structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
∞-Lie theory (higher geometry)
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 $Smooth \infty Grpd$ 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 $X$ a smooth manifold, say an open cover $\{U_i \to X\}$ is a differentiably good open cover if each non-empty finite intersection of the $U_i$ 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 $\{U_i \to X\}$ any open cover of a paracompact manifold also $\coprod_i U_i$ is paracompact. Hence we may find a differentiably good open cover $\{K_j \to \coprod_i U_i\}$. This is then a refinement of the original open cover of $X$.
Let CartSp${}_{smooth}$ 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${}_{smooth}$:
$Smooth \infty Grpd$ is a cohesive (∞,1)-topos.
The site CartSp${}_{smooth}$ 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 $\{U_i \to U\}$ 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 $Smooth \infty Grpd$ is equivalent to the hypercompletion $\hat Sh_{(\infty,1)}(SmoothMfd)$ of the (∞,1)-category of (∞,1)-sheaves on the large site SmoothMfd
By the above we have that CartSp${}_{smooth}$ 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 $Smooth\infty Grpd$ is a full and faithful (∞,1)-functor
We discuss the relation of $Smooth\infty Grpd$ to other cohesive (∞,1)-toposes.
The cohesive (∞,1)-topos ETop∞Grpd of Euclidean-topological ∞-groupoids has as site of definition CartSp${}_{top}$. There is a canonical forgetful functor
The functor $i$ extends to an essential (∞,1)-geometric morphism
such that the (∞,1)-Yoneda embedding is factored through the induced inclusion SmoothMfd $\stackrel{i}{\hookrightarrow}$ Mfd as
Using the observation that $i$ 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 $Smooth \infty Grpd$ factors through that of ETop∞Grpd
This follows from the essential uniqueness of the global section (∞,1)-geometric morphism and of adjoint (∞,1)-functors.
The functor $i_!$ here is the forgetful functor that forgets smooth structure and only remembers Euclidean topology-structure.
Observe that CartSp${}_{smooth}$ is (the syntactic category of) a Lawvere theory: the algebraic theory of smooth algebras ($C^\infty$-rings). Write $SmoothAlg := Alg(C)$ for the category of its algebras. Let $InfPoint \hookrightarrow SmoothAlg^{op}$ be the full subcategory on the infinitesimally thickened points.
Let CartSp${}_{synthdiff} \hookrightarrow SmoothAlg^{op}$ be the full subcategory on the objects of the form $U \times D$ with $D \in CartSp_{smooth} \hookrightarrow SmoothAlg^{op}$ and $D \in InfPoint \hookrightarrow SmoothAlg^{op}$. Write
for the canonical inclusion.
The inclusion exhibits an infinitesimal cohesive neighbourhood of $Smooth \infty Grpd$
where SynthDiff∞Grpd is the cohesive (∞,1)-topos of synthetic differential ∞-groupoids: the (∞,1)-category of (∞,1)-sheaves over $CartSp_{synthdiff}$.
This follows as a special case of this proposition after observing that $CartSp_{synthdiff}$ is an infinitesimal neighbourhood site of $CartSp_{smooth}$ 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
the sheaf topos on SmthMfd/CartSp discussed at smooth space.
The concrete objects in there
are precisely the diffeological spaces.
We discuss the general abstract structures in a cohesive (∞,1)-topos realized in $Smooth \infty Grpd$.
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 $(\infty,1)$-topos $Smooth \infty Grpd$ is discussed in section 3.3 of
A discussion of smooth $\infty$-groupoids as $(\infty,1)$-sheaves on $CartSp$ 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
and
which parallels
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