nLab
concrete smooth infinity-groupoid

This is a sub-entry of smooth ∞-groupoid – structures. See there for more context.


Context

Cohesive \infty-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

Backround

Definition

Presentation over a site

Structures in a cohesive (,1)(\infty,1)-topos

structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

infinitesimal cohesion?

Models

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

          In every cohesive (∞,1)-topos there is a notion of concrete cohesive ∞-groupoids. Here we discuss concrete smooth ∞-groupoids. These are the higher generalization of diffeological spaces.

          Definition

          The general abstract definition is:

          Definition

          A concrete smooth \infty-groupoid is a concrete cohesive ∞-groupoids in Smooth∞Grpd.

          We will compare this intrinsic definition with more concrete models:

          Definition

          A diffeological space is a concrete sheaf on the site CartSp smooth{}_{smooth}.

          Write

          DiffeolSpaceSh(CartSp) DiffeolSpace \hookrightarrow Sh(CartSp)

          for the full subcategory on diffeological spaces. Notice that is a quasitopos.

          Definition

          A diffeological groupoid is an internal groupoid in the category of diffeological spaces.

          (…)

          Special cases

          We discuss the special cases of n-truncated concrete smooth \infty-groupoids:

          Diffeological spaces

          Proposition

          Concrete smooth 0-groupoids are equivalently diffeological spaces.

          More in detail, write Conc(τ 0SmoothGrpd)Conc(\tau_{\leq 0} Smooth \infty Grpd) for the full subcategory on the concrete 0-truncated objects. This is equivalent to the category of diffeological spaces

          DiffeolSpConc(τ 0SmoothGrpd). DiffeolSp \simeq Conc(\tau_{\leq 0} Smooth \infty Grpd) \,.
          Proof

          Let XSh(CartSp)SmoothGrpdX \in Sh(CartSp) \hookrightarrow Smooth \infty Grpd be a sheaf on CartSpCartSp. The condition for it to be concrete is that the unit

          XcoDiscΓX X \to coDisc \Gamma X

          is a monomorphism. Since monomorphisms of sheaves are detected objectwise (see category of sheaves) this is equivalent to the statement that for all UCartSpU \in CartSp the morphism

          X(U)SmoothGrpd(U,X)SmoothGrpd(U,coDiscΓX)Grpd(ΓU,ΓX) X(U) \simeq Smooth\infty Grpd(U, X) \to Smooth \infty Grpd(U, coDisc \Gamma X) \simeq \infty Grpd(\Gamma U, \Gamma X)

          is a monomorphism of sets, where in the first step we used the (∞,1)-Yoneda lemma and in the last one the (ΓcoDisc)(\Gamma \dashv coDisc)-adjunction.

          That this morphism is indeed Γ:Sh(U,X)Set(Γ(U),Γ(X))Grpd(Γ(U),Γ(X))\Gamma : Sh(U,X) \to Set(\Gamma(U), \Gamma(X)) \hookrightarrow \infty Grpd(\Gamma(U), \Gamma(X)) follows by chasing the identity on ΓX\Gamma X through the adjunction naturality square for any morphism f:UXf : U \to X

          Set(ΓX,ΓX) Sh(X,coDiscΓX) Γ(f) * f * Set(ΓU,ΓX) Sh(U,coDiscΓX). \array{ Set(\Gamma X, \Gamma X) &\stackrel{\simeq}{\to}& Sh(X, coDisc \Gamma X) \\ \downarrow^{\mathrlap{\Gamma(f)^*}} && \downarrow^{\mathrlap{f^*}} \\ Set(\Gamma U, \Gamma X) &\stackrel{\simeq}{\leftarrow}& Sh(U, coDisc \Gamma X) } \,.

          So this is indeed the defining condition for concrete sheaves that defines diffeological spaces.

          Corollary

          The canonical embedding SmoothMfdSmoothGrpdSmoothMfd \hookrightarrow Smooth \infty Grpd from above factors through diffeological spaces: we have a sequence of full and faithful (∞,1)-functors

          SmoothMfdDiffeolSpSmoothGrpd. SmoothMfd \hookrightarrow DiffeolSp \hookrightarrow Smooth \infty Grpd \,.

          Diffeological groupoids

          We want to say the following

          Proposition

          Concrete 1-truncated smooth \infty-groupoids are equivalent to diffeological groupoids, def. .

          (…)

          Proof

          Let AA be 1-truncated and concrete. Then by definition there is a concrete 0-truncated object A 0A_0 and an effective epimorphism A 0AA_0 \to A – an atlas – , such that the (∞,1)-pullback

          A 1:=A 0× AA 0 A_1 := A_0 \times_A A_0

          is itself concrete.

          Since AA is assumed to be 1-truncated, it follows that A 1A_1 is 0-truncated. By Giraud's axioms in the (∞,1)-topos Smooth∞Grpd we have that AA is equivalent to the groupoid object in an (∞,1)-category A 1A 0A_1 \stackrel{\to}{\to} A_0:

          (1)Alim (A 1A 0). A \simeq \lim_\to (A_1 \stackrel{\to}{\to} A_0) \,.

          Now by prop. both A 0A_0 and A 1A_1 are diffeological spaces. Hence the above exhibits AA as equivalent to a diffeological groupoid.

          (…)

          Transgression of differential cocycles to mapping spaces

          For nn \in \mathbb{N}, write B nU(1) connSmoothGrpd\mathbf{B}^n U(1)_{conn} \in Smooth\infty Grpd for the smooth \infty-groupoid given under the Dold-Kan correspondence by the Deligne complex. Over smooth manifolds this is the coefficient object for circle n-bundles with connection.

          At ∞-Chern-Simons theory the following fact is proven:

          Proposition

          Let Σ\Sigma be a closed smooth manifold of dimension dimΣndim \Sigma \leq n. Then there is an equivalence

          hol:τ ndimΣH(Σ,B nU(1) conn)B ndimΣU(1) hol : \tau_{n - dim \Sigma} \mathbf{H}(\Sigma, \mathbf{B}^n U(1)_{conn}) \to B^{n-dim \Sigma} U(1)

          of discrete ∞-groupoids such that for dimσ=ndim \sigma = n this computes the nn-volume holonomy of circle nn-bundles with connection.

          Using concretization we want to refine this from discrete to smooth \infty-groupoids.

          Write [Σ,B nU(1) conn][\Sigma, \mathbf{B}^n U(1)_{conn}] for the internal hom in the (∞,1)-topos (see there).

          We first look at this for n=dimΣn = dim \Sigma

          Proposition

          For dimΣ=ndim \Sigma = n there is an equivalence

          hol:Concτ 0[Σ,B nU(1) conn]U(1) hol : Conc \tau_0 [\Sigma, \mathbf{B}^nU(1)_{conn}] \to U(1)

          in Smooth∞Grpd.

          Proof

          Since τ 0[Σ,B nU(1) conn]\tau_0 [\Sigma, \mathbf{B}^n U(1)_{conn}] is 0-truncated, hence a sheaf, concretification is that discussed at concrete sheaves:

          Conc(F)(U)=image(F(U)=H(U,F)H(U,coDiscΓF)=Set(Γ(U),Γ(F))). Conc(F)(U) = image( F(U) = \mathbf{H}(U,F) \to \mathbf{H}(U, coDisc \Gamma F) = Set(\Gamma(U), \Gamma(F)) ) \,.

          So for U=*U = * first of all we have by prop. that

          Concτ 0H(Σ,B nU(1) conn)(*)=U(1) Conc \tau_0 \mathbf{H}(\Sigma, \mathbf{B}^n U(1)_{conn})(*) = U(1)

          in SetSet.

          Generally Concτ 0H(Σ,B nU(1) conn)(U)Conc \tau_0 \mathbf{H}(\Sigma, \mathbf{B}^n U(1)_{conn})(U) is therefore a subset of the set of functions of sets UU(1)U \to U(1). We need to show that it is precisely the set of smooth such functions.

          But this is clear: holonomy of a smoth family of smoth circle nn-bundles is a smooth function. Moreover, every smooth function arises this way: for f:UU(1)f : U \to U(1) any smooth function, pick a trivial family of trivial circle nn-bundles with connection and then rescale the connection form using ff.

          References

          Many of the ideas involved here are due to Dave Carchedi.

          A writeup of some aspects is in section 3.3.1 of

          Last revised on June 29, 2011 at 12:13:50. See the history of this page for a list of all contributions to it.