it and a proof that, in the case of periodic modular flow, under some technical conditions (strict semi-finiteness of the lifting of the KMS state to the modular centralizer algebra and compactness of the resolvent of the modular generator with repect to the canonical trace on the centralizer algebra) the definition essentially reproduces the modular spectral triples axiomatized by A.Carey-J.Phillips-A.Rennie in their investigation of modular index theory. This somehow justifies the geometrical nature of a definition that is a priori only algebro-analytical.

Much more remains to be done in the future in order to understand the specific role of Connes-Takesaki duality and Falcone-Takesaki non-commutative flow of weights in modular non-commutative geometry. For this purpose it will be necessary to start a detailed investigation of how (modular) spectral triples behave under the action of dynamical systems and the study of "cross-products" of spectral geometries under one-parameter groups of automorphisms, a topic that has remained so far almost unexplored.

To test some of the previous ideas, we have started in direct collaboration with Matti Raasakka and Roberto Conti a research project that is planning to examine the role (and possibly the physical meaning) of modular spectral geometries beginning from the elementary case of finite dimensional C*-algebras and that will later consider the more interesting case of AF C*-algebras. 

The possibile interrelations between modular spectral geometries and standard Connes' real spectral triples is also under consideration: our guess being that modular spectral geometries provide a form of non-commutative geometry at the level of "phase-space". We suspect that generalizations of the process of spectral tensorial symmetrization (see section 5.2 in Modular Theory, Non-commutative Geometry and Quantum Gravity) might be relevant here.

The connection between spectral action and modular generator (with modular generator taking the role of square of the Dirac operator) seems also to indicate that "second order" spectral geometries (with Laplace in place of Dirac operators) might be relevant in this context.

The original motivation for this kind of research in "modular spectral geometry", is directly related to the foundation of quantum physics and originates from the following ideological input that is the basic assumption of our proposal for a theory of "modular algebraic quantum gravity".

A reconstruction of (non-commutative) space time in this context has not been achieved yet, but we put forward a speculative proposal to utilize (higher) C*-categories and (hyper) C*-algebras for a formulation of algebraic relational quantum theory and the recovery of (quantum) space-time form such categories (see the last section of the proceedings paper Categorical Operator Algebraic Foundations of Relational Quantum Theory).

Some interesting links with other approaches to quantum gravity appeared and in our paper Modular Theory, Non-commutative Geometry and Quantum Gravity, where we have already described:
In the near future we intend to explore the possibility to "polarize" modular spectral geometries to obtain geometries on configuration spaces (along the line of M.Pascke's "scalar quantum mechanics") and the physical meaning of the categorical structure of modular non-commutative geometries.

Next research goals include: