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
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
\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 }
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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)
higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
function algebras on ∞-stacks?
derived smooth geometry
$Diff$ (also called $Man$ or $Mfd$) is the category whose
morphisms are differentiable functions (smooth functions) between these.
The category $Diff$ becomes a large site by equipping it with the coverage consisting of open covers.
This is an essentially small site: a dense sub-site for $Diff$ is given by CartSp${}_{smooth}$.
The first statement follows trivially as for Top: the preimage of an open subset under a continuous function is again open (by definition of continuouss function).
For the second statement one needs that every paracompact manifold admits a differentially good open cover : an open cover by open balls that are diffeomorphic to a Cartesian spaces. The proof for this is spelled out at good open cover.
The sheaf topos over $Diff$ is a cohesive topos.
The hypercompletion of the (∞,1)-sheaf (∞,1)-topos over $Diff$ is a cohesive (∞,1)-topos.
For the first statement, use that by the comparison lemma discussed at dense sub-site we have an equivalence of categories
By the discussion at CartSp we have that $CartSp_{smooth}$ is a cohesive site. By the discussion there the claim follows.
For the second statement observe that the Joyal-Jardine model structure on simplicial sheaves $Sh(Diff)^{\Delta^{op}}_{loc}$ is a presentation for the hypercompletion of the (∞,1)-category of (∞,1)-sheaves $\hat Sh_{(\infty,1)}(Diff)$ (see presentations of (∞,1)-sheaf (∞,1)-toposes). By the above result it follows that there is an equivalence of (∞,1)-categories between the hypercompletions
Now CartSp${}_{smooth}$ is even an ∞-cohesive site. By the discussion there it follows that $Sh_{(\infty,1)}(CartSp_{smooth})$ (before hypercompletion) is a cohesive (∞,1)-topos. This means that it is in particular a local (∞,1)-topos. But this implies (as discussed there), that the (∞,1)-category of (∞,1)-sheaves already is the hypercomplete (∞,1)-topos. Therefore finally
The cohesive topos $Sh(Diff) \simeq Sh(CartSp_{smooth})$ is in particular the home of diffeological spaces. See there for more details.
is that of smooth ∞-groupoids. Discussed at Smooth∞Grpd.
The theory of differentiable stacks is that of geometric stacks in the (2,1)-sheaf (2,1)-topos
Last revised on May 8, 2017 at 03:57:37. See the history of this page for a list of all contributions to it.