Tag Archives: spaces with dilations

How non-commutative geometry does not work well when applied to non-commutative analysis

I expressed several times the belief that sub-riemannian geometry represents an example of a mathematically new phenomenon, which I call “non-commutative analysis”. Repeatedly happened that apparently general results simply don’t work well when applied to sub-riemannian geometry. This “strange” (not for me) phenomenon leads to negative statements, like rigidity results (Mostow, Margulis), non-rectifiability results (like for example the failure of the theory of metric currents for Carnot groups).  And now, to this adds the following,  arXiv:1404.5494 [math.OA]

“the unexpected result that the theory of spectral triples does not apply to the Carnot manifolds in the way one would expect. [p. 11] ”

i.e.

“We will prove in this thesis that any horizontal Dirac operator on an arbitrary Carnot manifold cannot be hypoelliptic. This is a big difference to the classical case, where any Dirac operator is elliptic. [p. 12]”

It appears that the author reduces the problems to the Heisenberg groups. There is a solution, then, to use

R. Beals, P.C. Greiner, Calculus on Heisenberg manifolds, Princeton University Press, 1988

which gives something resembling spectral triples, but not quite all works, still:

“and show how hypoelliptic Heisenberg pseudodifferential operators furnishing a spectral triple and detecting in addition the Hausdorff dimension of the Heisenberg manifold can be constructed. We will suggest a few concrete operators, but it remains unclear whether one can detect or at least estimate the Carnot-Caratheodory metric from them. [p. 12]”

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This seems to be an excellent article, more than that, because it is a phd dissertation  many things are written clearly.

I am not surprised at all by this, it just means that, as in the case with the metric currents, there is an ingredient in the spectral triples theory which introduces by the backdoor some commutativity, which messes then with the non-commutative analysis  (or calculus).

Instead I am even more convinced than ever that the minimal (!) description of sub-riemannian manifolds, as models of a non-commutative analysis, is given by dilation structures, explained most recently in arXiv:1206.3093 [math.MG].

A corollary of this is: sub-riemannian geometry (i.e. non-commutative analysis of dilation structures)  is more non-commutative than non-commutative geometry .

I’m waiting for a negative result concerning the application of quantum groups to sub-riemannian geometry.

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A less understood problem in sub-riemannian geometry (I)

A complete, locally compact riemannian manifold is a length metric space by the Hopf-Rinow theorem. The problem of intrinsic characterization of riemannian spaces asks for the recovery of the manifold structure and of the riemannian metric from the distance function coming from  to the length functional.

For 2-dim riemannian manifolds the problem has been solved by A. Wald in 1935. In 1948 A.D. Alexandrov  introduces his famous curvature (which uses comparison triangles) and proves that, under mild smoothness conditions on this curvature, one is capable to recover the differential structure and the metric of the 2-dim riemannian manifold. In 1982 Alexandrov proposes as a conjecture that a characterization of a riemannian manifold (of any dimension) is possible in terms of metric (sectional)  curvatures (of the type introduced by Alexandrov) and weak smoothness assumptions formulated in metric way (as for example Hölder smoothness).

The problem has been solved by Nikolaev in 1998, in the paper A metric characterization of Riemannian spaces. Siberian Adv. Math.   9,  no. (1999),  1-58.  The solution of Nikolaev can be summarized  like this: he starts with a locally compact length metric space (and some technical details), then

  •  he constructs a (family of) intrinsically defined tangent bundle(s) of the metric space, by using a generalization of the cosine formula for estimating a kind of a distance between two curves emanating from different points. This will lead him to a generalization of the tangent bundle of a riemannian manifold endowed with the canonical Sasaki metric.
  • He defines a notion of sectional curvature at a point of the metric space, as a limit of a function of nondegenerated geodesic triangles, limit taken as these triangles converge (in a precised sense)  to the point.
  • The sectional curvature function thus constructed is supposed to satisfy a Hölder continuity condition (thus a regularity formulated in metric terms)
  • He proves then that  the metric space is isometric with (the metric space associated to) a riemannian manifold of precise (weak) regularity (the regularity is related to the regularity of the sectional curvature function).

Sub-riemannian spaces are length metric spaces as well. Any riemannian space is a sub-riemannian one. It is not clear at first sight why the characterization of riemannian spaces does not extend to sub-riemannian ones. In fact, there are two problematic steps for such a program for extending Nikolaev result to sub-riemannian spaces:

  • the cosine formula, as well as the Sasaki metric on the tangent bundle don’t have a correspondent in sub-riemannian geometry (because there is, basically, no statement canonically corresponding to Pythagoras theorem);
  • the sectional curvature at a point cannot be introduced by means of comparison triangles, because sub-riemanian spaces do not behave well with respect to this comparison of triangle idea, as proved by Scott Pauls.

In 1996 M. Gromov formulates the problem of intrinsic characterization of sub-riemannian spaces.  He takes the Carnot-Caratheodory (or CC) distance (this is the name of the distance constructed on a sub-riemannian manifold from the differential geometric data we have, which generalizes the construction of the riemannian distance from the riemannian metric) as the only intrinsic object of a sub-riemannian space. Indeed, in the linked article, section 0.2.B. he writes:

If we live inside a Carnot-Caratheodory metric space V we may know nothing whatsoever about the (external) infinitesimal structures (i.e. the smooth structure on V, the subbundle H \subset T(V) and the metric g on H) which were involved in the construction of the CC metric.
He then formulates the goal:
Develop a sufficiently rich and robust internal CC language which would enable us to capture the essential external characteristics of our CC spaces.
He proposes as an example to recognize the rank of the horizontal distribution, but in my opinion this is, say, something much less essential than to “recognize” the “differential structure”, in the sense proposed here as the equivalence class under local equivalence of dilation structures.
As in Nikolaev solution for the riemannian case, the first step towards the goal is to have a well defined, intrinsic, notion of tangent bundle. The second step would be to be able to go to higher order approximations, eventually towards a curvature.
My solution is to base all on dilation structures. The solution is not “pure”, because it introduces another ingredient, besides the CC distance: the field of dilations. However, I believe that it is illusory to think that, for the general sub-riemannian case, we may be able to get a “sufficiently rich and robust” language without. As an example, even the best known thing, i.e. the fact that the metric tangent spaces of a (regular) sub-riemannian manifold are Carnot groups, was previously not known to be an intrinsic fact. Let me explain: all proofs, excepting the one by using dilation structures, use non-intrinsic ingredients, like differential calculus on the differential manifold which enters in the construction of the CC distance. Therefore, it is not known (or it was not known, even not understood as a problem) if this result is intrinsic or if it is an artifact of the proof method.
Well, it is not, it turns out, if we accept dilation structures as intrinsic.
There is a bigger question lingering behind, once we are ready to think about intrinsic properties of sub-riemannian spaces:  what is a sub-riemannian space? The construction of such spaces uses notions and results which are by no means intrinsic (again differential structures, horizontal bundles, and so on).
Therefore I understand Gromov’s stated goal as:
Give a minimal, axiomatic, description of sub-riemannian spaces.
[Adapted from the course notes Sub-riemannian geometry from intrinsic viewpoint.]

Dictionary from emergent algebra to graphic lambda calculus (III)

Continuing from   Dictionary from emergent algebra to graphic lambda calculus (II) , let’s introduce the following macro, which is called “relative \Upsilon gate”:

linear_1

If this macro looks involved, then we might express it with the help of emergent algebra crossing macros, like this:

linear_2

Notice that none of these graphs are in the emergent algebra sector, a priori! However, by looking at the graph from the  left, we recognize an old friend: a chora.  See arXiv:1103.6007 section 5, for a definition  of the chora construction, but only in relation with emergent algebra gates. Previously the chora diagram appeared as a convenient notation for the relative dilation gate, but here we see it appearing in a different context. It is already present in relation with lambda calculus in the lambda-scale article arXiv:1205.0139  , section 4.  It will be important later, when I shall give an exposition of the second paragraph after Question 1 from the post Graphic lambda calculus used for quantum programming (Towards qubits III) .

With the help of the relative \Upsilon macro, we can give a new look at the mystery move from the end of dictionary II post. Indeed, look at the following succession of moves:

linear_4

The use of the mystery move is to transform the relative \Upsilon gate into a usual \Upsilon gate! So, if we accept the mystery move for the emergent algebra sector, then the effect is that the relative \Upsilon gate is in the emergent algebra sector of the graphic lambda calculus and, moreover, it can be transformed into a \Upsilon gate.

With this information let’s go back to the graphs A and C from the dictionary II post.

linear_5

We know that A \leftrightarrow C in the emergent algebra sector is the right translation of the approximate associativity from emergent algebras (formalism using binary trees, for example). In the last post we arrived at the conclusion that we can prove A \leftrightarrow C by using the mystery move as a legal move in the emergent algebra sector of the graphic lambda calculus (in combination with the other valid moves for this sector, but not using GOBAL FAN-OUT).

Instead of the graph C, we may  introduce the graph C' and compare it with the graph C:

linear_6

The only difference between C and C' is  given by the appearance of the relative \Upsilon gate, in C', instead of the regular \Upsilon gate in C.

 

We can prove  that  C' \leftrightarrow A. In particular this proves that C' is in the emergent algebra sector, even if it contains a relative \Upsilon gate. Recall that  if we don’t accept the mystery move in the emergent algebra sector, then it’s not clear if it belongs to that sector.  The proof is not given, but it’s straightforward, using the moves CO-ASSOC, CO-COMM, LOC PRUNING, R2 and ext 2 (therefore without the mystery move).

We can pass from C' to C by using the fact that the mystery move allows to transform a relative \Upsilon gate into a regular one. So this is the way in which enters the mystery move in the equivalence A \leftrightarrow C: through A \leftrightarrow C' , which can be done without the mystery move, and C' \leftrightarrow C by the mystery move, under the form of transforming a relative gate into a regular one.

Dictionary from emergent algebra to graphic lambda calculus (II)

This post continues the Dictionary from emergent algebra to graphic lambda calculus (I).     Let us see  if we can prove the approximate associativity property (for the approximate sum) in graphic lambda calculus.  I am going to use the dictionary and see if I can make the translation.

With the dilation structures/tree formalism, the approximate associativity, with it’s proof, is this:

sum_1

Let’s translate the trees, by using the dictionary.

sum_b

At first sight, the graphs A and C are clearly in the emergent algebra sector of the graphic lambda calculus. For the graph B, which corresponds to the tree from the middle of the first figure, it’s not clear if it belongs to the same sector.

The next figure shows that it does, because we can pass from A to B by a succession of moves acceptable in the sector.

sum_c

Problems appear when we try to relate B and C. The reason of these problems, we shall see, is the fact that we renounced at having variables, by employing  the “fan-out” gate \Upsilon. But this gate is a fan-out only in a very weak sense, in fact is more like a co-commutative co-monoid operation.

Let’s see, we prepare B in order to be more alike C.

sum_d

There is a part of the final graph which is encircled by a green dashed line, pay attention to it.

We prepare now C a bit:

sum_e

The graph from the right has also an encircled part in green. Look close to this graph, in parallel with the last graph from the preparation of B. We remark that these two graphs are the same outside of the respective encircled regions. If we could transform one encircled region into the other then we would have a proof that B \leftrightarrow C.

In conclusion, the mystery move

sum_f

would solve the problem of proving the approximate associativity in the emergent algebra sector of the graphic lambda calculus. Remark that if \Upsilon would really be a fan-out gate, i.e. if we could apply to it GLOBAL FAN-OUT moves, then the mystery move would be true.

But I don’t want to use global moves in the emergent algebra sector, because in the tree formalism all moves are local. Moreover, even with global fan-out, in order to be able to apply it, it would be necessary that at the labels “x” and “u” to be grafted graphs which have no connection in common, therefore, strictly speaking, we succeed to prove an approximate associativity weaker than the approximate associativity from emergent algebras/tree formalism.  (The same phenomenon, but for another identity of emergent algebras, is explained in this post.) The reason is in the apparently innocuous but hard to manage fact that we renounce of having variables in graphic lambda calculus.

Next time, about the meaning of the mystery move.

Dictionary from emergent algebra to graphic lambda calculus (I)

Because I am going to explore in future posts the emergent algebra sector, I think it is good to know where we stand with using graphic lambda calculus for describing proofs in emergent algebra as computations.  In the big map of research paths, this correspond to the black path linking “Energent algebra sector” with “Emergent algebras”.

A dictionary seems a good way to start this discussion.

Let’s see, there are three formalisms there:

  • in the first paper on spaces with dilations, Dilatation structures I. Fundamentals arXiv:math/0608536  section 4,  is introduced a formalism using binary decorated trees in order to ease the manipulations of dilation structures,
  • emergent algebras are an abstraction of dilation structures, in the sense that they don’t need a metric space to live on. The first paper on the subject is Emergent algebras arXiv:0907.1520   (see also Braided spaces with dilations and sub-riemannian symmetric spaces  arXiv:1005.5031  for explanations of the connection between dilation structures and emergent algebras, as well as for braided symmetric spaces, sub-riemannian symmetric spaces, conical groups, items you can see on the big map mentioned before). Emergent algebras is a mixture of an algebraic theory with an important part of epsilon-delta analysis.  One of the goals of graphic lambda calculus is to replace this epsilon-delta part by a computational part.
  • graphic lambda calculus, extensively described here, has an emergent algebra sector (see  arXiv:1305.5786 , equally check out the series Emergent algebras as combinatory logic part I, part II, part IIIpart IV,  ). This is not an algebraic theory, but a formalism which contains lambda calculus.

The first figure describes a dictionary of objects which appear in these three formalisms. In the first column you find objects as they appear in dilation structures – emergent algebra formalism. In the second column you find the corresponding object in the binary trees formalism. In the third column there are the respective objects as they appear in the emergent algebra sector of the graphic lambda calculus.

sum_basic

Some comments: the first two rows  are about an object called

  • “dilation (of coefficient \varepsilon, with \varepsilon \in \Gamma, a commutative group)” in dilation structures,  which is a operation in emergent algebras, indexed by \varepsilon, (the second row is about dilations of coefficient \varepsilon^{-1}),
  • it is an elementary binary tree with the node decorated by white (for \varepsilon) or black (for \varepsilon^{-1})
  • it is one of the elementary gates in graphic lambda calculus.

The third row is about the “approximate sum” in dilation structures, which is a composite operation in emergent algebras, which is a certain graph in graphic lambda calculus.

The fourth row is about the “approximate difference” and the fifth about the “approximate inverse”.

For the geometric meaning of these objects see the series on  The origin of emergent algebras part I, part II, part III,   or go directly and read arXiv:1304.3694 .

What is different between these rows?

  • In the first row we have an algebra structure based on identities between composites of operations defined on a set.
  • In the second row we have trees with leaves decorated by labels from an alphabet (of formal variables) or terms constructed recursively from those  .
  • In the third row we have graphs with no variable names. (Recall that in graphic lambda calculus there are no variable names. Everything written in red can be safely erased, it is put there only for the convenience of the reader.)

Let’s see now the dictionary of identities/moves.

moves_basic

The most important comment is that identities in emergent algebras become moves in the other two formalisms. A succession of moves is in fact a proof for an identity.

The names of the identities or moves have been commented in many places, you see there names like “Reidemeister move” which show relations to knot diagrams, etc. See this post for the names of the moves and relations to knot diagrams, as well as section 6 from  arXiv:1305.5786 .

Let’s read the first column: it says that from an algebraic viewpoint an emergent algebra is a one parameter family (indexed by \varepsilon \in \Gamma) of idempotent right quasigroups. From the geometric point of view of dilation structures, it is a formalisation of properties expected from an object called “dilation”, namely that it preserves the base-point ( “x” in the figure), that a composition of dilations of coefficients \varepsilon, \mu, with the same base-point,  is again a dilation, of coefficient \varepsilon \mu, etc.

In the second column we see two moves, R1 and R2, which can be applied anywhere in a decorated binary tree, as indicated.

In the third column we see that these moves are among the moves from graphic lambda calculus, namely that R1 is in fact related to the oriented Reidemeister move R1a, so it has the same name.

The fact that the idempotent right quasigroup indexed by the neutral element of \Gamma, denoted by 1, is trivial, has no correspondent for binary trees, but it appears as the move (ext 2) in graphic lambda calculus. Through the intermediary of this move appears the univalent termination gate.

These are the common moves. To these moves add, for the part of the emergent algebra sector, the R1b move, the local fan-out moves and some pruning moves. There is also the global fan-out move which is needed, but we are going to replace it by a local move which has the funny name of “linearity of fan-out”, but that’s for later.

The local fan-out moves and the pruning moves are needed for the emergent algebra sector but not for the binary trees or emergent algebras. They are the price we have to pay for eliminating variable names. See the algorithm for producing graphs from lambda calculus terms, for what concerns their use for solving the same problem for untyped lambda calculus. (However, the emergent algebra sector is to be compared not with the lambda calculus sector, but with the combinatory logic sector, more about this in a further post.)

We don’t need all pruning moves, but only one which, together with the local fan-out moves, form a family which could be aptly called:

co_mono_y

(notice I consider a reversible local pruning)

Grouping moves like this makes a nice symmetry with the fact that \Gamma is a commutative group, as remarked here.

As concerns the R1b move, which is the one from the next figure, I shall use it only if really needed (for the moment I don’t). It is needed for the knot diagrams made by emergent algebra gates sector, but it is not yet clear to me if we need it for the emergent algebra sector.

r1bmove

However, there is a correspondent of this move for emergent algebras. Indeed, recall that a right quasigroup is a a quasigroup if the equation x \circ a = b  has a solution, which is unique. If our emergent algebra is in fact a (family of) quasigroup(s) , as happens for the cases of conical groups or for symmetric spaces in the sense of Loos (explained in arXiv:1005.5031 ), then in particular it follows that the equation  x \circ_{\varepsilon} a = a has only the solution x = a (for \varepsilon \not = 1). This last statement has the R1b move as a correspondent in the realm of the emergent algebra sector.

Until now we have only local moves in the emergent algebra sector. We shall see that we need a global move (the global fan-out) in order to prove that the dictionary works, i.e. for proving the fundamental identities of emergent algebras within the graphic lambda calculus formalism. The goal will be to replace the global fan-out move by a new local move (i.e. one which is not a consequence of the existing moves of graphic lambda calculus). This new move will turn out to be a familiar sight, because it is related to the way we see linearity in emergent algebras.

What group is this? (Parallel transport in spaces with dilations, II)

I continue from Parallel transport in spaces with dilations, I.   Recall that we have a set X , which could be see as the complete directed graph X^{2}. By a construction using binary decorated trees, with leaves in X, we obtain first a set of finite trees FinT(X), then we put an equivalence relation \sim on this set, namely two finite trees A and B are close A \sim B if A \bullet B is a finite tree. The class of finite points PoinT(X) is formed by the equivalence classes [A] of finite trees A  with respect to the closeness relation \sim.

Notice that the equality relation is \leftrightarrow , in this world.  This equality relation is generated by the “oriented Reidemeister moves”  R1a and R2a, which appear also as moves in graphic lambda calculus. (By the way, this construction can be made in graphic lambda calculus, which has the moves R1a and R2a. In this way we obtain a higher level of abstraction, because in the process we eliminate the set X. Graphic lambda calculus does not need variables. More about this at a future time.) If you are not comfortable with this equality relation than you can just factorize with it and replace it by equality.

It is clear that to any “point” x \in X is associated a finite point [x] \in PoinT(X). Immediate questions jump into the mind:

  • (Q1)  Is the function x \in X \mapsto [x] \in PoinT(X) injective? Otherwise said, can you prove that if x \not = y then x \bullet y is not a finite tree?
  • (Q2)  What is the cardinality of PoinT(X)? Say, if X is finite is then  PoinT(X) infinite ?

Along with these questions, a third one is almost immediate. To any two finite trees A and B is associated the function [AB] : [B] \rightarrow [A]  defined by

[AB](C) = A \circ (B \bullet C) .

The function is well defined: for any C \in [B] we have B \bullet C \in FinT(X), by definition. Therefore [AB](C) \in [A], because A \bullet \left( [AB](C) \right) \leftrightarrow B \bullet C .

Consider now the groupoid ParaT(X) with the set of objects PoinT(X) and the set of arrows generated by the arrows [AB] from [B] to [A].  The third question is:

  • (Q3)  What is the isotropy group of a finite point [A]   (in particular [x] ) in this groupoid? Call this isotropy group IsoT(X) and remark that because the groupoid ParaT(X) is connected, it follows that the isotropy groupoid does not depend on the object (finite point), in particular is the same at any point x \in X (seen of course as [x] \in PoinT(X) ).

In a future post I shall explain the answers to these questions, which I think they are the following:

  • Q1:  yes.
  • Q2: infinite.
  • Q3: a kind of free nilpotent group.

But feel free to contradict me, or to propose solutions. Of course, I shall cite any valuable contribution, even if it appears in a blog  (via +Graham Steel).

Parallel transport in spaces with dilations, I

I intended to call this series of posts “What group is this?”, but I switched to this more precise, albeit more bland name. In this first post of the series I take again, in more generality, the construction explained in the post   Towards geometric Plünnecke graphs.

The construction starts in the same way, almost. After I give this first part of the construction, an interpretation in term sof groupoids is provided.  We consider only the moves  R1a and R2a, like in the post “A roadmap to computing with space“:

roadmap_4

(The names “R1a”, “R2a” come from the names of oriented Reidemeister moves, see arXiv:0908.3127  by M. Polyak.)
Definition 1. The moves R1a, R2a  act on the set of binary trees  T(X) with nodes decorated with two colours (black and white) and leaves decorated with elements of a set of “variable names”  X which has at least two elements.  I shall denote by  A, B, C … such trees and by  x, y, z, u, v, w … elements of  X.

The edges of the trees are oriented upward. We admit  X to be a subset of  T(X), thinking about  x \in X as an edge pointing upwards which is also a leaf decorated with x.

The moves are local, i.e. they can be used for any portion of a tree from  T(X) which looks like one of the patterns from the moves, with the understanding that the rest of the respective tree is left unchanged.

We denote by A \leftrightarrow B the fact that the A can be transformed into B by a finite sequence of moves.
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Definition 2. The class of finite trees   FinT(X) \subset T(X) is the smallest subset of  T(X) with the  properties:

  •   X \subset FinT(X),
  • if A, B \in FinT(X) then  A \circ B \in FinT(X)  , where A \circ B is the tree

roadmap_7

  • if A, B, C \in FinT(X) then  Sum(A,B,C) \in FinT(X) and Dif(A,B,C) \in FinT(X), where  Sum(A,B,C) is the tree

roadmap_8

and Dif(A,B,C) is the tree

roadmap_10

  • if A \in FinT(X) and we can pass from  A to  B (i.e. A \leftrightarrow B )  by one of the moves then  B \in FinT(X).

_________________

Definition 3. Two graphs   A, B \in FinT(X)  are close, denoted by  A \sim B, if there is   C \in FinT(X) such that  B can be moved into   A \circ C.

_________________

Notice that A \leftrightarrow B then A \sim B.
Proposition 1. The closeness relation is an equivalence.

Proof.  I start with the remark that A \sim B if and only if A \bullet B \in FinT(X), where A \bullet B is the tree

roadmap_11

Indeed, A \sim B if there is C \in FinT(X) such that B \leftrightarrow A \circ C. Then

roadmap_12
which proves that A \bullet B \in FinT(X).  Then  A \sim A for any A \in FinT(X), because A \leftrightarrow A \bullet A, therefore A \bullet A \in FinT(X). Suppose now that  A \sim B. Then A \bullet B \in FinT(X). Notice that B \bullet A \leftrightarrow Dif(A, A \bullet B, A), by the following sequence of moves:

roadmap_13
But Dif(A, A \bullet B, A) \in FinT(X), from the hypothesis. Therefore B \bullet A \in FinT(X), which is equivalent with B \sim A.

Finally, suppose that A \sim B, B \sim C. Then B \sim A by the previous reasoning. Then there are A', C' \in FinT(X) such that A \leftrightarrow B \circ A' and C \leftrightarrow B \circ C'. It follows that A \bullet C \leftrightarrow Dif(B, A', C'), therefore A \bullet C \in FinT(X), which proves that A \sim C.

_________________
Definition 4. The class of finite points  of T(X) is  PoinT(X)  is the set of equivalence classes w.r.t  \sim.

_________________

Same construction, with groupoids.  We may see \leftrightarrow as being an equivalence relation. Let  T_{0}(X)  be the set of equivalence classes w.r.t \leftrightarrow. We can define on  T_{0}(X) the operations (A,B) \mapsto A \circ B and (A,B) \mapsto A \bullet B  (because the moves R1a, R2a are local). Then  (T_{0}(X), \circ, \bullet) is the free left idempotent right quasigroup generated by the set X.

Idempotent right quasigroups are the focus of the article arXiv:0907.1520, where emergent algebras are introduced as deformations of such objects. An idempotent right quasigroup (M, \circ, \bullet) is a non-empty set endowed with two operations, such that

  •   (idempotence) x \circ x = x \bullet x = x for any x \in M,
  •   (right quasigroup) x \bullet (x \circ y) = x \circ (x \bullet y) = y for any x, y \in M

Let T_{0}(X)^{2} be the trivial (pair) groupoid over T_{0}(X). This is the groupoid with objects which are elements of  T_{0}(X) and arrows of the form  (A,B) \in T_{0}(X) \times T_{0}(X). Equivalently, we see T_{0}(X)^{2} to be the set of it’s arrows, we identify objects with their identity arrows (in this case we identify A \in Ob T_{0}(X)^{2} with it’s identity arrow (A,A) \in T_{0}(X)^{2}). Seen like this, the trivial groupoid  T_{0}(X)^{2} is just the set  T_{0}(X) \times T_{0}(X), with the partially defined operation (composition of arrows)

(A,B) (B,C) = (A,C)

and with the unary inverse operation

(A,B)^{-1} = (B,A) .

Remark that the function  F: T_{0}(X)^{2} \rightarrow T_{0}(X)^{2}  defined by  F(B,A) = (A \circ B, A)  is a bijection of the set of arrows and moreover

  •   it preserves the objects F(A,A) = (A,A),
  • the inverse has the expression  F^{-1}(B,A) = (A \bullet B, A).

Define the groupoid  F \sharp T_{0}(X)^{2} by declaring F to be an isomorphism of groupoids. This means  F \sharp T_{0}(X)^{2} to be  the set of arrows  T_{0}(X)\times T_{0}(X), with the partially defined composition of arrows given by

(B,A) * (D,C) = F^{-1} \left( F(B,A) F(D,C)) \right)
for any pair of arrows (B,A), (D,C) such that F(B,A) can be composed in  T_{0}(X)^{2} with F(D,C), and unary inverse operation given by

(B,A)^{-1,*} = F^{-1} \left( \left( F(B,A) \right)^{-1} \right)  .

The groupoid  F \sharp T_{0}(X)^{2} has then the composition operation

(B, C \circ D) * (D,C) = (Sum(C,D,B), C) ,

the unary inverse operation

(B,A)^{-1,*} = (Dif(A,B,A), A \circ B)

and the set of objects Ob(F \sharp T_{0}(X)^{2}) = T_{0}(X) .

Consider the set  X^{2} = X \times X, seen as a subset of arrows of the groupoid   F \sharp T_{0}(X)^{2} .

The class of finite trees FinT(X) appears in the following way. First define  Fin_{0}T(X) to be the set of equivalence classes w.r.t  \leftrightarrow of elements in FinT(X).

Remark that \left( Fin_{0} T(X)\right)^{2} is a sub-groupoid of F \sharp T_{0}(X)^{2}, which moreover it contains X^{2} and is closed w.r.t. the application of F, seen this time as a function (which is not a morphism) from F \sharp T_{0}(X)^{2} to itself. In fact Fin_{0} T(X) is the smallest subset of T_{0}(X) with this property. Let’s give to the groupoid \left( Fin_{0} T(X)\right)^{2} the name   \langle X^{2} \rangle, seen as a sub-groupoid of  F \sharp T_{0}(X)^{2} .

Moreover  F\left( \langle X^{2} \rangle \right) is a sub-groupoid of the trivial groupoid  T_{0}(X)^{2}, with set of objects  Fin_{0}T(X). But sub-groupoids of the trivial groupoid are the same thing as equivalence relations. In this particular case (A,B) \in F\left( \langle X^{2} \rangle \right) if and only if  A, B \in Fin_{0}T(X) and A \sim B.

Next time you’ll see some groups (which are associated to parallel transport in dilation structures) which are in some sense universal, but I don’t know (yet) what structure they have. “What group is this?” I shall ask next time.

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Do you remark at which stage of this construction the map becomes the territory, thus creating points out of abstract nonsense?

To get a sense of this, replace the set of arrows X^{2} with a graph with nodes in X.