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\begin{document}

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\section*{A Survey of Elliptic Cohomology}

\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]]

This entry is about the text

\begin{itemize}%
\item [[Jacob Lurie]], \emph{A Survey of Elliptic Cohomology}, Algebraic Topology Abel Symposia Volume 4, 2009, pp 219-277 (\href{http://www.math.harvard.edu/~lurie/papers/survey.pdf}{pdf})

\end{itemize}
It

\begin{itemize}%
\item reviews basics of [[elliptic cohomology]]


\item indicates the construction of the [[tmf]]-[[spectrum]]


\item and discusses this in the context of [[equivariant cohomology]].



\end{itemize}
The central theorem is

\begin{itemize}%
\item the realization of the moduli space of [[elliptic curve]]s as a [[structured (∞,1)-topos]]


\item with a structure [[(∞,1)-sheaf]] of [[E-∞ ring]]-valued functions


\item such that the [[tmf]]-[[spectrum]] is the [[E-∞ ring]] of global sections of this structure sheaf.



\end{itemize}
See also \emph{[[Elliptic Cohomology I]]} and \emph{[[Chromatic Homotopy Theory]]}.

\hypertarget{table_of_contents}{}\section*{{Table of Contents}}\label{table_of_contents}

\begin{enumerate}%
\item \hyperlink{Summary}{Summary}

The following entry has some paragraphs that summarize central ideas.

\begin{enumerate}%
\item \hyperlink{gluingallellipticcohomologytheoriestothetmfspectrum}{gluing all elliptic cohomology theories to the tmf spectrum}


\item \hyperlink{interpretationintermsofhighertopostheory}{interpretation in terms of higher topos theory}



\end{enumerate}

\item Partial surveys

These links point to pages that contain notes on aspects of the theory that are in the style of and originate from \href{http://nd.edu/~rgrady/elliptic_seminar.html}{a seminar on A Survey of Elliptic Cohomology}:

\begin{enumerate}%
\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]]


\item [[A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves]]


\item [[A Survey of Elliptic Cohomology - towards a proof]]


\item [[A Survey of Elliptic Cohomology - compactifying the derived moduli stack]]


\item [[A Survey of Elliptic Cohomology - descent ss and coefficients]]



\end{enumerate}

\item towards geometric models

These links point to pages that have an exposition of the Stolz-Teichner program for constructing [[geometric models for elliptic cohomology]].



\end{enumerate}
\begin{itemize}%
\item Outline of the constructions and statements

\begin{itemize}%
\item [[Axiomatic field theories and their motivation from topology]]


\item [[(1,1)-dimensional Euclidean field theories and K-theory]]


\item [[(2,1)-dimensional Euclidean field theories and tmf]]



\end{itemize}

\item Definitions

\begin{itemize}%
\item [[bordism categories following Stolz-Teichner]]


\item [[(2,1)-dimensional Euclidean field theories]]



\end{itemize}

\item [[Geometric Models for Elliptic Cohomology|Geometric models for elliptic cohomology -- Axiomatic field theories and their motivation from topology]]



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

Here is the table of contents of the Survey reproduced. Behind the links are linked keyword lists for relevant terms.

\noindent\hyperlink{table_of_contents}{Table of Contents}\dotfill \pageref*{table_of_contents} \linebreak
\noindent\hyperlink{contents}{Contents}\dotfill \pageref*{contents} \linebreak
\noindent\hyperlink{Summary}{Summary}\dotfill \pageref*{Summary} \linebreak
\noindent\hyperlink{gluingallellipticcohomologytheoriestothetmfspectrum}{Gluing all elliptic cohomology theories to the tmf spectrum}\dotfill \pageref*{gluingallellipticcohomologytheoriestothetmfspectrum} \linebreak
\noindent\hyperlink{interpretationintermsofhighertopostheory}{Interpretation in terms of higher topos theory}\dotfill \pageref*{interpretationintermsofhighertopostheory} \linebreak
\noindent\hyperlink{equivariantellipticcohomology}{equivariant elliptic cohomology}\dotfill \pageref*{equivariantellipticcohomology} \linebreak
\noindent\hyperlink{Contents}{Contents}\dotfill \pageref*{Contents} \linebreak
\noindent\hyperlink{EllipticCohomology}{1. Elliptic Cohomology}\dotfill \pageref*{EllipticCohomology} \linebreak
\noindent\hyperlink{CohomologyTheories}{1.1 Cohomology Theories}\dotfill \pageref*{CohomologyTheories} \linebreak
\noindent\hyperlink{FormalgroupsFromCohomologyTheories}{1.2 Formal Groups from Cohomology Theories}\dotfill \pageref*{FormalgroupsFromCohomologyTheories} \linebreak
\noindent\hyperlink{EllipticCohomology}{1.3 Elliptic Cohomology}\dotfill \pageref*{EllipticCohomology} \linebreak
\noindent\hyperlink{DerivedAlgebraicCohomology}{2 Derived Algebraic geometry}\dotfill \pageref*{DerivedAlgebraicCohomology} \linebreak
\noindent\hyperlink{21__rings}{2.1 $E_\infty$ rings}\dotfill \pageref*{21__rings} \linebreak
\noindent\hyperlink{22_derived_schemes}{2.2 Derived Schemes}\dotfill \pageref*{22_derived_schemes} \linebreak
\noindent\hyperlink{3_derived_group_schemes_and_orientations}{3 Derived Group Schemes and Orientations}\dotfill \pageref*{3_derived_group_schemes_and_orientations} \linebreak
\noindent\hyperlink{31_orientations_of_the_multiplicative_group}{3.1 Orientations of the Multiplicative Group}\dotfill \pageref*{31_orientations_of_the_multiplicative_group} \linebreak
\noindent\hyperlink{32_orientations_of_the_additive_group}{3.2 Orientations of the Additive Group}\dotfill \pageref*{32_orientations_of_the_additive_group} \linebreak
\noindent\hyperlink{33_the_geometry_of_preorientations}{3.3 The Geometry of Preorientations}\dotfill \pageref*{33_the_geometry_of_preorientations} \linebreak
\noindent\hyperlink{34_equivariant_cohomology_for_abelian_groups}{3.4 Equivariant $A$-Cohomology for Abelian Groups}\dotfill \pageref*{34_equivariant_cohomology_for_abelian_groups} \linebreak
\noindent\hyperlink{35_the_nonabelian_case}{3.5 The Nonabelian Case}\dotfill \pageref*{35_the_nonabelian_case} \linebreak
\noindent\hyperlink{4_oriented_elliptic_curves}{4 Oriented Elliptic Curves}\dotfill \pageref*{4_oriented_elliptic_curves} \linebreak
\noindent\hyperlink{41_construction_of_the_moduli_stack}{4.1 Construction of the Moduli Stack}\dotfill \pageref*{41_construction_of_the_moduli_stack} \linebreak
\noindent\hyperlink{42_the_proof_of_theorem_44_the_local_case}{4.2 The Proof of Theorem 4.4: The Local Case}\dotfill \pageref*{42_the_proof_of_theorem_44_the_local_case} \linebreak
\noindent\hyperlink{43_elliptic_cohomology_near_}{4.3 Elliptic Cohomology near $\infty$}\dotfill \pageref*{43_elliptic_cohomology_near_} \linebreak
\noindent\hyperlink{5_applications}{5 Applications}\dotfill \pageref*{5_applications} \linebreak
\noindent\hyperlink{51_2equivariant_elliptic_cohomology}{5.1 2-Equivariant Elliptic Cohomology}\dotfill \pageref*{51_2equivariant_elliptic_cohomology} \linebreak
\noindent\hyperlink{52_loop_group_representations}{5.2 Loop Group Representations}\dotfill \pageref*{52_loop_group_representations} \linebreak
\noindent\hyperlink{53_string_orientation}{5.3 String Orientation}\dotfill \pageref*{53_string_orientation} \linebreak
\noindent\hyperlink{54_higher_equivariance}{5.4 Higher Equivariance}\dotfill \pageref*{54_higher_equivariance} \linebreak
\noindent\hyperlink{55_elliptic_cohomology_and_geometry}{5.5 Elliptic Cohomology and Geometry}\dotfill \pageref*{55_elliptic_cohomology_and_geometry} \linebreak
\noindent\hyperlink{furtherreferences}{Further references}\dotfill \pageref*{furtherreferences} \linebreak


\hypertarget{Summary}{}\subsection*{{Summary}}\label{Summary}

The text starts with showing or recalling that

\begin{itemize}%
\item the collection of all [[elliptic cohomology]] theories and


\item the gluing of their representing [[spectrum|spectra]] into the single [[tmf]] spectrum



\end{itemize}
is best understood in terms of global sections of the structure sheaf of functions on the refinement of the moduli [[space]] of all elliptic curves to a [[structured (∞,1)-topos]].

Then it uses this [[higher topos theory|higher topos theoretic]] [[derived algebraic geometry]] perspective to analyze further properties of [[elliptic cohomology]] theories, in particular their refinements to [[equivariant cohomology]].

\hypertarget{gluingallellipticcohomologytheoriestothetmfspectrum}{}\subsubsection*{{Gluing all elliptic cohomology theories to the tmf spectrum}}\label{gluingallellipticcohomologytheoriestothetmfspectrum}

The triple of [[generalized (Eilenberg-Steenrod) cohomology]] theories

\begin{enumerate}%
\item periodic ordinary [[integral cohomology]]


\item complex [[K-theory]]


\item [[elliptic cohomology]]



\end{enumerate}
constitutes the collection of all possible [[generalized (Eilenberg-Steenrod) cohomology]] theories with the extra [[stuff, structure, property|property]] that they are

\begin{itemize}%
\item [[multiplicative cohomology theory|multiplicative]]

\end{itemize}
and

\begin{itemize}%
\item [[periodic cohomology theory|periodic]].

\end{itemize}
It so happens that all multiplicative periodic generalized Eilenberg-Steenrod cohomology theories $A$ are characterized by the [[formal group]] (an [[infinitesimal space|infinitesimal]] [[group]]) whose [[ring]] of functions is the [[cohomology ring]] $A(\mathbb{C}P^\infty)$ obtained by evaluating $A$ on the complex projective space $\mathbb{C}P^\infty \simeq \mathcal{B} U(1)$ -- the [[classifying space]] for complex [[line bundle]]s -- and whose group product is induced from the morphism $\mathbb{C}P^\infty \times \mathbb{C}P^\infty \to \mathbb{C}P^\infty$ that representes the [[tensor product]] of complex [[line bundle]]s.

There are precisely three different types of such formal groups:

\begin{itemize}%
\item the additive formal group (a single one)


\item the multiplicative formal group (a single one)


\item a formal group defined by an [[elliptic curve]] (many).



\end{itemize}
The first case corresponds to periodic [[integral cohomology]]. The second corresponds to complex [[K-theory]]. Each element in the third family corresponds to one flavor of [[elliptic cohomology]].

It is therefore natural to subsume all elliptic cohomology theories into one single cohomology theory. This is the theory called [[tmf]].

It turns out that the right way to formalize what ``subsume'' means in the above sentence involves formulating the way in which an [[elliptic cohomology]] theory is associated to a given [[elliptic curve]] in the correct [[higher category theory|higher categorical]] language:

The collection of all 1-dimensional [[elliptic curve]]s forms a generalized [[space]] $M_{1,1}$ -- a [[stack]] -- defined by the property that it is the [[classifying space]] for elliptic curves in that elliptic curves over a ring $R$ correspond to classifying maps $\phi : Spec R \to M_{1,1}$.

Then the classical assignment of an [[elliptic cohomology]] theory to an [[elliptic curve]] is an assignment

\begin{displaymath}
\{\phi : Spec R \to M_{1,1}\}
  \to
  CohomologyTheories
  \,.
\end{displaymath}
We may think of maps $Spec R \to M_{1,1}$ as picking certain subsets of the generalized [[space]] $M_{1,1}$ and of morphisms

\begin{displaymath}
\itexarray{
     Spec(R) &&\to&& Spec(R')
     \\   
     & \searrow && \swarrow
     \\
     && M_{1,1}
  }
\end{displaymath}
as maps between such subsets. Hence the assignment of cohomology theories to elliptic curves is much like a [[sheaf]] of cohomology theories on the moduli [[space]] ([[stack]]) of [[elliptic curve]]s.

In order to \emph{glue} all elliptic cohomology theories in some way one would like to take something like the [[category of elements]] of this [[sheaf]], i.e. its [[homotopy limit]]. In order to say what that should mean, one has to specify the suitable nature of the codomain, the collection of ``all cohomology theories''.

As emphasized at [[generalized (Eilenberg-Steenrod) cohomology]], the best way to do this is to identify a generalized (Eilenberg-Steenrod) cohomology theory with the [[spectrum]] that represents it. It is and was well known how to do this for each elliptic curve separately. What is not so clear is how this can be done coherently for all elliptic curves at once: we need a lift of the above cohomology-theory-valued sheaf to a sheaf of representing [[spectrum|spectra]]

\begin{displaymath}
\itexarray{
    && Spectra
    \\
    & {}^{?}\nearrow & \;\;\;\downarrow^{represent}
    \\
    \{\phi : Spec R \to M_{1,1}\}
    &\to&
    CohomologyTheories
  }
  \,.
\end{displaymath}
In this generality this turns out to be a hard problem. But by definition here we are really interested just in the special case where all cohomology theories in question are [[multiplicative cohomology theory|multiplicative cohomology theories]] and where hence all spectra in question are [[commutative ring spectrum|commutative ring spectra]]

\begin{displaymath}
\itexarray{
    && CommRingSpectra
    \\
    & {}^{O_{M^{der}}}\nearrow & \downarrow
    \\
    \{\phi : Spec R \to M_{1,1}\}
    &\to&
    MultiplicativeCohomologyTheories
  }
  \,.
\end{displaymath}
As indicated, this problem does turn out to have a solution: [[Paul Goerss|Goerss]], [[Mike Hopkins|Hopkins]] and [[Haynes Miller|Miller]] showed that the desired lift denoted $O_{M^{der}}$ above exists -- the \emph{[[Goerss-Hopkins-Miller theorem]]}

Accordingly, one can then obtain the [[tmf]] spectrum as the [[homotopy limit]] of this sheaf of [[E-∞ ring]]s $O_{M^{der}}$. Recall from the discussion at [[limit in a quasi-category]] that such a homotopy limit computes global sections. It is an $\infty$-version of computing sections in a [[Grothendieck construction]], really, as described there.

\hypertarget{interpretationintermsofhighertopostheory}{}\subsubsection*{{Interpretation in terms of higher topos theory}}\label{interpretationintermsofhighertopostheory}

What is noteworthy about the above construction is that, as the notation above suggests, sheaves of [[E-infinity ring]]s generalize sheaves of rings as thery are familiar from the theory of [[ringed space]]s, where they are called \textbf{structure sheaves}.

Accordingly, the morphism $O_{M^{der}}$ makes the moduli [[space]] of elliptic curves into a [[structured (∞,1)-topos]].

This perspective embeds the theory of [[elliptic cohomology]] and of the [[tmf]] spectrum as an application into the general context of [[higher topos theory]] and [[derived algebraic geometry]].

\hypertarget{equivariantellipticcohomology}{}\subsection*{{equivariant elliptic cohomology}}\label{equivariantellipticcohomology}

\begin{itemize}%
\item [[equivariant elliptic cohomology]]

\end{itemize}
\hypertarget{Contents}{}\subsection*{{Contents}}\label{Contents}

\hypertarget{EllipticCohomology}{}\subsubsection*{{1. Elliptic Cohomology}}\label{EllipticCohomology}

\hypertarget{CohomologyTheories}{}\paragraph*{{1.1 Cohomology Theories}}\label{CohomologyTheories}

\begin{itemize}%
\item [[cohomology]]

\begin{itemize}%
\item [[generalized (Eilenberg-Steenrod) cohomology]]


\item [[integral cohomology]]


\item [[K-theory]]


\item [[elliptic cohomology]]



\end{itemize}

\item [[multiplicative cohomology theory]]


\item [[periodic cohomology theory]]


\item [[topological modular form]]



\end{itemize}
\hypertarget{FormalgroupsFromCohomologyTheories}{}\paragraph*{{1.2 Formal Groups from Cohomology Theories}}\label{FormalgroupsFromCohomologyTheories}

\begin{itemize}%
\item [[formal group]]

\end{itemize}
\hypertarget{EllipticCohomology}{}\paragraph*{{1.3 Elliptic Cohomology}}\label{EllipticCohomology}

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


\item [[elliptic cohomology]]


\item [[E-∞ ring]]



\end{itemize}
\hypertarget{DerivedAlgebraicCohomology}{}\subsubsection*{{2 Derived Algebraic geometry}}\label{DerivedAlgebraicCohomology}

\begin{itemize}%
\item [[derived algebraic geometry]]

\begin{itemize}%
\item [[higher topos theory]]


\item [[higher algebra]]



\end{itemize}


\end{itemize}
\hypertarget{21__rings}{}\paragraph*{{2.1 $E_\infty$ rings}}\label{21__rings}

\begin{itemize}%
\item [[commutative algebra in an (∞,1)-category]]


\item [[E-∞ ring]]


\item [[commutative ring spectrum]]



\end{itemize}
\hypertarget{22_derived_schemes}{}\paragraph*{{2.2 Derived Schemes}}\label{22_derived_schemes}

\begin{itemize}%
\item [[structured (∞,1)-topos]]


\item [[derived scheme]]



\end{itemize}
\hypertarget{3_derived_group_schemes_and_orientations}{}\subsubsection*{{3 Derived Group Schemes and Orientations}}\label{3_derived_group_schemes_and_orientations}

\begin{itemize}%
\item [[group scheme]]

\end{itemize}
\hypertarget{31_orientations_of_the_multiplicative_group}{}\paragraph*{{3.1 Orientations of the Multiplicative Group}}\label{31_orientations_of_the_multiplicative_group}

\hypertarget{32_orientations_of_the_additive_group}{}\paragraph*{{3.2 Orientations of the Additive Group}}\label{32_orientations_of_the_additive_group}

\hypertarget{33_the_geometry_of_preorientations}{}\paragraph*{{3.3 The Geometry of Preorientations}}\label{33_the_geometry_of_preorientations}

\hypertarget{34_equivariant_cohomology_for_abelian_groups}{}\paragraph*{{3.4 Equivariant $A$-Cohomology for Abelian Groups}}\label{34_equivariant_cohomology_for_abelian_groups}

\hypertarget{35_the_nonabelian_case}{}\paragraph*{{3.5 The Nonabelian Case}}\label{35_the_nonabelian_case}

\hypertarget{4_oriented_elliptic_curves}{}\subsubsection*{{4 Oriented Elliptic Curves}}\label{4_oriented_elliptic_curves}

\hypertarget{41_construction_of_the_moduli_stack}{}\paragraph*{{4.1 Construction of the Moduli Stack}}\label{41_construction_of_the_moduli_stack}

\hypertarget{42_the_proof_of_theorem_44_the_local_case}{}\paragraph*{{4.2 The Proof of Theorem 4.4: The Local Case}}\label{42_the_proof_of_theorem_44_the_local_case}

\hypertarget{43_elliptic_cohomology_near_}{}\paragraph*{{4.3 Elliptic Cohomology near $\infty$}}\label{43_elliptic_cohomology_near_}

\hypertarget{5_applications}{}\subsubsection*{{5 Applications}}\label{5_applications}

\hypertarget{51_2equivariant_elliptic_cohomology}{}\paragraph*{{5.1 2-Equivariant Elliptic Cohomology}}\label{51_2equivariant_elliptic_cohomology}

\begin{itemize}%
\item [[equivariant elliptic cohomology]]

\end{itemize}
\hypertarget{52_loop_group_representations}{}\paragraph*{{5.2 Loop Group Representations}}\label{52_loop_group_representations}

\begin{itemize}%
\item [[loop group]]

\end{itemize}
\hypertarget{53_string_orientation}{}\paragraph*{{5.3 String Orientation}}\label{53_string_orientation}

\begin{itemize}%
\item [[string structure]]


\item [[sigma-orientation]]



\end{itemize}
\hypertarget{54_higher_equivariance}{}\paragraph*{{5.4 Higher Equivariance}}\label{54_higher_equivariance}

\hypertarget{55_elliptic_cohomology_and_geometry}{}\paragraph*{{5.5 Elliptic Cohomology and Geometry}}\label{55_elliptic_cohomology_and_geometry}

\hypertarget{furtherreferences}{}\subsection*{{Further references}}\label{furtherreferences}

Lots of literature on [[modular form]]s is collected at

\begin{itemize}%
\item [[Nora Ganter]], \href{http://www.math.uiuc.edu/~ganter/talbot/index.html}{Topological modular forms literature list}

\end{itemize}
An introduction to and survey of the [[Goerss-Hopkins-Miller-Lurie theorem]] is in

\begin{itemize}%
\item [[Paul Goerss]], \emph{Topological modular forms (after Hopkins, Miller, and Lurie)} S\'e{}minaire BOURBAKI Mars 2009 61\`e{}me ann\'e{}e, 2008-2009, no 1005(2009)(\href{http://arxiv.org/abs/0910.5130}{arXiv})

\end{itemize}
which has grown out of

\begin{itemize}%
\item [[Topological Algebraic Geometry - A Workshop]].

\end{itemize}
A good bit of details is in

\begin{itemize}%
\item [[David Gepner]], \emph{[[Homotopy topoi and equivariant elliptic cohomology]]}, 1999 (\href{http://www.ms.unimelb.edu.au/~nganter/talbot/gepner.thesisformat.pdf}{pdf})

\end{itemize}


\end{document}