smooth infinity-groupoid


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



A smooth \infty-groupoid is an ∞-groupoid equipped with cohesion in the form of smooth structure. Examples include smooth manifolds, Lie groups, Lie groupoids and generally Lie infinity-groupoids, but also for instance moduli spaces of differential forms, moduli stacks of principal connections and generally of cocycles in differential cohomology.

The (∞,1)-topos SmoothGrpdSmooth \infty Grpd of all smooth \infty-groupoids is a cohesive (∞,1)-topos. It realizes a higher geometry version of differential geometry.

Many properties of smooth \infty-groupoids are inherited from the underlying Euclidean-topological ∞-groupoids. See ETop∞Grpd for more.

There is a refinement of smooth \infty-groupoids to synthetic differential ∞-groupoids. See SynthDiff∞Grpd for more on that.



For XX a smooth manifold, say an open cover {U iX}\{U_i \to X\} is a differentiably good open cover if each non-empty finite intersection of the U iU_i is diffeomorphic to a Cartesian space.


Every paracompact smooth manifold admits a differentiably good open cover.


This is a folk theorem. A detailed proof is at good open cover.


Let SmoothMfd be the large site of paracompact smooth manifolds with smooth functions between them and equipped with the coverage of differentiably good open covers.


This does indeed define a coverage. The Grothendieck topology that is generated from it is the standard open cover topology.


For {U iX}\{U_i \to X\} any open cover of a paracompact manifold also iU i\coprod_i U_i is paracompact. Hence we may find a differentiably good open cover {K j iU i}\{K_j \to \coprod_i U_i\}. This is then a refinement of the original open cover of XX.


Let CartSp smooth{}_{smooth} be the site of Cartesian spaces with smooth functions between them and equipped with the coverage of differentiably good open covers.


The (∞,1)-topos of smooth \infty-groupoids is the (∞,1)-category of (∞,1)-sheaves on CartSp smooth{}_{smooth}:

SmoothGrpd:=Sh (,1)(CartSp smooth). Smooth \infty Grpd := Sh_{(\infty,1)}(CartSp_{smooth}) \,.




SmoothGrpdSmooth \infty Grpd is a cohesive (∞,1)-topos.


The site CartSp smooth{}_{smooth} is (as discussed there) an ∞-cohesive site. See there for the implication.


Let SmoothMfd be the large site of paracompact smooth manifolds with smooth functions between them and equipped with the coverage whose covering families are differentiably good open covers : open covers {U iU}\{U_i \to U\} where each non-empty open intersection is diffeomorphic to a Cartesian space.


This does indeed define a coverage and the Grothendieck topology generated by it is the standard open cover topology.


This is discussed in detail at good open cover.


The (∞,1)-topos SmoothGrpdSmooth \infty Grpd is equivalent to the hypercompletion Sh^ (,1)(SmoothMfd)\hat Sh_{(\infty,1)}(SmoothMfd) of the (∞,1)-category of (∞,1)-sheaves on the large site SmoothMfd

SmoothGrpdSh^ (,1)(SmoothMfd). Smooth \infty Grpd \simeq \hat Sh_{(\infty,1)}(SmoothMfd) \,.

By the above we have that CartSp smooth{}_{smooth} is a dense sub-site of SmoothMfd. With this the claim follows as in the analogous discussion at ETop∞Grpd.


The canonical embedding of smooth manifolds as 0-truncated objects in SmoothGrpdSmooth\infty Grpd is a full and faithful (∞,1)-functor

SmoothMfdSmoothGrpd,. SmoothMfd \hookrightarrow Smooth \infty Grpd ,.

Relative cohesion

We discuss the relation of SmoothGrpdSmooth\infty Grpd to other cohesive (∞,1)-toposes.

Continuous cohesion

The cohesive (∞,1)-topos ETop∞Grpd of Euclidean-topological ∞-groupoids has as site of definition CartSp top{}_{top}. There is a canonical forgetful functor

i:CartSp smoothCartSp top i : CartSp_{smooth} \to CartSp_{top}

The functor ii extends to an essential (∞,1)-geometric morphism

(i !i *i *):SmoothGrpdi *i *i !ETopGrpd (i_! \dashv i^* \dashv i_*) : Smooth\infty Grpd \stackrel{\overset{i_!}{\to}}{\stackrel{\overset{i^*}{\leftarrow}}{\underset{i_*}{\to}}} ETop\infty Grpd

such that the (∞,1)-Yoneda embedding is factored through the induced inclusion SmoothMfd i\stackrel{i}{\hookrightarrow} Mfd as

SmoothMfd SmoothGrpd i i ! Mfd ETopGrpd \array{ SmoothMfd &\hookrightarrow& Smooth\infty Grpd \\ \downarrow^{\mathrlap{i}} && \downarrow^{\mathrlap{i_!}} \\ Mfd &\hookrightarrow& ETop\infty Grpd } \,

Using the observation that ii preserves coverings and pullbacks along morphism in covering families, the proof follows precisely the steps of the proof of this proposition.

(Both of these are special cases of a general statement about morphisms of (∞,1)-sites, which should eventually be stated in full generality somewhere).


The essential global section (∞,1)-geometric morphism of SmoothGrpdSmooth \infty Grpd factors through that of ETop∞Grpd

(Π SmoothDisc SmoothΓ Smooth):SmoothGrpdi *i *i !ETopGrpdΓ ETopDisc ETopΠ ETopGrpd (\Pi_{Smooth} \dashv Disc_{Smooth} \dashv \Gamma_{Smooth}) : Smooth \infty Grpd \stackrel{\overset{i_!}{\to}}{\stackrel{\overset{i^*}{\leftarrow}}{\underset{i_*}{\to}}} ETop\infty Grpd \stackrel{\overset{\Pi_{ETop}}{\to}}{\stackrel{\overset{Disc_{ETop}}{\leftarrow}}{\underset{\Gamma_{ETop}}{\to}}} \infty Grpd

This follows from the essential uniqueness of the global section (∞,1)-geometric morphism and of adjoint (∞,1)-functors.


The functor i !i_! here is the forgetful functor that forgets smooth structure and only remembers Euclidean topology-structure.

Infinitesimal cohesion

Observe that CartSp smooth{}_{smooth} is (the syntactic category of) a Lawvere theory: the algebraic theory of smooth algebras (C C^\infty-rings). Write SmoothAlg:=Alg(C)SmoothAlg := Alg(C) for the category of its algebras. Let InfPointSmoothAlg opInfPoint \hookrightarrow SmoothAlg^{op} be the full subcategory on the infinitesimally thickened points.


Let CartSp synthdiffSmoothAlg op{}_{synthdiff} \hookrightarrow SmoothAlg^{op} be the full subcategory on the objects of the form U×DU \times D with DCartSp smoothSmoothAlg opD \in CartSp_{smooth} \hookrightarrow SmoothAlg^{op} and DInfPointSmoothAlg opD \in InfPoint \hookrightarrow SmoothAlg^{op}. Write

i:CartSp smoothCartSp synthdiff i : CartSp_{smooth} \hookrightarrow CartSp_{synthdiff}

for the canonical inclusion.


The inclusion exhibits an infinitesimal cohesive neighbourhood of SmoothGrpdSmooth \infty Grpd

(i !i *i *i !):SmoothGrpdSynthDiffGrpd, (i_! \dashv i^* \dashv i_* \dashv i^!) : Smooth \infty Grpd \hookrightarrow SynthDiff\infty Grpd \,,

where SynthDiff∞Grpd is the cohesive (∞,1)-topos of synthetic differential ∞-groupoids: the (∞,1)-category of (∞,1)-sheaves over CartSp synthdiffCartSp_{synthdiff}.


This follows as a special case of this proposition after observing that CartSp synthdiffCartSp_{synthdiff} is an infinitesimal neighbourhood site of CartSp smoothCartSp_{smooth} in the sense defined there.

In SynthDiff∞Grpd we have ∞-Lie algebras and ∞-Lie algebroids as actual infinitesimal objects. See there for more details.


The (1,1)-topos on the 0-truncated smooth \infty-groupoids is

Sh(CartSp)SmoothGrpd 0SmoothGrpd, Sh(CartSp) \simeq Smooth \infty Grpd_{\leq 0} \hookrightarrow Smooth\infty Grpd \,,

the sheaf topos on SmthMfd/CartSp discussed at smooth space.

The concrete objects in there

SmoothGrpd 0 concSmoothGrpd Smooth\infty Grpd_{\leq 0}^{conc} \hookrightarrow Smooth \infty Grpd

are precisely the diffeological spaces.

Structures in the cohesive (,1)(\infty,1)-topos SmoothGrpdSmooth \infty Grpd

We discuss the general abstract structures in a cohesive (∞,1)-topos realized in SmoothGrpdSmooth \infty Grpd.

This section is at

Smooth \infty-groupoids and related cohesive structures play a central role in the discussion at


For standard references on differential geometry and Lie groupoids see there.

The (,1)(\infty,1)-topos SmoothGrpdSmooth \infty Grpd is discussed in section 3.3 of

A discussion of smooth \infty-groupoids as (,1)(\infty,1)-sheaves on CartSpCartSp and the presentaton of the \infty-Chern-Weil homomorphism on these is in

For references on Chern-Weil theory in Smooth∞Grpd and connection on a smooth principal ∞-bundle, see there.

The results on differentiable Lie group cohomology used above are in

  • P. Blanc, Cohomologie différentiable et changement de groupes Astérisque, vol. 124-125 (1985), pp. 113-130.


which parallels

  • Graeme Segal, Cohomology of topological groups , Symposia Mathematica, Vol IV (1970) (1986?) p. 377

A review is in section 4 of

Classification of topological principal 2-bundles is discussed in

and the generalization to classification of smooth principal 2-bundles is in

Further discussion of the shape modality on smooth \infty-groupoids is in

Revised on August 30, 2016 03:40:10 by Urs Schreiber (