higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
A derived smooth manifold is the generalization of a smooth manifold in derived differential geometry: the derived geometry over the Lawvere theory for smooth algebras ($C^\infty$-rings):
it is a structured (∞,1)-topos whose structure sheaf of functions is a smooth (∞,1)-algebra.
According to the general logic of derived geometry, passing from smooth manifolds to derived smooth manifold serves to correct certain limits that do exist in Diff but do not have the correct cohomological behaviour. This concerns notably pullbacks along smooth functions that are not transversal maps.
For $X$ a compact smooth manifold, by the Pontrjagin-Thom construction there is a smooth function $f : S^n \to M O$ from an $n$-sphere to the Thom spectrum such that if chosen transversal to the zero-section $B \hookrightarrow M O$ the pullback $f^* B$
is a manifold cobordant to $X$, so that $[X] \simeq [f^* B]$ in the cobordism ring $\Omega$.
By using derived smooth manifolds instead of ordinary smooth manifolds here, the condition that $f$ be transversal to $B$ could be dropped.
(…)
(…)
The following definition characterizes the design criterion for derived smooth manifolds as being objects for which homotopy-intersections
preserve the cup product in the cobordism ring
We say an (∞,1)-category $C$ supports derived cup products for cobordisms if
it is equipped with a full and faithful functor
embedding the category of smooth manifolds into it;
for any two submanifolds $A \to X \leftarrow B$ (transversal or not) the (∞,1)-pullback
exists in $C$;
if $A \to X \leftarrow B$ happen to be transverse maps then
with the image under $i$ of the ordinary pullback in Diff on the left;
$i$ preserves the terminal object;
(…nice interaction with underlying topological spaces…)
for each $X \in Diff$ there is a derived cobordism ring $\Omega(X)$ such that …
for any submanifolds $A \to X \leftarrow B$ we have
in $\Omega(X)$
(…)
A central statement about derived smooth manifolds will be
The $(\infty,1)$-category of derived smooth manifolds has derived cup products for cobordisms.
This is (Spivak, theorm 1.8).
The definition of derived smooth manifolds is indicated at the very end of
A detailed construction and discjussion in terms of the model category presentation by homotopy T-algebras is in
Something roughly related is discussed in
There is also
Seminar notes on differential derived geometry in general and derived smooth manifolds in particular are in