concrete smooth infinity-groupoid

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


Cohesive \infty-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory



Presentation over a site

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

structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

infinitesimal cohesion?


Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

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



smooth space


The magic algebraic facts




tangent cohesion

differential cohesion

graded differential 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& 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 }


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)


Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids




\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras



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.


The general abstract definition is:


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

We will compare this intrinsic definition with more concrete models:


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


DiffeolSpaceSh(CartSp) DiffeolSpace \hookrightarrow Sh(CartSp)

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


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


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) \,.

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.


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


Concrete 1-truncated smooth \infty-groupoids are equivalent to diffeological groupoids, def. 3.



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. 1 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:


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


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.


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. 3 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.


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

A writeup of some aspects is in section 3.3.1 of

Revised on June 29, 2011 12:13:50 by Urs Schreiber (