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“:


(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.
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


  • 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


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


  • 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


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

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:

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.


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.


6 thoughts on “Parallel transport in spaces with dilations, I”

  1. Can we eventually think of all of this as a way to define transformations between arbitrarily large and random graphs?

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