The origin of emergent algebras (part III)

I continue from the post “The origin of emergent algebras (part II)“.   Recall that in the first post on this subject  is mentioned a paragraph by Bellaiche, where he explains that the structure of the tangent bundle is “concealed in dilations”. This is now clear, I hope, but there is something left to do. Bellaiche also mentions the classical construction of the tangent bundle made by Connes, where the tangent bundle of the space X  appears as a completion of the trivial pair groupoid X \times X.  What I want to explain now is how the dilations interact with the trivial groupoid to give the tangent bundle, in a sort of generalization of Connes construction. See arXiv:1107.2823 for all details.

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1. The trivial pair groupoid over a set X is the set X \times X with  the partially defined operation:

(x,u) (u,v) = (x,v)

This partially defined operation is the composition of arrows in the groupoid which has X as the set of objects and X \times X as the set of arrows.  The source of the arrow (x,y) is y = \alpha(x,y), the target of that arrow is x = \omega(x,y).  (This may seem strange, but we have to define the source and target like this in order to  see (x,u) (u,v) = (x,v) as composition of arrows in a groupoid).

The inverse of the arrow (x,y) is (y,x) = (x,y)^{-1}.

The addition of arrows (i.e. composition) is

add [(x,u), (u,v)] = (x,v)

and it is defined for pairs of arrows ((x,u), (u,v)) such that $\alpha(x,u) = \omega(u,v)$.

It will be useful further to introduce the difference of two arrows:

dif [(u,x), (v,x)] = (u,x) \, (v,x)^{-1} = (u,x) (x,v) = (u,v)

The difference is a partially defined operation which  makes sense (is defined for) for pairs of arrows with the same source.

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2. Dilations from the groupoid point of view.  Let \Gamma = (0,+\infty) be the multiplicative group of strictly positive reals.  For any \varepsilon \in \Gamma we define the dilation of arrows of coefficient \varepsilon to be:

\delta_{\varepsilon} (x,y) = (\delta_{\varepsilon}^{y} x , y)

where \delta_{\varepsilon}^{y} x is the intrinsic dilation of coefficient \varepsilon, of the point x with respect to the basepoint y.  Again, as in the case of the definition of source and target of an arrow, here the definition of the dilation of arrows is turned on its head with respect to the writing convention from left to right.

For simplicity we don’t care about the domain of definition of the intrinsic dilations, or about the domain of definition of the dilation of arrows. Enough is to say that the dilation of arrows is defined for small enough arrows. What could that mean ?It is simple: think about the distance function d: X \times X \rightarrow [0,+\infty) as if it is defined on the trivial groupoid. Then, the distance function appears as a kind of a norm on the set of arrows of the groupoid. The norm of the arrow (x,y) is simply d(x,y)  (first time remarked by Lawvere). Then, a short enough arrow is one with a small norm.

Dilations of arrows have the following properties:

  • they preserve the source of arrows:  \alpha (\delta_{\varepsilon} (x,y)) = y = \alpha(x,y),
  • upon the identification x \equiv (x,x) of objects with their respective identity arrows, any dilation of arrows preserves the objects: \delta_{\varepsilon}(x,x) = (x,x), as a consequence of the fact that \delta_{\varepsilon}^{x}x = x,
  • finally, they form a one-parameter group: \delta_{\varepsilon} ( \delta_{\mu} (x,y)) = \delta_{\varepsilon \mu}(x,y), because of the relation \delta^{x}_{\varepsilon} \delta^{x}_{\mu} y = \delta^{x}_{\varepsilon \mu} y.

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3. Deforming the trivial groupoid.  I want to use the trivial groupoid in order to explain the recipe for the approximate sum, given by the red dots in the figure

bellaiche

which appears also in the last post. With the notations from emergent algebras we see that E = \Sigma^{x}_{\varepsilon} (y,z).  There is an equivalent construction for the approximate difference of two points, with respect to a basepoint:

bellaiche_2

Again with the notations from emergent algebras we see that E' = \Delta^{x}_{\varepsilon}(z,y).

By using the trivial groupoid and its deformations by dilations of arrows, it  is easier to explain the origin of the approximate difference. That is because the difference of two arrows is defined on pairs of arrows which have the same source and because dilations of arrows preserve the sources of arrows.

Take two arrows with the same source, say g = (y,x) and h = (z,x). Then g_{\varepsilon} = \delta_{\varepsilon}(y,x) and h_{\varepsilon} = \delta_{\varepsilon}(z,x) have the same source. We can define then the difference of those deformed arrows:

dif(g_{\varepsilon}, h_{\varepsilon}) = (\delta^{x}_{\varepsilon} y , \delta^{x}_{\varepsilon} z)

We define now the deformed difference dif_{\varepsilon}, which is defined on pairs of arrows with the same source, by the relation

\delta_{\varepsilon} dif_{\varepsilon} (g,h) = dif(g_{\varepsilon}, h_{\varepsilon})

A short computation gives

dif_{\varepsilon} [ (z,x) , (y,x) ] = \, ( \Delta^{x}_{\varepsilon}(z,y) , \delta_{\varepsilon}^{x} z)

or, with the notations from the last figure

dif_{\varepsilon} [ (z,x) , (y,x) ] = \, (E', D')

Finally we can pass to the limit with \varepsilon, as in the tangent bundle construction by Connes.

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