nLab
path groupoid

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous *

          \array{ && id &\dashv& id \ && \vee && \vee \ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \ && \bot && \bot \ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }

          </semantics></math></div>

          Models

          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)

          \infty-Lie theory

          ∞-Lie theory (higher geometry)

          Background

          Smooth structure

          Higher groupoids

          Lie theory

          ∞-Lie groupoids

          ∞-Lie algebroids

          Formal Lie groupoids

          Cohomology

          Homotopy

          Examples

          \infty-Lie groupoids

          \infty-Lie groups

          \infty-Lie algebroids

          \infty-Lie algebras

          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

          Last revised on October 7, 2012 at 17:33:18. See the history of this page for a list of all contributions to it.