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\section*{A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves}

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

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

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

[[!include higher geometry - contents]]

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

\begin{quote}%
\textbf{Abstract} We sketch some basic ideas ([[Jacob Lurie]]`s ideas, that is) about [[higher geometry]] motivated from the [[higher category theory|higher]] version of the [[moduli stack]] of [[elliptic curve]]s: the derived moduli stack of [[derived elliptic curve]]s. We survey aspects of the theory of [[generalized scheme]]s and then sketch how the derived moduli stack of derived elliptic curves is an example of a generalized scheme modeled on the formal dual of [[E-∞ ring]]s.


\end{quote}
This is a sub-entry of

\begin{itemize}%
\item [[A Survey of Elliptic Cohomology]]

\end{itemize}
see there for background and context.

For fully appreciating the details of the main theorem here the material discussed in the previous sessions (and a little bit more) is necessary, but our exposition of [[generalized scheme]]s is meant to be relatively self-contained (albeit necessarily superficial).

This are the entries on the previous sessions:

\begin{itemize}%
\item [[A Survey of Elliptic Cohomology - cohomology theories]]


\item [[A Survey of Elliptic Cohomology - formal groups and cohomology]]


\item [[A Survey of Elliptic Cohomology - E-infinity rings and derived schemes]]


\item [[A Survey of Elliptic Cohomology - elliptic curves]]


\item [[A Survey of Elliptic Cohomology - equivariant cohomology]]


\item [[A Survey of Elliptic Cohomology - derived group schemes and (pre-)orientations]]


\item [[A Survey of Elliptic Cohomology - A-equivariant cohomology]]



\end{itemize}

\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{the_derived_moduli_stack_of_derived_elliptic_curves}{}\section*{{the derived moduli stack of derived elliptic curves}}\label{the_derived_moduli_stack_of_derived_elliptic_curves}

\noindent\hyperlink{motivation_and_statement}{Motivation and Statement}\dotfill \pageref*{motivation_and_statement} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{NotionsOfSpace}{Notions of Space}\dotfill \pageref*{NotionsOfSpace} \linebreak
\noindent\hyperlink{spaces_probeable_by_model_spaces_stacks}{spaces probeable by model spaces: $\infty$-stacks}\dotfill \pageref*{spaces_probeable_by_model_spaces_stacks} \linebreak
\noindent\hyperlink{concrete_spaces_coprobeable_by_model_spaces_structured_toposes}{concrete spaces co-probeable by model spaces: structured $(\infty,1)$-toposes}\dotfill \pageref*{concrete_spaces_coprobeable_by_model_spaces_structured_toposes} \linebreak
\noindent\hyperlink{spaces_locally_like_model_spaces_generalized_schemes}{Spaces locally like model spaces: generalized schemes}\dotfill \pageref*{spaces_locally_like_model_spaces_generalized_schemes} \linebreak
\noindent\hyperlink{the_derived_moduli_space_of_elliptic_curves}{The derived moduli space of elliptic curves}\dotfill \pageref*{the_derived_moduli_space_of_elliptic_curves} \linebreak
\noindent\hyperlink{derived_elliptic_curves}{Derived elliptic curves}\dotfill \pageref*{derived_elliptic_curves} \linebreak
\noindent\hyperlink{the_derived_moduli_stack}{The derived moduli stack}\dotfill \pageref*{the_derived_moduli_stack} \linebreak
\noindent\hyperlink{the_classification_of_derived_elliptic_curves}{the classification of derived elliptic curves}\dotfill \pageref*{the_classification_of_derived_elliptic_curves} \linebreak


\hypertarget{motivation_and_statement}{}\subsection*{{Motivation and Statement}}\label{motivation_and_statement}

In the context of [[elliptic cohomology]] one assigns to every [[elliptic curve]] $\phi$ over a [[ring]] $R$ a [[cohomology theory]] [[Brown representability theorem|represented]] by an [[E-∞ ring]] [[spectrum]] $E_\phi$.

Since, by definition, we may identify the [[elliptic curve]] $\phi$ over $R$ with a patch $\phi : Spec R \to \mathcal{M}_{1,1}$ of the [[moduli stack]] $\mathcal{M}_{1,1}$ of elliptic curves, this assignment

\begin{displaymath}
\mathcal{O}^{Der} : \phi \mapsto E_\phi
\end{displaymath}
looks like an [[E-∞ ring]] valued [[structure sheaf]] on $\mathcal{M}_{1,1}$.

There is a very general theory of [[higher geometry]] for [[nLab:generalized scheme|generalized spaces]] with generalized [[nLab:structure sheaf|structure sheaves]]. Using this one may regard the pair

\begin{displaymath}
(\mathcal{M}_{1,1}, \mathcal{O}^{Der})
\end{displaymath}
as a [[structured (infinity,1)-topos|structured space]] that is a ``derived'' [[Deligne-Mumford stack]].

The central theorem about [[elliptic cohomology]] of [[Jacob Lurie]], refining the Goerss-Hopkins-Miller theorem says that

\textbf{the central theorem, first version}

The [[moduli stack]] $\mathcal{M}_{1,1}$ of [[elliptic curve]]s equipped with the [[E-∞ ring]]-valued [[structure sheaf]] $\mathcal{O}^{Der}$ may be regarded as the \emph{derived moduli stack} of [[derived elliptic curve]]s in that for any [[E-∞ ring]] $R$ the space of derived stack morphisms

\begin{displaymath}
(Spec R, R) \to (\mathcal{M}_{1,1}, \mathcal{O}^{Der})
\end{displaymath}
is equivalent to the space of [[derived elliptic curve]]s over $R$.

After we have looked at some concepts in [[higher geometry]] a bit more closely below, we will restate this in slightly nicer fashion.

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

A sketch of what this theorem means and how it is proven is part of the content of

\begin{itemize}%
\item [[Jacob Lurie]], [[A Survey of Elliptic Cohomology]].

\end{itemize}
and goes back to Jacob Lurie's PhD thesis (listed [[Jacob Lurie|here]]).

The general theory for the context of [[higher geometry]] invoked here has later been spelled out in

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

\end{itemize}
The special case of the general theory that is needed here, where the coefficient objects of structure sheaves are [[E-∞ ring]]s, is described in

\begin{itemize}%
\item [[Jacob Lurie]], [[Spectral Schemes]],

\end{itemize}
while the general theory of [[E-∞ ring]]s themselves, in the [[(∞,1)-category]] theory context needed here, is developed in

\begin{itemize}%
\item [[Jacob Lurie]], [[higher algebra|Commutative geometry]].

\end{itemize}
\hypertarget{NotionsOfSpace}{}\subsection*{{Notions of Space}}\label{NotionsOfSpace}

The statement that we are after really lives in the context of [[higher geometry]] (often called ``derived geometry''). Here is an outline of the central aspects.

The \textbf{central ingredient} which we choose at the beginning to get a theory of [[higher geometry]] going is an [[(∞,1)-category]] $\mathcal{G}$ whose objects we think of as \textbf{model spaces} : the simplest objects exhibiting the geometric structures that we mean to consider.

\textbf{Examples for categories of model spaces}

\begin{itemize}%
\item with smooth structure

\begin{itemize}%
\item $\mathcal{G} =$ [[Diff]], the category of smooth [[manifold]]s;


\item $\mathcal{G} = \mathbb{L}$, the category of [[smooth locus|smooth loci]];



\end{itemize}

\item without smooth structure

\begin{itemize}%
\item $\mathcal{G} = (C Ring^{fin})^{op}$, the formal dual of [[CRing]]: the category of (finitely generated) algebraic [[affine scheme]]s;


\item $\mathcal{G} = (sC Ring^{fin})^{op}$, the formal dual of [[simplicial object]]s in [[CRing]];


\item $\mathcal{G} = (E_\infty Ring^{fin})^{op}$, the formal dual of [[E-∞ ring]]s: the category of (finitely generated) algebraic derived [[affine scheme]]s.



\end{itemize}


\end{itemize}
These [[(∞,1)-category|(∞,1)-categories]] $\mathcal{G}$ are naturally equipped with the structure of a [[site]] (and a bit more, which we won't make explicit for the present purpose). Following [[Jacob Lurie]] we call such a $\mathcal{G}$ a \textbf{[[geometry (for structured (infinity,1)-toposes)|geometry]]} .

We want to be talking about generalized spaces \emph{modeled on} the objects of $\mathcal{G}$. There is a hierarchy of notions of what that may mean:

\textbf{Hierarchy of generalized spaces modeled on $\mathcal{G}$}

\begin{displaymath}
\itexarray{
    \mathcal{G} 
    &\stackrel{Spec^{\mathcal{G}}}{\hookrightarrow}&   
    Sch(\mathcal{G}) 
    &\hookrightarrow&
    \mathcal{L}Top(\mathcal{G})^{op} 
    &\hookrightarrow&      
    Sh_{(\infty,1)}(Pro(\mathcal{G}))
    \\
    \\
    model spaces
    &&
    spaces locally like model spaces
    &&
    concrete spaces coprobeable by model spaces
    &&
    spaces probeable by model spaces
    \\
    \\
    affine\;\mathcal{G}-schemes
    &&
    \mathcal{G}-schemes
    &&
    \mathcal{G}-structured\;(\infty,1)-toposes
    &&
    \infty-stacks\;on\;\mathcal{G}
    \\
    \stackrel{tame\;but\;restrictive}{\leftarrow} & 
    &&&&
    &
    \stackrel{versatile\;but\;possibly\;wild}{\to}
  }
\end{displaymath}
We explain what this means from right to left.

\hypertarget{spaces_probeable_by_model_spaces_stacks}{}\subsubsection*{{spaces probeable by model spaces: $\infty$-stacks}}\label{spaces_probeable_by_model_spaces_stacks}

An object $X$ \emph{probeable} by objects of $\mathcal{G}$ should come with an assignment

\begin{displaymath}
X : (U \in \mathcal{G}) \mapsto (X(U) \in \infty Grpd)
\end{displaymath}
of an [[∞-groupoid]] of possible ways to probe $X$ by $U$, for each possible $U$, natural in $U$. More precisely, this should define an object in the [[(∞,1)-category of (∞,1)-presheaves]] on $\mathcal{G}$

\begin{displaymath}
X \in PSh(\mathcal{G}) 
   = 
   Funct(\mathcal{G}^{op}, \infty Grpd)
\end{displaymath}
But for $X$ to be \emph{consistently} probeable it must be true that probes by $U$ can be reconstructed from overlapping probes of pieces of $U$, as seen by the [[coverage|topology]] of $\mathcal{G}$. More precisely, this should mean that the [[(∞,1)-presheaf]] $X$ is actually an object in an [[(∞,1)-category of (∞,1)-sheaves]] on $\mathcal{G}$

\begin{displaymath}
X \in 
  Sh(\mathcal{G}) \stackrel{}{\hookrightarrow}
  PSh(\mathcal{G})
  \,.
\end{displaymath}
Such objects are called [[∞-stack]]s on $\mathcal{G}$. The [[(∞,1)-category]] $Sh(\mathcal{G})$ is called an [[∞-stack]] [[(∞,1)-topos]].

A supposedly pedagogical discussion of the general philosophy of [[∞-stacks]] as probebable spaces is at [[motivation for sheaves, cohomology and higher stacks]].

The [[∞-stack]]s on $\mathcal{G}$ that are used in the following are those that satisfy [[descent]] on [[?ech cover]]s. But we will see [[(∞,1)-topos]]es of [[∞-stack]]s that may satisfy different descent conditions, in particular with respect to [[hypercover]]s. Every [[∞-stack]] [[(∞,1)-topos]] has a [[hypercompletion]] to one of this form.

For concretely working with [[hypercomplete (∞,1)-topos]]es it is often useful to use [[models for ∞-stack (∞,1)-toposes]] in terms of the [[model structure on simplicial presheaves]].

\begin{displaymath}
\itexarray{
    Sh^{hc}_{(\infty,1)}(C) 
   &\stackrel{\stackrel{\;\;\;\;\;lex\;\;\;\;\;\;}{\leftarrow}}  
  {\hookrightarrow}&
     PSh_{(\infty,1)}(C)
    && \text{abstract nonsense def of (∞,1)-topos}
    \\
    \uparrow^{\simeq} &&
    \uparrow^{\simeq}
    && \text{Lurie's theorem}
    \\
    ([C^{op}, SSet]_{loc})^\circ
    &\stackrel{\stackrel{Bousfield\;loc.}{\leftarrow}}{\to}&
    ([C^{op}, SSet]_{glob})^\circ
    &&
    \text{model category of simplicial presheaves}
  }
\end{displaymath}
\begin{uremark}
This discussion here is glossing over all set-theoretic size issues. See [[Structured Spaces|StSp, warning 2.4.5]].

\end{uremark}
\hypertarget{concrete_spaces_coprobeable_by_model_spaces_structured_toposes}{}\subsubsection*{{concrete spaces co-probeable by model spaces: structured $(\infty,1)$-toposes}}\label{concrete_spaces_coprobeable_by_model_spaces_structured_toposes}

Spaces probeable by $\mathcal{G}$ in the above sense can be very general. They need not even have a \emph{concrete underlying space} , even for general definitions of what \emph{that} might mean.

\textbf{(Counter-)Example} For $\mathcal{G} =$ [[Diff]], for every $n \in \mathbb{N}$ we have the [[∞-stack]] $\Omega_{cl}^n(-)$ (which happens to be an ordinary [[sheaf]]) that assigns to each manifold $U$ the set of closed [[differential form|n-form]]s on $U$. This is important as a generalized space: it is something like the rational version of the [[Eilenberg-MacLane space]] $K(\mathbb{Z}, n)$. But at the same time this is a ``wild'' space that has exotic properties: for instance for $n=3$ this space has just a single point, just a single curve in it, just a single surface in it, but has many nontrivial probes by 3-dimensional manifolds.

In the classical theory for instance of [[ringed space]]s or [[diffeological space]]s a \emph{concrete underlying space} is taken to be a [[topological space]]. But this in turn is a bit \emph{too} restrictive for general purposes: a topological space is the same as a [[localic topos]]: a [[category of sheaves]] on a [[category of open subsets]] of a [[topological space]]. The obvious generalization of this to [[higher geometry]] is: an [[n-localic (∞,1)-topos]] $\mathcal{X}$.

This makes us want to say and make precise the statement that

An \textbf{concrete [[∞-stack]]} $X$ is one which has an \emph{underlying} [[(∞,1)-topos]] $\mathcal{X}$:

the collection of $U$-probes of $X$ is a [[subobject]] of the collection of [[(∞,1)-topos]]-morphisms from $U$ to $\mathcal{X}$:

\begin{displaymath}
X(U) \subset \mathcal{L}Top(\mathcal{G})^{op}(Sh_{\infty}(U),\mathcal{X})
\end{displaymath}
We think of $\mathcal{X}$ as the [[(∞,1)-topos]] of [[∞-stack]]s on a category of open subsets of a would-be space $X$, only that this would be space $X$ might not have an independent existence as a space apart from $\mathcal{X}$. The available entity closest to it is the [[terminal object]] ${*}_{\mathcal{X}} \in \mathcal{X}$.

To say that $\mathcal{X}$ is \emph{modeled on $\mathcal{G}$} means that among all the [[∞-stack]]s on the would-be space a [[structure sheaf]] of functions with values in objects of $\mathcal{G}$ is singled out: for each object $V \in \mathcal{G}$ there is a [[structure sheaf]] $\mathcal{O}(-,V) \in \mathcal{X}$, naturally in $V$.

This yields an [[(∞,1)-functor]]

\begin{displaymath}
\mathcal{O} : \mathcal{G} \to \mathcal{X}
  \,.
\end{displaymath}
We think of $X$ as being a concrete space \emph{co-probebale} by $\mathcal{G}$ (we can map from the concrete $X$ into objects of $\mathcal{G}$).

Such an $\mathcal{O}$ is a \emph{consistent} collection of coprobes if coprobes with values in $V$ may be reconstructed from co-probes with values in pieces of $V$.

More precisely:

\begin{udef}
\textbf{($\mathcal{G}$-structure, [[Structured Spaces|StSp, def. 1.2.8]])}

An [[(∞,1)-functor]] $\mathcal{O} : \mathcal{G} \to \mathcal{X}$ is a \textbf{$\mathcal{G}$-valued structure sheaf} on the [[(∞,1)-topos]] if

\begin{itemize}%
\item it preserves finite [[limit]]s


\item and sends covering coproducts $(\coprod_i U_i) \to U$ to [[effective epimorphism]]s.



\end{itemize}
A pair $(\mathcal{X}, \mathcal{O})$ of an [[(∞,1)-topos]] $\mathcal{X}$ equipped with $\mathcal{G}$-valued [[structure sheaf]] $\mathcal{O} : \mathcal{G} \to \mathcal{X}$ we call a [[structured (∞,1)-topos]].

\end{udef}
In summary:

A \textbf{concrete [[∞-stack]] $X$ modeled on $\mathcal{G}$} is

\begin{itemize}%
\item an [[(∞,1)-topos]] $\mathcal{X}$ (``of $\infty$-stacks on $X$'')


\item equipped with a $\mathcal{G}$-valued structure sheaf $\mathcal{O}$ in the form of a finite limits and cover preserving functor $\mathcal{O} : \mathcal{G} \to \mathcal{X}$.



\end{itemize}
The fundamental \textbf{example} for [[structured (∞,1)-topos]]es are provided by the objects of $\mathcal{G}$ themselves, which are canonically equipped with a $\mathcal{G}$-structure as follows.

\begin{utheorem}
\textbf{([[Structured Spaces|StSp, thm. 2.1.1]])}

Let $f : \mathcal{G} \to \mathcal{G}'$ be a morphism of [[geometry (for structured (infinity,1)-toposes)|geometries]], then the obvious [[(∞,1)-functor]] $f^* : \mathcal{L}Top(\mathcal{G}) \to \mathcal{L}Top(\mathcal{G}')$ admits a [[left adjoint]]

\begin{displaymath}
f^*
  :
  \mathcal{L}Top(\mathcal{G}') 
   \stackrel{\leftarrow}{\to} 
  \mathcal{L}Top(\mathcal{G}) 
  :
  Spec_{\mathcal{G}}^{\mathcal{G}'}
\end{displaymath}
called the \textbf{relative spectrum functor}.

\end{utheorem}
For $\mathcal{G}$ any [[geometry (for structured (infinity,1)-toposes)|geometry]], write $\mathcal{G}_{disc}$ for the [[geometry (for structured (infinity,1)-toposes)|geometry]] obtained from this by forgetting its [[coverage|Grothendieck topology]] and instead using the discrete topology where only equivalences cover.

Notice that we may identify $\mathcal{G}_{disc}$-structures on the archetypical [[(∞,1)-topos]] [[∞Grpd]], being finite [[limit]]-preserving functors $\mathcal{G}_{disc}^{op} \to \infty Grpd$ with [[ind-object]]s in $\mathcal{G}^{op}$, hence with the opposite of [[pro-object]]s in $\mathcal{G}$. This gives a canonical inclusion

\begin{displaymath}
Pro(\mathcal{G}) 
    \hookrightarrow 
  \mathcal{L}Top(\mathcal{G})^{op}
  \,.
\end{displaymath}
\begin{udef}
\textbf{([[Structured Spaces|StSp, def. 2.1.2]])}

The composite [[(∞,1)-functor]]

\begin{displaymath}
Spec^{\mathcal{G}}
  : 
  Pro(\mathcal{G})^{op}
  \hookrightarrow 
  \mathcal{L}Top(\mathcal{G}_{disc})
  \stackrel{Spec_{\mathcal{G}}^{\mathcal{G}_{disc}}}{\to}
  \mathcal{L}Top(\mathcal{G})
\end{displaymath}
we call the \textbf{absolute spectrum functor}

\end{udef}
This [[category theory|abstract nonsense]] is reassuring, but we want a more concrete definition of what such $Spec^{\mathcal{G}} U$ is like:

\begin{udef}
\textbf{([[Structured Spaces|StSp, def. 2.2.9]])}

For every $U \in \mathcal{G}$ there is naturally induced a [[coverage|topology]] on the [[over category]] $Pro(\mathcal{G})/U$. Define the [[(∞,1)-topos]]

\begin{displaymath}
Spec U := Sh_{(\infty,1)}(Pro(\mathcal{G})/U)
  \,,
\end{displaymath}
naturally to be thought of as the [[(∞,1)-topos]] of [[∞-stack]]s \emph{on $U$} .

This is canonically equipped with a [[(∞,1)-functor]]

\begin{displaymath}
\mathcal{O}_{Spec X}
  : 
  \mathcal{G} \to Spec X
  \,.
\end{displaymath}
\end{udef}
And this is indeed the concrete underlying space produced by the absolute spectrum functor:

\begin{utheorem}
\textbf{[[Structured Spaces|StSp, prop. 2.2.11, thm. 2.2.12]])}

For every $U \in \mathcal{G}$ the pair $(Spec U, \mathcal{O}_{Spec U})$ is indeed a [[structured (∞,1)-topos]] and is indeed equivalent to the $Spec^{\mathcal{G}} U$ defined more abstractly above.

\end{utheorem}
\textbf{Example} For $\mathcal{G} = (C Ring^{fin})^{op}$ with the standard [[coverage|topology]] we have that 0-localic $\mathcal{G}$-structured spaces are \emph{[[locally ringed space]]s} , while $\mathcal{G}_{disc}$-structured 0-localic spaces are just arbitrary [[ringed space]]s. Applying the above machinery to this situaton gives a spectrum functor that takes a [[ring]] $R$ first to the [[ringed space]] $({*,R})$ and this then to the [[locally ringed space]] $(Spec R, R)$.

\hypertarget{spaces_locally_like_model_spaces_generalized_schemes}{}\subsubsection*{{Spaces locally like model spaces: generalized schemes}}\label{spaces_locally_like_model_spaces_generalized_schemes}

We have seen that $\mathcal{G}$-[[structured (∞,1)-topos]]es are those general spaces modeled on $\mathcal{G}$ that are well-behaved in that at least they do have an ``underlying topological structure'' in the form of an underlying [[(∞,1)-topos]]. But such concrete spaces may still be very different from the model objects in $\mathcal{G}$.

In parts this is desireable: many objects that one would naturally build out of the objects in $\mathcal{G}$, such as mapping spaces $[\Sigma,X]$, are much more general than objects in $\mathcal{G}$ but do live happily in $\mathcal{L}Top(\mathcal{G})^{op}$.

But in many situations one would like to regard $\mathcal{G}$-[[structured (∞,1)-topos]]es that are not globally but \emph{locally} equivalent to objects in $\mathcal{G}$. This is supposed to be captured by the following definition.

\begin{udef}
\textbf{[[Structured Spaces|StSp, def. 2.3.9]]}

A [[structured (∞,1)-topos]] $(\mathcal{X}, \mathcal{O})$ is a \textbf{$\mathcal{G}$-generalized scheme} if

\begin{itemize}%
\item there exists a collection $\{V_i \in \mathcal{X}\}$


\item such that

\begin{itemize}%
\item this covers $\mathcal{X}$ in that the canonical morphism

\begin{displaymath}
(\coprod_i V_i) \to {*}_{\mathcal{X}}
\end{displaymath}
to the [[terminal object]] in $\mathcal{X}$ is an [[effective epimorphism]]


\item the [[structured (∞,1)-topos]]es \newline $(\mathcal{X}/V_i, \mathcal{O}|_{V_i})$ induced by the $V_i$ are model spaces in that there exists $\{U_i \in \mathcal{G}\}$ and equivalences

\begin{displaymath}
(\mathcal{X}/V_i, \mathcal{O}|_{V_i})
  \simeq
  Spec^{\mathcal{G}} U_i
\end{displaymath}


\end{itemize}


\end{itemize}
\end{udef}
\textbf{Examples}

\begin{quote}%
\textbf{warning} the following statements really pertain to pregeometries, not geometries. for the moment this here is glossing over the difference between the two. See [[geometry (for structured (∞,1)-toposes)]] for the details.


\end{quote}
\begin{itemize}%
\item ordinary smooth [[manifold]]s are [[n-localic (infinity,1)-topos|0-localic]] [[Diff]]-[[generalized scheme]]s (Structured Spaces|StSp, ex. 4.5.2])


\item ordinary [[schemes]] are those $(CRing^{fin})^{op}$-[[generalized scheme]]s whose underlying [[(∞,1)-topos]] is [[n-localic (infinity,1)-topos|0-localic]] and whose [[structure sheaf]] is [[n-truncated object of an (infinity,1)-category|0-truncated]] (Structured Spaces|StSp, prop. 4.2.9])


\item [[Deligne-Mumford stack]]s are 1-localic $(CRing^{fin})_{et}^{op}$-[[generalized scheme]]s (Structured Spaces|StSp, prop. 4.2.9])


\item This last statement is then the basis for calling a general $(CRing^{fin})_{et}^{op}$-[[generalized scheme]] a \textbf{derived Deligne-Mumford stack}


\item Finally, to make contact with the application to the derived moduli stack of derived elliptic curves, it seems that in [[Spectral Schemes]] a derived Deligne-Mumford stack (with derived in the sense of having replaced ordinary commutative rings by [[E-∞ ring]]s) is gonna be a 1-localic $(E_\infty Ring^{fin})^{op}$-[[generalized scheme]].



\end{itemize}
\hypertarget{the_derived_moduli_space_of_elliptic_curves}{}\subsection*{{The derived moduli space of elliptic curves}}\label{the_derived_moduli_space_of_elliptic_curves}

With the above machinery for [[higher geometry]] in hand, we now set out to describe the particular application that we are interested in: the study of the derived [[moduli stack]] of [[derived elliptic curve]]s.

\hypertarget{derived_elliptic_curves}{}\subsubsection*{{Derived elliptic curves}}\label{derived_elliptic_curves}

\begin{itemize}%
\item [[derived elliptic curve]]

\end{itemize}
\begin{displaymath}
A \mapsto E(A)
\end{displaymath}
\hypertarget{the_derived_moduli_stack}{}\subsubsection*{{The derived moduli stack}}\label{the_derived_moduli_stack}

Lurie's discussion of the derived moduli stack $(\mathcal{M}_{1,1}, \mathcal{O}^{Der})$ is more than a re-interpretation of the Goerss-Miller-Hopkins theorem. It is in particular a re-derivation of this result, from the following perspective

\textbf{the central statement, conceptually}

\textbf{Input} We have the $(E_\infty Ring)^{op}$-probeable space

\begin{displaymath}
(E : R \mapsto \{derived\;elliptic\;curves\;over\;R\})
  \in Sh(E_\infty Ring^{op})
  \,.
\end{displaymath}
\textbf{Question}: Does this happen to even be a $E_\infty Ring^{op}$-[[generalized scheme]]?

\textbf{Answer} Yes. It is actually a [[derived Deligne-Mumford stack]].

Let $\mathcal{M}_{1,1}$ be the ordinary [[moduli stack]] of [[elliptic curve]]s.

Using constructions in [[elliptic cohomology]] we may associate to each [[elliptic curve]] over $R$, i.e. each morphism $\phi : Spec R \to \mathcal{X}$, an [[E-infinity ring]] $E_\phi$ -- the multiplicative spectrum that represents the elliptic cohomology theory given by $T$.

This gives an $E_\infty$-ring valued structure sheaf

\begin{displaymath}
\mathcal{O}^{Der} : (\phi : Spec R \to \mathcal{X}) \mapsto
  E_\phi
  \,.
\end{displaymath}
\textbf{Question} What, if anything, is this derived stack a derived [[moduli stack]] of?

\hypertarget{the_classification_of_derived_elliptic_curves}{}\subsubsection*{{the classification of derived elliptic curves}}\label{the_classification_of_derived_elliptic_curves}

The big theorem is that the derived space $(\mathcal{X}, \mathcal{O}^{Der})$ classifies derived elliptic curves over $E_\infty$-rings

This is the theorem that we said above we wanted to consider, stated now a little bit more precisely.

\begin{utheorem}
\textbf{(J. Lurie)}

For

\begin{itemize}%
\item $A$ any [[E-∞ ring]]


\item and $E(A)$ is the space of [[derived elliptic curve]]s over $A$ (the realization of the topological category of elliptic curves over $A$).



\end{itemize}
we have an equivalence

\begin{displaymath}
Hom(Spec A, (\mathcal{X}, \mathcal{O}^{Der}))  
  \simeq
  E(A)
\end{displaymath}
naturally in $A$.

\end{utheorem}
\begin{proof}
Jacob Lurie writes that the proof proceeds alonmg these steps. Details will be discussed in the next session.

\begin{enumerate}%
\item consider the presheaf of preoriented ellitptic curves $E'(A)$ first


\item observe that this restricted to ordinary rings produces the ordinary moduli stack


\item notice that every oo-stack with good deformation theory that restricts this way is a derived Deligne-Mumford stack $(\mathcal{X}, \mathcal{O}')$ that assigns connective $E_\infty$-rings over affines


\item let $\omega$ be the line bundle on $\mathcal{M}_{1,1}$ regarded as a coheren sheaf. There is then from the preorientation of the universal curve over $(\mathcal{M}, \mathcal{O}')$ a morphism

\begin{displaymath}
\beta : \omega \to \pi_2 \mathcal{O}'
\end{displaymath}

\item let $\mathcal{O}$ be the sheaf obtained from $\mathcal{O}'$ by inverting $\beta$


\item show that

\begin{enumerate}%
\item for $n = 2 k$ we have an isomorphism $\omega^k \to \pi_{2 k}\mathcal{O}$


\item for $n = 2 k + 1$ we have an isomorphism $0 \to \pi_{2k +1}\mathcal{O}$

strategy: reduce to neighbourhood of a point



\end{enumerate}

\item notice that this implies the desired statement



\end{enumerate}
\end{proof}
[[!redirects derived moduli stack of derived elliptic curves]]



\end{document}