# Infinity-categories, symbolic dynamical systems, and mathematical physics

This is a page to keep track of our working groups and readings related to the project Infinity-categories, symbolic dynamics, and mathematical physics.

# Seminars

Oct 6, 2011 Thomas Seiller. Introduction to C*-algebras.

Sept 29, 2011 Guillaume Theyssier talked about some aspects of symbolic dynamics that might benefit from a categorical approach.

## C*-algebras and dynamical systems

### C*-algebras associated with lambda-synchronizing subshifts and flow equivalence

This is the paper we started with. It was probably a bit early to attack, and Tim quickly switched the focus to earlier, more accessible papers.

### Cuntz-Krieger algebras of directed graphs

The authors specify, via a universal property, a C*-algebra associated to any given directed graph satisfying a condition called row finiteness (all vertices have finite input degree). The idea is that the C*-algebra is a subalgebra of the algebra of bounded operators on an infinite-dimensional Hilbert space. A natural construction, starting from any infinite-dimensional Hilbert space, starts by choosing a specific isomorphism $H \cong \bigoplus_{v \in V} H$. That is, we specify a way of seeing $H$ as a direct sum of itself over all vertices of the given graph. Then, for each $v$, choose now a specific isomorphism $H \cong \bigoplus_{t(e) = v} H$. That is, specify a way of seeing $H$ as a direct sum of itself over all input edges of $v$.

Then, each vertex $v$ is interpreted as being the projection to the $v$th component of the direct sum. Each edge is interpreted as a partial isometry. That means an operator $u$ such that both $u u^*$ and $u^* u$ are projections. For an edge $e \colon v \to v'$, we pick the partial isometry projecting on the $v$th component, and then injecting into the $e$th component of the $v'$th component.

We then take the smallest subalgebra generated by these projections and partial isometries, to obtain the universal C*-algebra satisfying the specified property.

### A family of 2-graphs arising from two-dimensional subshifts

Pask, Raeburn, and Weaver. Ergodic Theory and Dynamical Systems, to appear. Link.

In the same vein as the previous paper, but starting from a more complicated notion of dynamics than just directed graphs. Here, the considered systems are based on a rather restricted class of labellings of the discrete plane. Acceptable labellings are determined by a local constraint, to be satisfied by all occurrences of a basic tile in the plane. Such labellings are viewed as the infinite path space of a dynamical system, I'm not so sure what the states are exactly now. But the essential idea is that providing a path from an occurrence v of the basic tile to another, say v', roughly consists in labelling the points between v and v', respecting the local constraint.

We stopped reading this in depth, because there is a hypothesis ensuring that in the direction orthogonal to $y = x$, labellings always extend in a deterministic way. In a sense, this makes the dynamics 1-dimensional. Guillaume argued that this is a very restricted class of dynamical systems.

### Graphs, groupoids, and Cuntz-Krieger algebras

Kumjian, Pask, Raeburn, and Renault. J. Funct. Anal. 144 (1997), 505–541. Link.

We briefly had a look at this paper for getting an idea of the groupoid-based approach to constructing C*-algebras associated to dynamical systems.

## C*-algebras and sheaves

### Topos theory and complex analysis

In any sheaf topos, there is a natural numbers object, given by the constant sheaf mapping any object of the underlying site to $\mathbb{N}$. By adequately quotienting $\mathbb{N} \times \mathbb{N}$, we get an object $\mathbb{Q}$ of rationals, which similarly maps any object to (the usual) $\mathbb{Q}$. There are then at least two natural ways of defining an object of real numbers: as limits of Cauchy sequences, or as Dedekind cuts. In case the site is spatial, i.e., is the poset of open sets of a space $X$, Dedekind cuts nicely yield the sheaf of continuous maps to $\mathbb{R}$. Anyway, from such an object of real numbers, one may construct an object of complex numbers. Rousseau actually axiomatises such an object, and proceeds to perform a few constructions of analysis.

What might be of interest to us is that the object of complex numbers bears a striking resemblance to C*-algebras. To start with, the object of complex numbers is a complete metric space object, as well as a ring object. In particular, if the underlying space is compact, then its global sections are precisely the elements of the classical C*-algebra of continuous maps $X \to \mathbb{C}$. Furthermore, the internal ring operations agree with the C*-algebra's. And, although Thomas would certainly recall better than me, a Cauchy sequence converges for the usual norm iff it converges for the internal norm (hence in the internal language).

In the general case, continuous maps to $\mathbb{C}$ do not form a C*-algebra, and one usually resorts to some compactification procedure. This is not necessary in the sheaf-theoretic approach: the norm of an internal complex number is a continuous map to $\mathbb{C}$, i.e., pointwise norm.

## Dynamical systems

### Cellular automata and groups

T. Ceccherini-Silberstein and M. Coornaert. Springer, 2010. Link.

Guillaume referred to this book in his talk, notably for concepts such as entropy, surjunctivity, and amenability.

# Ideas

We need a place to put idiot ideas! Here it is: Once an idea has been aired it would be good, (eventually) to delete the entry otherwise it will get too long.

## Homotopy of graphs and infinity categories.(19/10/2011)

There is a reasonable theory of homotopy for graphs (several in fact). As homotopy is increasingly being seen as a form of $(\infty,1)$ category theory it may pay to look at some of these ideas, even if only to reject that line! Some of this is for finite graphs but if one is looking at infinite or row finite' graphs then a form of proper homotopy' might work. (Tim)

## Topology on locally finite metric spaces. (14/11/2011)

Valerio Capraro has asked a good question on MathOverflow and has a theory of Topology on locally finite metric spaces that may be linked to what interests us. He has an archive preprint: (http://arxiv.org/abs/1111.0268). He is interested in operator algebras, amenability etc. and has some good ideas. (Tim)

## Homotopy theory of C*-algebras (2/12/2011)

K. Anderson and Jesper Grodal wrote a note in 1999 that may be useful. They looked at A BAUES FIBRATION CATEGORY STRUCTURE ON BANACH AND C^∗ -ALGEBRAS : [1]

They followed this up with a nice short set of notes on E-theory:[2].

The great thing about both these is that they are fairly short and fairly elementary in their assumptions about prior knowledge.

There is a fairly recent arxiv article that may be of interest:

Homotopy Theory for C^*-algebras: [3]

Authors: Otgonbayar Uuye (Submitted on 12 Nov 2010 (v1), last revised 30 Nov 2011 (this version, v2))

Abstract: Category of fibrant objects is a convenient framework to do homotopy theory, introduced and developed by Ken Brown. In this paper, we apply it to the category of C^{*}-algebras. In particular, we get a unified treatment of (ordinary) homotopy theory for C^{*}-algebras, KK-theory and E-theory, as all of these can be expressed as the homotopy category of a category of fibrant objects.

# Questions

## Analogues of covering spaces?

In the symbolic dynamics theory is there some sort of covering space or unfolding construction? (Compare for instance with the covering space theory of classical topology and the recent attempts to get one in directed homotopy theory. There is such a theory for graphs and it comes in in the modelling of certain types of modal logics.)(Tim:20/10/2011)

## Homotopy theory of C*-algebras

• Should a list of references be built up for this?
• A good question is to see what starting with a dynamical system then passing to its C*-algebra (which is the C*-algebra of a simple topological groupoid associated to the system), what does the homotopy type of that C*-algebra tell one about the system?
• Another ... if you start with a higher dimensional dynamical system, what is the gadget corresponding to the groupoid that Raeburn and others consider?

## nLab entries

• Homotopy theory of operator algebras [4]

## Books

• Paul Arne Østvær, Homotopy theory of C^*-algebras, Frontiers in Mathematics, Springer Basel, 2010, see also arxiv/0812.0154,[5].

## Articles

• Symbolic dynamics and the category of graphs

Terrence Bisson and Aristide Tsemo

Symbolic dynamics is partly the study of walks in a directed graph. By a walk, here we mean a morphism to the graph from the Cayley graph of the monoid of non-negative integers. Sets of these walks are also important in other areas, such as stochastic processes, automata, combinatorial group theory, $C^*$-algebras, etc. We put a Quillen model structure on the category of directed graphs, for which the weak equivalences are those graph morphisms which induce bijections on the set of walks. We determine the resulting homotopy category. We also introduce a "finite-level" homotopy category which respects the natural topology on the set of walks. To each graph we associate a basal graph, well defined up to isomorphism. We show that the basal graph is a homotopy invariant for our model structure, and that it is a finer invariant than the zeta series of a finite graph. We also show that, for finite walkable graphs, if $B$ is basal and separated then the walk spaces for $X$ and $B$ are topologically conjugate if and only if $X$ and $B$ are homotopically equivalent for our model structure.

Keywords: category of graphs, Quillen model structure, walks, symbolic dynamics, coverings

2000 MSC: 05C20, 18G55, 55U35, 37B10

Theory and Applications of Categories, Vol. 25, 2011, No. 22, pp 614-640.

Published 2011-12-31.

## ArXiv submissions

• Dynamical systems of type (m,n) and their C*-algebras,

Authors: Pere Ara, Ruy Exel, Takeshi Katsura