nLab derived smooth manifold

Idea

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.

Motivation: correction of limits

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.

Pontrjagin-Thom construction

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 $f^* B$

$$\array{ f^* B &\to& B \\ \downarrow && \downarrow \\ S^n &\stackrel{f}{\to}& M O }$$

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.

String topology

(…)

Floer homology

(…)

Definition

The following definition characterizes the design criterion for derived smooth manifolds as being objects for which homotopy-intersections

$$A \cap_X B := A \times_X^h B$$

preserve the cup product in the cobordism ring

$$[A] \smile [B] \simeq [A \cap_X B] \,.$$

Definition

We say an (∞,1)-category $C$ supports derived cup products for cobordisms if

• it is equipped with a full and faithful functor

  $$i : Diff \hookrightarrow C$$

  embedding the category of smooth manifolds into it;

• for any two submanifolds $A \to X \leftarrow B$ (transversal or not) the (∞,1)-pullback

  $$A \cap_X B := i(A) \times_{i(X)} i(B)$$

  exists in $C$;

• if $A \to X \leftarrow B$ happen to be transverse maps then

  $$i(A \times_X B) \simeq i(A) \times_{i(X)} i(B) \,,$$

  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

  $$[A] \smile [B] = [A \cap_X B]$$

  in $\Omega(X)$

  (…)

A central statement about derived smooth manifolds will be

Theorem

The $(\infty,1)$-category of derived smooth manifolds has derived cup products for cobordisms.

This is (Spivak, theorem 1.8).

Related concepts

• dg-manifold

References

The definition of derived smooth manifolds is indicated at the very end of

• Jacob Lurie, Structured Spaces .

A detailed construction and discjussion in terms of the model category presentation by homotopy T-algebras is in

• David Spivak, Derived smooth manifolds, Duke Math. J. 153 1 (2010) 22-128 [arXiv:0810.5174, doi:10.1215/00127094-2010-021]

Something similar (derived orbifolds):

• Dominic Joyce, D-orbifolds, Kuranishi spaces, and polyfolds talk notes (Jan 2010) [pdf, pdf]

Seminar notes:

• Urs Schreiber, Seminar on derived differential geometry (2010)

Further developments:

• Dennis Borisov, Justin Noel, Simplicial approach to derived differential manifolds (arXiv:1112.0033)

• David Carchedi, Pelle Steffens, On the universal property of derived manifolds [arXiv:1905.06195]

  (universal property of the (infinity,1)-category of derived smooth manifolds)

• David Carchedi, Derived Manifolds as Differential Graded Manifolds [arXiv:2303.11140]

• Gregory Taroyan, Equivalent models of derived stacks [arXiv:2303.12699]

• Pelle Steffens, Derived $C^\infty$-Geometry I: Foundations [arXiv:2304.08671]