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\title[Deformation theory of algebraic structures] % (optional, use only with long paper titles)
{Deformation theory of algebraic structures}

\institute[Bruno Vallette] % (optional, but mostly needed)
{ Second Congr\`es Canada-France }

\date % (optional)
{June 2008}

\subject{Talks}

%\AtBeginSection[]
%{
%  \begin{frame}&lt;beamer&gt;
%    \frametitle{Overview}
%    \tableofcontents[currentsection]
%  \end{frame}
%}


\setbeamercovered{invisible}

\begin{document}

\begin{frame}
  \titlepage
\end{frame}


%\begin{frame}
%  \frametitle{Overview}
%  \tableofcontents
%\end{frame}

\begin{frame}

{\Large \bf Data: }
 \begin{itemize}
\item An algebraic structure $\Po$

\item A vector space $A$
\end{itemize}
\sip

{\Large \bf Goal: }  {\bf Study the deformation theory of $\Po$-structures on $A$}

\sip

\begin{itemize}
\item  Define a dg Lie algebra $\mathfrak{g}(\Po, A)$ [Deligne]

\item  Maurer-Cartan elements $\longleftrightarrow$ $\Po$-structures on $A$

\item  $H^\bullet(\mathfrak{g}(\Po, A))$ : obstruction space for the possible deformations
\end{itemize}
\end{frame}


\begin{frame}
{\Large \bf Paradigm: Associative algebras}

\footnotesize


$\bullet$ Let $A$ be a $\KK$-module, consider $\End_A:=\lbrace
\Hom(A^{\ot n}, A)\rbrace_{n \ge 0}$.

\smallskip

For $f \in \Hom(A^{\ot n}, A)$, $g \in \Hom(A^{\ot m}, A)$
$$f\star g := \sum_{i=1}^n \pm
\vcenter{\xymatrix@R=10pt@C=10pt@M=3pt{&amp;&amp; \\ &amp;
*+[F-,]{g}\ar@{-}[ul] \ar@{-}[ur] \ar@{-}[d]^{i} &amp;
\\&amp;*+[F-,]{f} \ar@{-}[ul]\ar@{-}[ur] \ar@{-}[d]&amp;
\\&amp;&amp;}}
=\sum_{i=1}^n \pm f \circ_i g.$$

\begin{thm}[Gerstenhaber]
$$(f\star g) \star h - f \star ( g \star h)=(f\star h) \star g - f \star (h \star g) $$
$(\End V, \star)$ is a preLie algebra
$\Longrightarrow$ with $[f, g]:= f\star g - (-1)^{|f|.|g|} g \star
f$,

\begin{center}
{\bf $(\End V, [\ ,\, ])$ is a Lie algebra.}
\end{center}
\end{thm}
\end{frame}

\begin{frame}
\footnotesize
$\bullet$ Associative algebra structure on $A$ :
$$\mu \ : \ A^{\ot 2} \to A \quad, \quad     \Y$$
$$\vcenter{\xymatrix@M=0pt@R=6pt@C=6pt{
&amp; &amp; &amp; &amp; \\ \ar@{-}[ddrr] &amp;
&amp;\ar@{-}[dl] &amp; &amp; \ar@{-}[ddll]  \\
&amp;  &amp; &amp;  &amp; \\
&amp; &amp;\ar@{-}[d] &amp; &amp; \\
&amp; &amp; }} - \vcenter{\xymatrix@M=0pt@R=6pt@C=6pt{&amp; &amp; &amp; &amp; \\
\ar@{-}[dr] &amp;
&amp;\ar@{-}[dr] &amp; &amp; \ar@{-}[dl]  \\
&amp; \ar@{-}[dr] &amp; &amp;\ar@{-}[dl]  &amp; \\
&amp; &amp;\ar@{-}[d] &amp; &amp; \\
&amp; &amp; }} =0 \quad \textrm{in} \quad \Hom(A^{\ot 3}, A)\Longleftrightarrow \boxed{\mu \star \mu =0} \Longleftrightarrow \boxed{[\mu,
\mu] =0}$$
In this case,
$$d_\mu(f):= [\mu, f] \quad \textrm{satisfies} \quad d_\mu(f)^2=0.$$
Explicitly, for $f\in \Hom(A^{\ot n}, A)$
$$d_\mu(f)=\sum_{i=1}^n \pm \vcenter{\xymatrix@R=10pt@C=10pt@M=3pt{*{}&amp;*{}&amp;*{} \\ &amp; *+[F-,]{\mu}\ar@{-}[ul]
\ar@{-}[ur] \ar@{-}[d]^{i} &amp;
\\&amp;*+[F-,]{f} \ar@{-}[ul]\ar@{-}[ur] \ar@{-}[d]&amp;
\\*{}&amp;*{}&amp;*{}}} \pm \vcenter{\xymatrix@R=10pt@C=10pt@M=3pt{*{}&amp;*{}&amp;*{}&amp;*{} \\ &amp;*+[F-,]{f}\ar@{-}[ul]\ar@{-}[u]\ar@{-}[ur]&amp;&amp; \\
 &amp;&amp;*+[F-,]{\mu}\ar@{-}[ul]\ar@{-}[ur] \ar@{-}[d]&amp;
\\ *{}&amp;*{}&amp;*{}&amp;*{}}} \pm \vcenter{\xymatrix@R=10pt@C=10pt@M=3pt{*{}&amp;*{}&amp;*{}&amp;*{}\\&amp;&amp;*+[F-,]{f}\ar@{-}[u]\ar@{-}[ur]\ar@{-}[ul]&amp;\\
&amp;*+[F-,]{\mu}\ar@{-}[ul]\ar@{-}[ur] \ar@{-}[d]&amp;&amp; \\*{}&amp;*{}&amp;*{}&amp;*{}}} \in
\mathop\Hom(A^{\ot n+1}, A)$$

$$\Hom(\KK, A)\xrightarrow{d_\mu}\Hom(A, A)\xrightarrow{d_\mu}
\Hom(A^{\ot 2}, A)\xrightarrow{d_\mu} \cdots $$

$\rightarrow$ {\bf Hochschild cohomology of ``$A$ with coefficients into itself''}

\end{frame}

\begin{frame}

\footnotesize
$\bullet$ Deformation complex of the associative algebra structure $\mu$ on $A$ :
\begin{center}
 $\mathop{C}^\bullet(\Ass, A):=(End_A , d_\mu,
[\, ,\, ])$ dg Lie algebra (twisted by $\mu$).
\end{center}


$\triangleright$ Interpretation of $\textrm{H}^0, \textrm{H}^1, \textrm{H}^2$ in
terms of deformations of $\mu$

$\quad \rightarrow$ {\bf Deformation Quantization} [Kontsevich]

\bip

$\bullet$ More operations on $\mathop{C}^\bullet(\Ass, A)$ :

\begin{itemize}
\item[$\triangleright$] Cup product $\cup$ : associative operation

\item[$\triangleright$] Braces operations
\end{itemize}

$\quad \rightarrow$ {\bf Deligne Conjecture}
\end{frame}


\begin{frame}

\footnotesize
$\bullet$ $(A, d)$ dg module $\Longrightarrow$ $\End_A$  dg module

$$D(f) := \sum_{i=1}^n \vcenter{\xymatrix@R=10pt@C=10pt@M=3pt{&amp; {d} \ar@{-}[d]^{i}&amp; \\  &amp; *+[F-,]{f}
\ar@{-}[ur] \ar@{-}[d] \ar@{-}[ul]&amp; \\ &amp;&amp;}} -(-1)^{|f|}
\vcenter{\xymatrix@R=10pt@C=10pt@M=3pt{&amp;&amp; \\  &amp; *+[F-,]{f}
\ar@{-}[ur] \ar@{-}[d] \ar@{-}[ul]&amp; \\ &amp;d&amp;}}  $$

$(\End_A, D, \star)$ is a dg preLie algebra and  $(\End_A, D, [\
,\, ])$ is a dg Lie algebra.
\smallskip

$(A, d, \mu)$ is a dg associative algebra $\Longleftrightarrow$

$$\boxed{D \mu + \mu \star \mu=0}\Longleftrightarrow \boxed{D \mu + \frac{1}{2}[\mu, \mu]=0} :
\textbf{Maurer-Cartan equation} $$
\end{frame}

\begin{frame}
\footnotesize
{\bf General solutions:}


$$\mu \in \lbrace \Hom(V^{\ot n}, V) \rbrace_{n\ge 2}, \quad
\mu_n : V^{\ot n} \to V$$

$D \mu + \mu \star \mu=0 \Longleftrightarrow$

$$ \underline{n=2} \quad : \quad  {\vcenter{\xymatrix@R=6pt@C=6pt{
d \ar@{-}[dr] &amp;  &amp;\ar@{-}[dl]  \\
 &amp; *{} \ar@{-}[d] &amp;  \\  &amp; &amp;}}} + {\vcenter{\xymatrix@R=6pt@C=6pt{
\ar@{-}[dr] &amp;  &amp;\ar@{-}[dl] d  \\
 &amp; *{} \ar@{-}[d] &amp;  \\  &amp; &amp;}}} ={\vcenter{\xymatrix@R=6pt@C=6pt{
\ar@{-}[dr] &amp;  &amp;\ar@{-}[dl]  \\
 &amp; *{} \ar@{-}[d] &amp;  \\  &amp; d &amp;}}} : \quad d\  \textrm{is a derivation for}\  \mu_2$$



$$ \underline{n=3} \quad : \quad  \vcenter{\xymatrix@M=0pt@R=6pt@C=6pt{
&amp; &amp; &amp; &amp; \\ \ar@{-}[ddrr] &amp;
&amp;\ar@{-}[dl] &amp; &amp; \ar@{-}[ddll]  \\
&amp;  &amp; &amp;  &amp; \\
&amp; &amp;\ar@{-}[d] &amp; &amp; \\
&amp; &amp; }} - \vcenter{\xymatrix@M=0pt@R=6pt@C=6pt{&amp; &amp; &amp; &amp; \\
\ar@{-}[dr] &amp;
&amp;\ar@{-}[dr] &amp; &amp; \ar@{-}[dl]  \\
&amp; \ar@{-}[dr] &amp; &amp;\ar@{-}[dl]  &amp; \\
&amp; &amp;\ar@{-}[d] &amp; &amp; \\
&amp; &amp; }} =D(\mu_3): \quad \mu_2 \ \textrm{is associative up to the homotopy} \  \mu_3 $$


$$ \underline{n} \quad : \quad  \sum_{i+j=n+1 \atop i, j \ge 2} \pm \vcenter{\xymatrix@M=0pt@R=10pt@C=10pt{
&amp;  &amp;  &amp; &amp; \\ \ar@{-}[ddrr]^(0.93){j} &amp; \ar@{-}[d]^(0.19){i}
&amp;\ar@{-}[dl]&amp; &amp; \ar@{-}[ddll]  \\
&amp;  &amp; &amp;  &amp; \\
&amp; &amp;\ar@{-}[d] &amp; &amp; \\
&amp; &amp; }} =D(\mu_n) $$


\end{frame}


\begin{frame}\footnotesize
\begin{dei}[Stasheff]
$\ $

Maurer-Cartan element $\mu$ in $(\End_A, D, \star)$ $\Longleftrightarrow$

\emph{associative algebra up
to homotopy} or \emph{$\mathop{A}_\infty$-algebra} structure on
$(A, d, \mu =\lbrace \mu_n \rbrace_n)$.
\end{dei}

\sip

{ \bf Viewpoint :} An associative algebra $=$ very particular
$\mathop{A}_\infty$-algebra.

\sip

Once again, $d_\mu(f):= D(f)+[\mu, f]$ $\ $ satisfies $\ $
$d_\mu^2=0$.

$\Longrightarrow \  (\End_A, d_\mu, [\, ,\, ])$ dg Lie algebra
(twisted by $\mu$)

$\quad \rightarrow \ $ {\bf deformation complex of  $\mathop{A}_\infty$-algebra structures}

\end{frame}

\begin{frame}
\footnotesize
$\bullet$ Other kind of algebraic structures :
\begin{itemize}
\item[$\triangleright$] Lie algebras, commutative algebras, Poisson algebras, Gerstenhaber algebras,
PreLie algebras, Batalin-Vilkovisky algebras, ...

\item[$\triangleright$] Lie bialgebras, Frobenius bialgebras, associative bialgebras, ...
\end{itemize}
\bip

For any algebraic structure $\Po$:

$$\xymatrix@C=10pt{ \textrm{dg module} \ A  \ar[d] &amp; \\ \textrm{dg Lie algebra}\ : \   \mathfrak{g}(\Po, A) \ar[d] \\
\textrm{Maurer-Cartan
elements}= \textrm{ homotopy}\ \Po-\textrm{algebra structure on}\  A \ar[d] \\
\textrm{twisted dg Lie : deformation complex
(cohomology)}  }$$


\end{frame}

\begin{frame}
\footnotesize
\begin{tabular}{|p{1.5cm}|c|c|c|c|}
\hline \textbf{\scriptsize Operations}  &amp;
$\vcenter{\xymatrix@M=0pt@R=6pt@C=6pt{&amp;&amp; \\  &amp; \ar@{-}[d]&amp;  \\
&amp; \bullet \ar@{-}[d] &amp; \\ &amp; &amp; \\ &amp;&amp;}}$  &amp;
$\vcenter{\xymatrix@M=0pt@R=6pt@C=6pt{&amp;&amp; \\ \ar@{-}[dr]  &amp; \ar@{-}[d]&amp; \ar@{-}[dl]  \\
&amp; \bullet \ar@{-}[d] &amp; \\ &amp; &amp; \\ &amp;&amp;}} $ &amp;  $\vcenter{\xymatrix@M=0pt@R=6pt@C=6pt{&amp;&amp; \\ \ar@{-}[dr]  &amp; \ar@{-}[d]&amp; \ar@{-}[dl]  \\
&amp; \bullet \ar@{-}[dl] \ar@{-}[dr] &amp; \\ &amp; &amp; \\ &amp;&amp;}}$ &amp;
 $\vcenter{\xymatrix@M=0pt@R=6pt@C=6pt{&amp;&amp; \\ \ar@{-}[dr]  &amp; \ar@{-}[d]&amp; \ar@{-}[dl]  \\
&amp; \bullet \ar@{-}[dl] \ar@{-}[dr] &amp; \\ &amp; &amp; \\ &amp;&amp;}}$\\
\hline \textbf{\scriptsize Composition}  &amp;  $\vcenter{\xymatrix@M=0pt@R=6pt@C=6pt{&amp;&amp; \\  &amp; \ar@{-}[d]&amp;  \\
\ar@{..}[r]&amp; \bullet \ar@{-}[d]\ar@{..}[r]  &amp; \\\ar@{..}[r] &amp; \bullet \ar@{-}[d] \ar@{..}[r]&amp; \\ &amp; &amp; \\
&amp;&amp;}}$ &amp;
$\vcenter{\xymatrix@M=0pt@R=6pt@C=6pt{&amp;&amp;&amp;&amp;&amp;&amp; \\ \ar@{-}[dr]  &amp;
\ar@{-}[d]&amp; \ar@{-}[dl] &amp; \ar@{-}[d] &amp; \ar@{-}[dr]
&amp; &amp; \ar@{-}[dl] \\
\ar@{..}[r] &amp; \bullet \ar@{-}[drr] \ar@{..}[rr] &amp; &amp;
\bullet \ar@{-}[d] \ar@{..}[rr] &amp; &amp;\bullet   \ar@{-}[dll] \ar@{..}[r] &amp;  \\
\ar@{..}[rrr] &amp; &amp; &amp; \bullet \ar@{-}[d] \ar@{..}[rrr]&amp; &amp;  &amp;   \\
&amp; &amp; &amp; &amp; &amp; &amp;   } } $ &amp; $\vcenter{\xymatrix@M=0pt@R=6pt@C=6pt{&amp;&amp;&amp;&amp; \\ \ar@{-}[dr] &amp; &amp; \ar@{-}[dl] &amp; \ar@{-}[d] &amp; \\
\ar@{..}[r]&amp;\bullet \ar@{-}[ddrr]   \ar@{-}@/_/[dd] \ar@{..}[rr]
&amp;&amp;\bullet \ar@{-}[ddll]\ar@{-}@/^/[dd] \ar@{..}[r] &amp;
\\ &amp;&amp;&amp;&amp;
\\ \ar@{..}[r] &amp;\bullet \ar@{-}[d] \ar@{..}[rr]&amp;&amp;\bullet \ar@{-}[dl] \ar@{-}[d] \ar@{-}[dr] \ar@{..}[r] &amp;
\\&amp;&amp;&amp;&amp;
\\&amp;&amp;&amp;&amp; }}$ &amp; $\vcenter{\xymatrix@M=0pt@R=6pt@C=6pt{
&amp;&amp;&amp;&amp; &amp;&amp;\\
 \ar@{-}[dr] &amp; &amp; \ar@{-}[dl] &amp; \ar@{-}[d] &amp; \ar@{-}[dr]&amp;\ar@{-}[d]&amp;\ar@{-}[dl]\\
\ar@{..}[r]&amp;\bullet \ar@{-}[ddrr]   \ar@{-}@/_/[dd] \ar@{..}[rr]
&amp;&amp;\bullet \ar@{-}[ddll]\ar@{-}@/^/[dd] \ar@{..}[rr] &amp;&amp;\bullet
\ar@{-}@/_/[dd]
\ar@{-}@/^/[dd] \ar@{..}[r] &amp;\\
&amp;&amp;&amp;&amp;&amp;&amp;\\
 \ar@{..}[r] &amp;\bullet \ar@{-}[d] \ar@{..}[rr]&amp;&amp;\bullet
\ar@{-}[dl] \ar@{-}[d] \ar@{-}[dr] \ar@{..}[rr] &amp;&amp;\bullet \ar@{-}[d] \ar@{..}[r]&amp;\\
&amp;&amp;&amp;&amp;&amp;&amp;\\
&amp;&amp;&amp;&amp; &amp;&amp;}}$  \\
\hline \textbf{\scriptsize Monoidal category} &amp; $(\textrm{Vect},
\otimes)$  &amp; $(\Sy\textrm{-Mod},
\circ)$   &amp; $(\Sy\textrm{-biMod},
\boxtimes_c)$ &amp;
$(\Sy\textrm{-biMod}, \boxtimes)$  \\
\hline \textbf{\scriptsize Monoid}  &amp;  $R\ot R \to R \atop \textrm{Associative Algebra}$ &amp;  $\Po \circ \Po \to \Po \atop \textbf{Operad}$     &amp;
$\Po \boxtimes_c \Po \to \Po \atop \textbf{Properad}$ &amp; $\Po \boxtimes \Po \to \Po \atop \textbf{Prop}$    \\
\hline \textbf{\scriptsize Modules}  &amp; $\textrm{Modules}$   &amp;
$\textrm{Algebras}$ &amp; $\textrm{Bialgebras}$  &amp;  $\textrm{Bialgebras}$  \\
\hline \textbf{\scriptsize Examples}  &amp; &amp;
{$\textrm{Associative, Lie,}\atop \textrm{Gerstenhaber algebras}$}  &amp; $\textrm{Lie, associative} \atop
\textrm{bialgebras}$ &amp;   \\
\hline \textbf{\scriptsize Free monoid}  &amp; ${\textrm{Ladders} \atop
\textrm{(Tensor module)}}$ &amp; ${\textrm{Trees}}$ &amp; ${\textrm{Connected} \atop \textrm{graphs}}$ &amp;  ${\textrm{Graphs}}$ \\
\hline
\end{tabular}
\end{frame}

\begin{frame}
{\Large \bf Properadic homological algebra}
\footnotesize




$\Co$ dg coproperad $\Co \to \Co \bc_c \Co$ , $\Po$ a dg properad $\Po \bc_c \Po \to \Po$.

\begin{thm}[Merkulov-V.]$\ $

\begin{itemize}
\item[$\triangleright$] $\Hom(\Co, \Po):=\lbrace \Hom(\Co(m,n),\Po(m,n) \rbrace_{m,n}$ is a dg properad : \textbf{convolution properad}

\item[$\triangleright$] $(\Hom_\Sy(\Co, \Po), D, [\, ,\,])$ is a dg Lie algebra
\end{itemize}
\end{thm}

$\Hom_\Sy(\Co, \Po)$ is a ``symmetric non-differential $B_\infty$-algebra'' :

$\rightarrow$ non-symmetric operations $\lbrace x_1, \ldots, x_k\rbrace \lbrace y_1,\ldots , y_l \rbrace$.

\sip

Solution of the Maurer-Cartan equation : \textbf{twisting morphism} $\textrm{Tw}(\Co, \Po)$.
\end{frame}

\begin{frame}

\begin{thm}[Merkulov-V., V.]
$$\Hom_{\textrm{dg properads}}(\Omega(\Co), \Po) \cong \textrm{Tw}(\Co, \Po) \cong
\Hom_{\textrm{dg coproperads}}(\Co, B(\Po))$$
\end{thm}

\begin{proo}
\emph{Cobar construction of a coproperad} :
$\Omega(\Co):=(\F(s^{-1}\bar{\Co}), d)$, where

\begin{itemize}
\item $\F$ : free properad (graphs)

\item $s^{-1}$ : homological desuspension

\item $\bar{\Co}$ : augmentation coideal

\end{itemize}
 $d$  : unique coderivation which extends
$$\vcenter{\xymatrix@R=6pt@C=6pt{ &amp;&amp; \\ &amp; *+[F-,]{\ {\Co} \ }
\ar@{-}[ul] \ar@{-}[ur]\ar@{-}[u]\ar@{-}[dr] \ar@{-}[d]  \ar@{-}[dl] &amp; \\
&amp;&amp;}}   \xrightarrow{\Delta'}
\vcenter{\xymatrix@R=4pt@C=4pt{&amp;&amp;&amp;&amp;\\
&amp; &amp; &amp; *+[F-,]{\ {\Co} \ } \ar@{-}[u] \ar@{-}[dr] \ar@{=}[ddll] \ar@{-}[ur] &amp; \\
&amp;&amp;&amp;&amp; \\
&amp; *+[F-,]{\ {\Co} \ }  \ar@{-}[ul] \ar@{-}[dr] \ar@{-}[dl] &amp; &amp; &amp;
\\ &amp;&amp;&amp;&amp;}}
$$
\end{proo}
\end{frame}

\begin{frame}
\begin{thm}[V.]
Canonical bar-cobar resolution (counit of adjunction)

$$\Omega(B(\Po))\xrightarrow{\sim}
\Po$$
\end{thm}

\begin{dei}

\begin{itemize}

\item \emph{Quadratic resolution} : $\Po_\infty=\Omega(\Co)\xrightarrow{\sim}
\Po$

\item \emph{Homotopy $\Po$-algebra} : $\Po_\infty$-algebra
\end{itemize}


\end{dei}
\end{frame}



\begin{frame}
{\Large \bf Deformation dg Lie algebra}

\footnotesize

$$\xymatrix@R=18pt@C=18pt{\Po_\infty=\Omega(\Co) \ar[r]^(0.65){\sim} \ar[dr]&amp; \Po \ar@{..&gt;}[d]\\
&amp; \End_A}$$

$$\boxed{\mathfrak{g}(\Po, A):=(\Hom_\Sy(\Co, \End_A), D, [\, , \, ])}$$


$\textrm{MC}(\mathfrak{g}(\Po, A))=\textrm{Tw}(\Co, \End_A)$ $\longleftrightarrow$ homotopy $\Po$-algebra structures on $A$

{\sc Particular case : } $\alpha : \Co_0 \to \End_A$, $\alpha(\Co_n)=0$ $\longleftrightarrow$ strict $\Po$-algebra structures on $A$


{\bf Example :}
$\Po=\Ass$, $\Co=\Ass^*$

$$\Delta'\left(\vcenter{\xymatrix@M=0pt@R=10pt@C=10pt{\ar@{-}[drr] &amp;\ar@{-}[dr] &amp; &amp;\ar@{-}[dl] &amp;\ar@{-}[dll] \\ &amp; &amp;\ar@{-}[d] &amp; &amp; \\ &amp; &amp; &amp; &amp; }} \right)=\sum_{i+j=n+1 \atop i, j \ge 2} \pm \vcenter{\xymatrix@M=0pt@R=10pt@C=10pt{
&amp;  &amp;  &amp; &amp; \\ \ar@{-}[ddrr]^(0.93){j} &amp; \ar@{-}[d]^(0.19){i}
&amp;\ar@{-}[dl]&amp; &amp; \ar@{-}[ddll]  \\
&amp;  &amp; &amp;  &amp; \\
&amp; &amp;\ar@{-}[d] &amp; &amp; \\
&amp; &amp; }} $$

$$\Longrightarrow \mathfrak{g}(\Ass, A)\cong \End_A \ : \ \textrm{Gerstenhaber dg Lie algebra}$$


\end{frame}


\begin{frame}

\footnotesize






{\bf\Large  Problems: }
\medskip
\begin{itemize}
\item Well defined ?

\item How to make  $\Co$ or $\Po_\infty$ explicit?

\end{itemize}



\end{frame}



\begin{frame}
{\Large \bf Homotopy theory for properads}

\footnotesize



{\bf Problem : } No model category structure on $\Po$-bialgebras over a properad (no coproduct)


\begin{thm}[Merkulov-V.]
$\ $


 The category of dg prop(erad)s has a cofibrantly generated
model category structure\\


$\Longrightarrow$ $\mathfrak{g}(\Po, A)$ well defined in the homotopy category of dg Lie algebras

\end{thm}

\end{frame}



\begin{frame}
\begin{dei}
\emph{Quasi-free properad} $(\F(X), d)$

\begin{itemize}
\item $\F(X)$  free properad


\item $d$ derivation
$\longleftrightarrow$ $d_{|X} \ : X \to \F(X) $
\end{itemize}

$$d_{|X} \ : \ \vcenter{\xymatrix@M=0pt@R=6pt@C=6pt{&amp;&amp; \\ \ar@{-}[dr]  &amp; \ar@{-}[d]&amp; \ar@{-}[dl]  \\
&amp; \bullet \ar@{-}[dl] \ar@{-}[dr] \ar@{-}[d] &amp; \\ &amp; &amp; \\ &amp;&amp;}}
\mapsto  \sum \vcenter{\xymatrix@M=0pt@R=6pt@C=6pt{&amp;&amp;&amp;&amp; \\ \ar@{-}[dr] &amp; &amp; \ar@{-}[dl] &amp; \ar@{-}[d] &amp; \\
&amp;\bullet \ar@{-}[ddrr]   \ar@{-}@/_/[dd]  &amp;&amp;\bullet
\ar@{-}[ddll]\ar@{-}@/^/[dd]  &amp;
\\ &amp;&amp;&amp;&amp;
\\  &amp;\bullet \ar@{-}[d] &amp;&amp;\bullet \ar@{-}[dl]  \ar@{-}[dr]  &amp;
\\&amp;&amp;&amp;&amp;
\\&amp;&amp;&amp;&amp; }}
\quad : \quad \textbf{not only two vertices}    $$
\end{dei}


\begin{thm}[Merkulov-V.]
$\ $

\begin{itemize}
\item Cofibrant properads = rectract of quasi-free properads

$\Longrightarrow$ Quasi-free properads are cofibrant

\item Any properad admits a quasi-free resolution
\end{itemize}

\end{thm}


\end{frame}

\begin{frame}
{\Large \bf Explicit quasi-free resolution}


\footnotesize



$\bullet$ {\bf \large Koszul duality theory}

\sip

Associative algebras [Priddy], Operads
[Ginzburg-Kapranov], Properads [V.])


When $\Po=\F(V)/(R)$ with $R\subset \F(V)^{(2)}$ : quadratic relations

\begin{itemize}

\item $\Co={\Po^!}^*=$, with  $\Po^!:=\F(V^*)/(R^\perp)$

\item $d$ $\longleftrightarrow$ $\Delta'_\Co=\gamma_{\Po^{\ac}}^t$

\item small criterion
\end{itemize}


{\bf Examples }

\begin{itemize}
\item $\Po=\Ass$, $\Co=\Ass^*$ :  Hochschild cohomology of associative algebras



\item $\Po=\Li$, $\Co=\Com^*$ : Chevalley-Eilenberg cohomology of Lie algebras


\item $\Po=\Com$, $\Co=\Li^*$  : Harrison cohomology of commutative algebras


\item $\Po=\BLi$, $\Co=\Frob_\Diamond^*$
 : Lecomte-Roger cohomology of Lie bialgebras
\end{itemize}

\end{frame}


\begin{frame}
\footnotesize

$\bullet$ {\bf \large Linear Koszul duality theory} {  [Galvez-Tonks-V.]}

\sip
When $\Po=\F(V)/(R)$ with $R\subset \F(V)^{(2)}\oplus \F(V)^{(1)}$ : quadratic and linear relations


{\bf Example : } Batalin-Vilkovisky algebras

$$
\underbrace{\vcenter{\xymatrix@M=2pt@R=8pt@C=8pt{
 \ar@{-}[dr] &amp; &amp;\ar@{-}[dl]   \\
 &amp;  {\scriptstyle \bullet }     \ar@{-}[d] &amp;  \\
&amp; {\scriptstyle \Delta} &amp; }} -
\vcenter{\xymatrix@M=2pt@R=8pt@C=8pt{
  \ar@{-}[dr]{ \scriptstyle \Delta} &amp; &amp;\ar@{-}[dl]   \\
 &amp;  {\scriptstyle \bullet }     \ar@{-}[d] &amp;  \\
&amp;  &amp; }} -
\vcenter{\xymatrix@M=2pt@R=8pt@C=8pt{
  \ar@{-}[dr]&amp; &amp; {\scriptstyle \Delta} \ar@{-}[dl]   \\
 &amp;  {\scriptstyle \bullet }     \ar@{-}[d] &amp;  \\
&amp;  &amp; }}}_{\bf 2}  =
\underbrace{\vcenter{\xymatrix@M=2pt@R=8pt@C=8pt{
 \ar@{-}[dr] &amp; &amp;\ar@{-}[dl]   \\
 &amp;  {\scriptstyle [\, , \, ]}     \ar@{-}[d] &amp;  \\
&amp; &amp; }}}_{\bf 1}
$$

\end{frame}


\footnotesize

\begin{frame}
\footnotesize
$\bullet$ {\bf \large Homotopy Koszul}
\sip

$(\F(\Co_\infty),d)$ quasi-free properad $\Longrightarrow$ $\Co_\infty$ is a {\bf homotopy coproperad} $\Co_\infty \to \F(\Co_\infty)$

\sip
When $\Po=\F(V)/(R)$ with $R\subset \F(V)^{(\ge 2)}$


\sip

{\bf Example :} $\Po=\Bi$
\sip

\begin{thm}[Merkulov-V.]
$\mathfrak{g}(\Bi, A) \cong $ Gerstenhaber-Shack bicomplex
\end{thm}



\end{frame}


\begin{frame}
{\Large \bf Deformation $L_\infty$-algebra}


\footnotesize


$\xymatrix@R=18pt@C=18pt{\Po_\infty=\F(\Co_\infty) \ar[r]^(0.65){\sim} \ar[dr]&amp; \Po \ar@{..&gt;}[d]\\
&amp; \End_A}$

$$\mathfrak{g}(\Po, A):=(\Hom_\Sy(\Co_\infty, \End_A), D, \lbrace l_n\rbrace_n) \ :  \ L_\infty-\textbf{algebra}$$

Well defined in the homotopy category of $L_\infty$-algebras



{\bf Example :}
$\Po=\Bi$

\begin{thm}[Merkulov-V.]
$L_\infty$-algebra structure on Gerstenhaber-Shack bicomplex
\end{thm}

\end{frame}


\begin{frame}
{\Large \bf Deformation theory of morphisms of properads}


\footnotesize



$\xymatrix@R=18pt@C=18pt{\Po_\infty=\F(\Co_\infty) \ar[r]^(0.65){\sim} \ar[dr]&amp; \Po \ar@{..&gt;}[d]\\
&amp; \Qo}$


$$\mathfrak{g}(\Po, \Qo):=(\Hom_\Sy(\Co_\infty, \Qo), D, \lbrace l_n\rbrace_n) \ :  \ L_\infty-\textbf{algebra}$$


Extension of Quillen-Illusie deformation theory of morphisms
\begin{eqnarray*}
\Der(\Po_\infty, \Qo)\cong\Hom_\Sy(\Co_\infty, \Qo)&amp;\cong &amp; \Hom_{\Po-\textrm{bimodules}}(\underbrace{\mathbb{L}_{\Po_\infty}}_{\textbf{Cotangent complex}}, \Qo)
 \end{eqnarray*}

based on the module of K\"ahler
differentials of the properad $\Po$.




{\bf Example :}
$\Po=\Ass$, $\Qo=\mathcal{P}oisson$ :  Invariant of knots [Vassiliev, Turchine]


\end{frame}


\end{document}


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