structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
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${}_{smooth}$. This is equivalently an 1-truncated smooth infinity-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 $\mathbb{R}^n, n \in \mathbb{N}$ and smooth functions between them, equipped with the standard coverage of good open covers.
We write
$\;\;\;$SmoothGrpd $\coloneqq Sh_{(2,1)}(CartSp) \simeq L_{lhe} Func(CartSp^{op}, Grpd)$
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 $\coloneqq Sh_{(\infty,1)}(CartSp) \simeq L_{lhe} Func(CartSp^{op}, KanCplx)$.
We often write $\mathbf{H} \coloneqq$ 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
For $\mathcal{G}_\bullet \in Grpd_\infty(\mathbf{H})$ a groupoid object we write
for its (∞,1)-colimiting cocone, hence $\mathcal{G} \in \mathbf{H}$ (without subscript decoration) denotes the realization of $\mathcal{G}_\bullet$ as a single object in $\mathbf{H}$.
By the Giraud-Rezk-Lurie axioms of the (∞,1)-topos $\mathbf{H}$ this morphism $\mathcal{G}_0 \to \mathcal{G}$ is a 1-epimorphism – an atlas – and its construction establishes is an equivalence of (∞,1)-categories $Grpd_\infty(\mathbf{H}) \simeq \mathbf{H}^{\Delta^1}_{1epi}$, hence morphisms $\mathcal{G}_\bullet \to \mathcal{K}_\bullet$ in $Grpd_\infty(\mathbf{H})$ are equivalently diagrams in $\mathbf{H}$ of the form
This is evidently more constrained that just morphisms
by themselves. The latter are the “generalized” or Morita morphisms between the groupoid objects $\mathcal{G}_\bullet$, $\mathcal{K}_\bullet$. These can be modeled as $\mathcal{G}_\bullet$-$\mathcal{K}_\bullet$-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 $G$ a smooth group, its delooping $\mathbf{B}G$ is a smooth groupoid, the moduli stack of smooth $G$-principal bundles.
Discussion of smooth stacks as target spaces for sigma-model quantum field theories is in
Tony Pantev, Eric Sharpe, String compactifications on Calabi-Yau stacks, Nucl.Phys. B733 (2006) 233-296, (arXiv:hep-th/0502044)
Tony Pantev, Eric Sharpe, Gauged linear sigma-models for gerbes (and other toric stacks), (arXiv:hep-th/0502053)
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