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\begin{document}

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\section*{derived smooth manifold}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry}

[[!include higher geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{motivation_correction_of_limits}{Motivation: correction of limits}\dotfill \pageref*{motivation_correction_of_limits} \linebreak
\noindent\hyperlink{pontrjaginthom_construction}{Pontrjagin-Thom construction}\dotfill \pageref*{pontrjaginthom_construction} \linebreak
\noindent\hyperlink{string_topology}{String topology}\dotfill \pageref*{string_topology} \linebreak
\noindent\hyperlink{floer_homology}{Floer homology}\dotfill \pageref*{floer_homology} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{derived smooth manifold} is the generalization of a [[smooth manifold]] in [[derived differential geometry]]: the [[derived geometry]] over the [[Lawvere theory]] for [[smooth algebra]]s ($C^\infty$-rings):

it is a [[structured (∞,1)-topos]] whose [[structure sheaf]] of functions is a [[smooth (∞,1)-algebra]].

\hypertarget{motivation_correction_of_limits}{}\subsection*{{Motivation: correction of limits}}\label{motivation_correction_of_limits}

According to the general logic of [[derived geometry]], passing from [[smooth manifold]]s to \emph{derived smooth manifold} serves to \emph{correct} certain [[limit]]s that do exist in [[Diff]] but do not have the correct [[cohomology|cohomological behaviour]]. This concerns notably [[pullback]]s along [[smooth function]]s that are not [[transversal map]]s.

\hypertarget{pontrjaginthom_construction}{}\subsubsection*{{Pontrjagin-Thom construction}}\label{pontrjaginthom_construction}

For $X$ a [[compact space|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 map|transversal]] to the zero-[[section]] $B \hookrightarrow M O$ the [[pullback]] $f^* B$

\begin{displaymath}
\itexarray{
     f^* B &\to& B
     \\
     \downarrow && \downarrow
     \\
     S^n &\stackrel{f}{\to}& M O
  }
\end{displaymath}
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.

\hypertarget{string_topology}{}\subsubsection*{{String topology}}\label{string_topology}

(\ldots{})

\hypertarget{floer_homology}{}\subsubsection*{{Floer homology}}\label{floer_homology}

(\ldots{})

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

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

\begin{displaymath}
A \cap_X B := A \times_X^h B
\end{displaymath}
preserve the [[cup product]] in the [[cobordism ring]]

\begin{displaymath}
[A] \smile [B] \simeq [A \cap_X B]
  \,.
\end{displaymath}
\begin{udefn}
We say an [[(∞,1)-category]] $C$ supports \textbf{derived cup products for cobordisms} if

\begin{itemize}%
\item it is equipped with a [[full and faithful functor]]

\begin{displaymath}
i : Diff \hookrightarrow C
\end{displaymath}
embedding the category of [[smooth manifold]]s into it;


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

\begin{displaymath}
A \cap_X B := i(A) \times_{i(X)} i(B)
\end{displaymath}
exists in $C$;


\item if $A \to X \leftarrow B$ happen to be [[transverse map]]s then

\begin{displaymath}
i(A \times_X B) \simeq i(A) \times_{i(X)} i(B)
  \,,
\end{displaymath}
with the image under $i$ of the ordinary [[pullback]] in [[Diff]] on the left;


\item $i$ preserves the [[terminal object]];


\item (\ldots{}nice interaction with underlying topological spaces\ldots{})


\item for each $X \in Diff$ there is a \emph{derived cobordism ring} $\Omega(X)$ such that \ldots{}


\item for any submanifolds $A \to X \leftarrow B$ we have

\begin{displaymath}
[A] \smile [B] = [A \cap_X B]
\end{displaymath}
in $\Omega(X)$

(\ldots{})



\end{itemize}
\end{udefn}
A central statement about derived smooth manifolds will be

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

\end{utheorem}
This is (\hyperlink{Spivak}{Spivak, theorm 1.8}).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[dg-manifold]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

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

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Structured Spaces]]} .

\end{itemize}
A detailed construction and discjussion in terms of the [[model category]] [[presentable (∞,1)-category|presentation]] by [[homotopy T-algebra]]s is in

\begin{itemize}%
\item [[David Spivak]], \emph{Derived smooth manifolds} (\href{http://arxiv.org/abs/0810.5174}{arXiv:0810.5174})

\end{itemize}
Something roughly related is discussed in

\begin{itemize}%
\item [[Dominic Joyce]], \emph{D-orbifolds, Kuranishi spaces, and polyfolds} talk notes (Jan 2010) (\href{http://people.maths.ox.ac.uk/joyce/dmtalk.pdf}{pdf})

\end{itemize}
There is also

\begin{itemize}%
\item Dennis Borisov, Justin Noel, \emph{Simplicial approach to derived differential manifolds} (\href{http://arxiv.org/abs/1112.0033}{arXiv:1112.0033})

\end{itemize}
Seminar notes on differential [[derived geometry]] in general and derived smooth manifolds in particular are in

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:Seminar on derived differential geometry]]}

\end{itemize}
[[!redirects derived smooth manifolds]]

[[!redirects derived manifold]] [[!redirects derived manifolds]]



\end{document}