nLab path groupoid

Contents

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

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

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& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

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

Related topics

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

References

For “path groupoids” in simplicial sets, aka Dwyer-Kan loop groupoids see there, such as:

Last revised on April 26, 2023 at 08:05:36. See the history of this page for a list of all contributions to it.