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path groupoid

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differential geometry

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Contents

Idea

For XX a smooth space, there are useful refinements of the fundamental groupoid Π 1(X)\Pi_1(X) which remember more than just the homotopy class of paths, i.e. whose morphisms are (piecewise, say) smooth paths in XX modulo an equivalence relation still strong enough to induce a groupoid structure, but weaker than dividing out homotopies relative to endpoints.

Definition

Let XX be a smooth manifold.

Definition

For γ 1,γ 2:[0,1]X\gamma_1, \gamma_2 : [0,1] \to X two smooth maps, a thin homotopy γ 1γ 2\gamma_1 \Rightarrow \gamma_2 is a smooth homotopy, i.e. a smooth map

Σ:[0,1] 2X \Sigma : [0,1]^2 \to X

with

  • Σ(0,)=γ 1\Sigma(0,-) = \gamma_1
  • Σ(1,)=γ 2\Sigma(1,-) = \gamma_2
  • Σ(,0)=γ 1(0)=γ 2(0)\Sigma(-,0) = \gamma_1(0) = \gamma_2(0)
  • Σ(,1)=γ 1(1)=γ 2(1)\Sigma(-,1) = \gamma_1(1) = \gamma_2(1)

which is thin in that it doesn’t sweep out any surface: every 22-form pulled back to it vanishes:

  • BΩ 2(X):Σ *B=0\forall B \in \Omega^2(X)\colon \Sigma^* B = 0.
Definition

A path γ:[0,1]X\gamma\colon [0,1] \to X has sitting instants if there is a neighbourhood of the boundary of [0,1][0,1] such that γ\gamma is locally constant restricted to that.

Definition

The path groupoid P 1(X)P_1(X) is the diffeological groupoid that has

  • Obj(P 1(X))=XObj(P_1(X)) = X
  • P 1(X)(x,y)={P_1(X)(x,y) = \{thin-homotopy classes of paths γ:xy\gamma\colon x \to y with sitting instants}\}.

Composition of paths comes from concatenation and reparameterization of representatives. The quotient by thin-homotopy ensures that this yields an associative composition with inverses for each path.

This definition makes sense for XX any generalized smooth space, in particular for XX a sheaf on Diff.

Moreover, P 1(X)P_1(X) is always itself naturally a groupoid internal to generalized smooth spaces: if XX is a Chen space or diffeological space then P 1(X)P_1(X) is itself internal to that category. However, even if XX is a manifold, P 1(X)P_1(X) will not be a manifold, see smooth structure of the path groupoid for details.

There are various generalizations of the path groupoid to n-groupoids and ∞-groupoids. See

Remarks

If GG is a Lie group, then internal (i.e. smooth) functors from the path groupoid to the one-object Lie groupoid corresponding to GG are in bijection to Lie(G)Lie(G)-valued differential forms on XX. With gauge transformations regarded as morphisms between Lie-algebra values differential forms, this extends naturally to an equivalence of categories

[P 1(X),BG]Ω 2(X,Lie(G)) [P_1(X), \mathbf{B}G] \simeq \Omega^2(X, Lie(G))

where on the left the functor category is the one of internal (smooth) functors.

More generally, smooth anafunctors from P 1(X)P_1(X) to BG\mathbf{B}G are canonically equivalent to smooth GG-principal bundles on XX with connection:

Ana(P 1(X),BG)GBund (X). Ana(P_1(X), \mathbf{B}G) \simeq G Bund_\nabla(X) \,.

See also

Revised on October 7, 2012 17:33:18 by Urs Schreiber (89.204.137.246)