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\section*{Deligne conjecture}

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

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

[[!include higher algebra - contents]]

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

[[!include cohomology - contents]]

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{GeometricInterpretation}{Geometric interpretation}\dotfill \pageref*{GeometricInterpretation} \linebreak
\noindent\hyperlink{for_higher_algebras_and_higher_monoidal_categories}{For higher algebras and higher monoidal categories}\dotfill \pageref*{for_higher_algebras_and_higher_monoidal_categories} \linebreak
\noindent\hyperlink{history}{History}\dotfill \pageref*{history} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\hypertarget{statement}{}\subsubsection*{{Statement}}\label{statement}

The original \emph{Deligne conjecture} on the structure of [[Hochschild cohomology]] of an [[associative algebra]] and more generally an [[A-∞ algebra]] states that Hochschild cohomology naturally has the structure of a [[BV-algebra]] and that moreover the full Hochschild cohomology complex has the structure of an [[algebra over an operad]] over the ([[framed little disk operad|framed]]) [[little disk operad]]. (The second statement implies the first since ([[BV-algebra|BV]]-)[[nLab:Gerstenhaber algebra]]s are [[algebras over an operad]] over the (framed) little disk operad.)

Proofs of the conjecture have been given first by \hyperlink{McClureSmith}{McClure-Smith} and others, then also by \hyperlink{KontsevichSoibelman}{Kontsevich-Soibelman}, and \hyperlink{Tamarkin}{Tamarkin}.

A proof for general [[Ek-algebra]]s in [[symmetric monoidal (∞,1)-categories]] is in (\hyperlink{Lurie}{Lurie, section 2.5}).

\hypertarget{GeometricInterpretation}{}\subsubsection*{{Geometric interpretation}}\label{GeometricInterpretation}

A general geometric ([[higher geometry|higher geometric]]) interpretation has been indicated in \hyperlink{Ben-ZviFrancisNadler}{Ben-ZviFrancisNadler}. They observed (see [[Hochschild cohomology]] for details) that the Hochschild \emph{homology} of $\mathcal{O}(X)$ is naturally interpreted as the algebra of functions on the [[derived loop space]] $\mathcal{L}X$. In the presence of good [[geometric ∞-function theory]] this naturally induces an action of the [[little disk operad]] on $\mathcal{O}(\mathcal{L}X)$. Since a [[Gerstenhaber algebra]] is an algebra over the [[homology]] of the [[little disk operad]], this immediately explains the existence of this structure.

See also \href{Hochschild+cohomology#DeligneConjectureViaDerivedMappingSpaces}{this prop}.

\hypertarget{for_higher_algebras_and_higher_monoidal_categories}{}\subsubsection*{{For higher algebras and higher monoidal categories}}\label{for_higher_algebras_and_higher_monoidal_categories}

More generally, analogs of the statement of the Deligne conjecture exist and work for $(\infty,n)$-algebras: [[k-tuply monoidal n-categories]]. This is closely related to the statement (and proofs) of the [[delooping hypothesis]].

This case is discussed in (\hyperlink{Francis}{Francis}) and (\hyperlink{Lurie}{Lurie}).

In (\hyperlink{KockToen}{KockToen}) it is shown that in a [[monoidal model category]] that is also a [[simplicial model category]] the [[derived hom-space]] $\mathbb{R} Hom(I,I)$ from the tensor unit to itself is a [[Ek-algebra|E-2 algebra]].

\hypertarget{history}{}\subsubsection*{{History}}\label{history}

Historically it was first found that there is the structure of a [[Gerstenhaber algebra]] on $HH^\bullet(A,A)$. By (\hyperlink{Cohen}{Cohen}) it was known that Gerstenhaber algebras arise as the [[homology]] of [[E2-algebra]]s in [[chain complex]]es. In a letter in 1993 Deligne wondered whether the [[Gerstenhaber algebra]] structure on the Hochschild cohomology $HH^\bullet(A,A)$ lifts to an [[E2-algebra]]-structure on the cochain complex $C^\bullet(A,A)$.

In \hyperlink{GerstenhaberVoronov}{GerstenhaberVoronov (1994)} a resolution of the Gerstenhaber algebra structure was given, but the relationship to $E_2$-algebras remained unclear.

In (\hyperlink{Tamarkin}{Tamarkin (1998)}) a genuine resolution in the [[model structure on operads]] of the Gerstenhaber operad was given and shown to act via the Gerstenhaber-Voronov construction on $C^\bullet(A,A)$. This proved Deligne's conjecture.

Various authors later further refined this result. A summary of this history can be found in (\hyperlink{Hess}{Hess}).

In \hyperlink{HuKrizVoronov}{Hu-Kriz-Voronov (2003)} it was further shown that for $A$ an [[En-algebra]], $C^\bullet(A,A)$ is an $E_{n+1}$-algebra.

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

Direct proofs of the Deligne conjecture have been given in.

\begin{itemize}%
\item James McClure, Jeffrey Smith, \emph{A solution of Deligne's conjecture} (\href{http://arxiv.org/abs/math/9910126}{arXiv:math/9910126})

\end{itemize}
\begin{itemize}%
\item [[Maxim Kontsevich]], [[Yan Soibelman]], \emph{Deformations of algebras over operads and Deligne's conjecture} (\href{http://arxiv.org/abs/math/0001151}{arXiv:math/0001151})

\end{itemize}
\begin{itemize}%
\item [[Dmitry Tamarkin]], \emph{Another proof of M. Kontsevich formality theorem} (\href{http://arxiv.org/abs/math/9803025}{arXiv:math/9803025})

\end{itemize}
\begin{itemize}%
\item [[Clemens Berger]], [[Benoit Fresse]], \emph{Combinatorial operad actions on cochains} (\href{http://arxiv.org/abs/math/0109158}{arXiv:math/0109158})

\end{itemize}
A review is in

\begin{itemize}%
\item [[Kathryn Hess]], \emph{Deligne's Hochschild cohomology conjecture} (\href{http://sma.epfl.ch/~hessbell/Pub_DeligneColloq.pdf}{pdf})

\end{itemize}
\begin{itemize}%
\item Cohen

\end{itemize}
\begin{itemize}%
\item Gertstenhaber-Voronov

\end{itemize}
\begin{itemize}%
\item Hu, Kriz, Voronov

\end{itemize}
A transparent [[higher geometry|higher geometric]] interpretation in a suitably dualizable context is indicated in

\begin{itemize}%
\item [[David Ben-Zvi]], [[John Francis]], [[David Nadler]], \emph{[[geometric infinity-function theory|Integral transforms and Drinfeld centers in derived algebraic geometry]]} (\href{http://arxiv.org/abs/0805.0157}{arXiv:0805.0157})

\end{itemize}
\begin{itemize}%
\item [[John Francis]], PhD thesis (\href{http://dspace.mit.edu/handle/1721.1/43792}{web})

\end{itemize}
Section 2.5 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Ek-Algebras]]}

\end{itemize}
A construction in [[monoidal model categories]] is in

\begin{itemize}%
\item [[Joachim Kock]], [[Bertrand Toen]], \emph{Simplicial localization of monoidal structures, and a non-linear version of Deligne's conjecture} Compositio Math. 141 (2005), 253-261 (\href{http://arxiv.org/abs/math.AT/0304442}{arXiv})

\end{itemize}
[[!redirects Deligne conjectures]]



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