smooth 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

differential geometry

synthetic differential geometry






\infty-Lie theory

∞-Lie theory


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



The notion of smooth groupoid is the first generalization of the notion of smooth space to higher differential geometry. A smooth groupoid is a stack on the site CartSp smooth{}_{smooth}. This is equivalently an 1-truncated smooth infinity-groupoid.

This concepts contains notably the deloopings of Lie groups and more generally of diffeological groups, and the concept of Lie groupoid.


A smooth stack or smooth groupoid is a stack on the site SmoothMfd of smooth manifolds or equivalently (and often more conveniently) on its dense subsite CartSp of just Cartesian spaces n,n\mathbb{R}^n, n \in \mathbb{N} and smooth functions between them, equipped with the standard coverage of good open covers.

We write

\;\;\; SmoothGrpd Sh (2,1)(CartSp)L lheFunc(CartSp op,Grpd)\coloneqq Sh_{(2,1)}(CartSp) \simeq L_{lhe} Func(CartSp^{op}, Grpd)

for the (2,1)-category of stacks on this site, equivalently the result of taking groupoid-valued presheaves and then universally turning local (as seen by the coverage) equivalences of groupoids into global equivalence in an (infinity,1)-category.

By generalizing here groupoids to general Kan complexes and equivalences of groupoids to homotopy equivalences of Kan complexes, one obtains smooth ∞-stacks or smooth ∞-groupoids, which we write

\;\;\; Smooth∞Grpd Sh (,1)(CartSp)L lheFunc(CartSp op,KanCplx)\coloneqq Sh_{(\infty,1)}(CartSp) \simeq L_{lhe} Func(CartSp^{op}, KanCplx) .

We often write H\mathbf{H} \coloneqq Smooth∞Grpd for short.

By the (∞,1)-Yoneda lemma there is a sequence of faithful inclusions

\;\;\; SmoothMfd \hookrightarrow SmoothGrpd \hookrightarrow Smooth∞Grpd.

This induces a corresponding inclusion of simplicial objects and hence of groupoid objects

LieGrpdGrpd (SmoothMfd)Grpd (SmoothGrpd). LieGrpd \hookrightarrow Grpd_\infty(SmoothMfd) \hookrightarrow Grpd_\infty(Smooth\infty Grpd) \,.

For 𝒢 Grpd (H)\mathcal{G}_\bullet \in Grpd_\infty(\mathbf{H}) a groupoid object we write

𝒢 0𝒢lim n𝒢 n \mathcal{G}_0 \to \mathcal{G} \coloneqq \underset{\longrightarrow}{\lim}_{n} \mathcal{G}_n

for its (∞,1)-colimiting cocone, hence 𝒢H\mathcal{G} \in \mathbf{H} (without subscript decoration) denotes the realization of 𝒢 \mathcal{G}_\bullet as a single object in H\mathbf{H}.

By the Giraud-Rezk-Lurie axioms of the (∞,1)-topos H\mathbf{H} this morphism 𝒢 0𝒢\mathcal{G}_0 \to \mathcal{G} is a 1-epimorphism – an atlas – and its construction establishes is an equivalence of (∞,1)-categories Grpd (H)H 1epi Δ 1Grpd_\infty(\mathbf{H}) \simeq \mathbf{H}^{\Delta^1}_{1epi}, hence morphisms 𝒢 𝒦 \mathcal{G}_\bullet \to \mathcal{K}_\bullet in Grpd (H)Grpd_\infty(\mathbf{H}) are equivalently diagrams in H\mathbf{H} of the form

𝒢 0 𝒦 0 𝒢 𝒦. \array{ \mathcal{G}_0 &\to& \mathcal{K}_0 \\ \downarrow &\swArrow& \downarrow \\ \mathcal{G} &\to& \mathcal{K} } \,.

This is evidently more constrained that just morphisms

𝒢𝒦 \mathcal{G} \to \mathcal{K}

by themselves. The latter are the “generalized” or Morita morphisms between the groupoid objects 𝒢 \mathcal{G}_\bullet, 𝒦 \mathcal{K}_\bullet. These can be modeled as 𝒢 \mathcal{G}_\bullet-𝒦 \mathcal{K}_\bullet-bibundles.


Every Lie groupoid presents a smooth groupoids. Those of this form are also called differentiable stacks.

A 0-truncated smooth groupoid is equivalently a smooth space.

For GG a smooth group, its delooping BG\mathbf{B}G is a smooth groupoid, the moduli stack of smooth GG-principal bundles.


Discussion of diffeological groups (such as diffeomorphism groups and quantomorphism groups) goes back to

  • Jean-Marie Souriau, Groupes différentiels, in Differential Geometrical Methods in Mathematical Physics (Proc. Conf., Aix-en-Provence/Salamanca, 1979), Lecture Notes in Math. 836, Springer, Berlin, (1980), pp. 91–128. (MathScinet)

Discussion of smooth stacks as target spaces for sigma-model quantum field theories is in

Discussion of geometric Langlands duality in terms of 2d sigma-models on stacks (moduli stacks of Higgs bundles over a given algebraic curve) is in

Revised on March 6, 2015 07:16:04 by Urs Schreiber (