\begin{theorem}\label{thm:main}
    The properad $\mathscr{Y}^{(n)}_{prop}$ is Koszul contractible, and therefore it is a Koszul properad.
\end{theorem}

In this note we work over a field $\kk$ of characteristic zero, and everything is in $\kk$-vector spaces. A \emph{properad} is an object that encodes a type of algebra whose operations have multiple inputs and outputs; one allows composition along any connected graph, and imposes relations that involve compositions given by graphs of any genera. In a \emph{dioperad}, one only records the data of compositions along trees, and is constrained to impose relations that only involve compositions along trees. A \emph{planar dioperad} is an even more restrictive notion, where one requires these trees to be embedded in the plane in a coherent manner. There are functors
\[ \PDiop \xrightarrow{\Phi} \Diop \xrightarrow{F} \mathsf{Properads}\]
that, starting from a planar dioperad, freely adjoin compositions along any connected graphs, quotienting out by planar genus zero relations.

If $\mathscr{Q}$ is a quadratic dioperad that is Koszul (as a dioperad), it is not true in general that $F\mathscr{Q}$ will be Koszul as a properad; this is notably the case with non-commutative Frobenius algebras \cite{merkulov2009deformation} and $\scrV^{(n)}$-algebras \cite{poirier2019koszuality}. There is, however, a class of dioperads for which dioperadic Koszulity implies properadic Koszulity; this happens for a quadratic dioperad $\mathscr{Q}$ when the Koszul dual coproperad of $F\mathscr{Q}$ is in fact a codioperad, that is, its decomposition map does not generate any higher genus terms; in this case, one says that the properad is \emph{contractible}.

The properad $\scrY^{(n)}$ was defined in \cite{emprin2025properadic}, and one of its uses is to encode the data of an orientation on a space $X$ into the chains on the based loop space. This properad is defined by planar genus zero relations, that is, it satisfies
\[ \mathscr{Y}^{(n)}_{prop} = F \mathscr{Y}^{(n)} = F\Phi(\mathscr{Y}^{(n)}_{pl})\]
for a certain planar dioperad $\mathscr{Y}^{(n)}_{pl}$. An algebra $A$ over this planar dioperad has
\begin{itemize}
    \item A product $\mu \colon A \otimes A \to A$ of degree zero,
    \item A coproduct $\alpha \colon A \to A \otimes A$ of homological degree $(1-n)$,
    \item A cyclically symmetric rank three tensor $\beta \in A \otimes A \otimes A$ of homological degree $(2-2n)$,
\end{itemize}
satisfying the following relations.

\small\begin{align*}
    &\tikzfig{
        \node [vertex] (bot) at (0,-0.2) {$\mu$};
        \node [vertex] (top) at (-0.5,0.5) {$\mu$};
        \draw [->-] (-1,1) to (top);
        \draw [->-] (0,1) to (top);
        \draw [->-] (top) to (bot);
        \draw [->-] (1,1) to (bot);
        \draw [->-] (bot) to (0,-1);
    } ~ - ~ \tikzfig{
        \node [vertex] (bot) at (0,-0.2) {$\mu$};
        \node [vertex] (top) at (0.5,0.5) {$\mu$};
        \draw [->-] (-1,1) to (bot);
        \draw [->-] (0,1) to (top);
        \draw [->-] (top) to (bot);
        \draw [->-] (1,1) to (top);
        \draw [->-] (bot) to (0,-1);
    } ~ \quad ; \quad \tikzfig{
        \node [vertex] (top) at (0,0.5) {$\mu$};
        \node [vertex] (bot) at (0,-0.5) {$\alpha$};
        \draw [->-] (-1,1) to (top);
        \draw [->-] (top) to (bot);
        \draw [->-] (1,1) to (top);
        \draw [->-] (bot) to (-1,-1);
        \draw [->-] (bot) to (1,-1);
    } ~ - ~ \tikzfig{
        \node [vertex] (top) at (0.5,0) {$\alpha$};
        \node [vertex] (bot) at (-0.5,0) {$\mu$};
        \draw [->-] (-1,1) to (bot);
        \draw [->-] (top) to (bot);
        \draw [->-] (1,1) to (top);
        \draw [->-] (bot) to (-1,-1);
        \draw [->-] (top) to (1,-1);
    } ~ - ~ \tikzfig{
        \node [vertex] (top) at (-0.5,0) {$\alpha$};
        \node [vertex] (bot) at (0.5,0) {$\mu$};
        \draw [->-] (-1,1) to (top);
        \draw [->-] (top) to (bot);
        \draw [->-] (1,1) to (bot);
        \draw [->-] (top) to (-1,-1);
        \draw [->-] (bot) to (1,-1);
    } ~;\\
    &\tikzfig{
        \node [vertex] (left) at (-0.5,0) {$\alpha$};
        \node [vertex] (right) at (0.5,0) {$\mu$};
        \draw [->-] (0,1) to (left);
        \draw [->-] (0,-1) to (right);
        \draw [->-] (left) to (-1.2,0);
        \draw [->-] (left) to (right);
        \draw [->-] (right) to (1.2,0);
    } ~ - ~ \tikzfig{
        \node [vertex] (left) at (-0.5,0) {$\mu$};
        \node [vertex] (right) at (0.5,0) {$\alpha$};
        \draw [->-] (0,1) to (right);
        \draw [->-] (0,-1) to (left);
        \draw [->-] (left) to (-1.2,0);
        \draw [->-] (right) to (left);
        \draw [->-] (right) to (1.2,0);
    } ~ + (-1)^n \left(\tikzfig{
        \node [vertex] (left) at (-0.5,0) {$\alpha$};
        \node [vertex] (right) at (0.5,0) {$\mu$};
        \draw [->-] (0,1) to (right);
        \draw [->-] (0,-1) to (left);
        \draw [->-] (left) to (-1.2,0);
        \draw [->-] (left) to (right);
        \draw [->-] (right) to (1.2,0);
    } ~ - ~ \tikzfig{
        \node [vertex] (left) at (-0.5,0) {$\mu$};
        \node [vertex] (right) at (0.5,0) {$\alpha$};
        \draw [->-] (0,1) to (left);
        \draw [->-] (0,-1) to (right);
        \draw [->-] (left) to (-1.2,0);
        \draw [->-] (right) to (left);
        \draw [->-] (right) to (1.2,0);
    }\right) ~;
\end{align*}
\begin{align*}
    &\tikzfig{
        \node [vertex] (top) at (0,0.4) {$\mu$};
        \node [vertex] (bot) at (0,-0.6) {$\beta$};
        \draw [->-] (bot) to (top);
        \draw [->-] (bot) to (-1.2,-1);
        \draw [->-] (bot) to (1.2,-1);
        \draw [->-] (-1.2,0.4) to (top);
        \draw [->-] (top) to (0,1);
    } ~ - \qquad \tikzfig{
        \node [vertex] (top) at (0,0.1) {$\beta$};
        \node [vertex] (bot) at (-0.7,-0.5) {$\mu$};
        \draw [->-] (top) to (0,1);
        \draw [->-] (bot) to (-1.2,-1);
        \draw [->-] (top) to (1.2,-1);
        \draw [->-] (-1.2,0.4) to (bot);
        \draw [->-] (top) to (bot);
    } ~ + ~ \tikzfig{
        \node [vertex] (top) at (0,0.4) {$\alpha$};
        \node [vertex] (bot) at (0,-0.6) {$\alpha$};
        \draw [->-] (top) to (bot);
        \draw [->-] (bot) to (-1.2,-1);
        \draw [->-] (bot) to (1.2,-1);
        \draw [->-] (-1.2,0.4) to (top);
        \draw [->-] (top) to (0,1);
    } ~ - (-1)^{n-1} ~ \tikzfig{
        \node [vertex] (top) at (0,0.1) {$\alpha$};
        \node [vertex] (bot) at (-0.7,-0.5) {$\alpha$};
        \draw [->-] (top) to (0,1);
        \draw [->-] (bot) to (-1.2,-1);
        \draw [->-] (top) to (1.2,-1);
        \draw [->-] (-1.2,0.4) to (bot);
        \draw [->-] (bot) to (top);
    } ~; \\ \\
    &\tikzfig{
        \node [vertex] (top) at (0,0.5) {$\alpha$};
        \node [vertex] (bot) at (0,-0.5) {$\beta$};
        \draw [->-] (bot) to (top);
        \draw [->-] (top) to (-1,1);
        \draw [->-] (top) to (1,1);
        \draw [->-] (bot) to (-1,-1);
        \draw [->-] (bot) to (1,-1);
    } ~ - ~ \tikzfig{
        \node [vertex] (left) at (-0.5,0) {$\alpha$};
        \node [vertex] (right) at (0.5,0) {$\beta$};
        \draw [->-] (right) to (left);
        \draw [->-] (left) to (-1,1);
        \draw [->-] (left) to (-1,-1);
        \draw [->-] (right) to (1,-1);
        \draw [->-] (right) to (1,1);
    } ~ + (-1)^n \left( ~ \tikzfig{
        \node [vertex] (top) at (0,0.5) {$\beta$};
        \node [vertex] (bot) at (0,-0.5) {$\alpha$};
        \draw [->-] (top) to (bot);
        \draw [->-] (top) to (-1,1);
        \draw [->-] (top) to (1,1);
        \draw [->-] (bot) to (-1,-1);
        \draw [->-] (bot) to (1,-1);
    } ~ - ~ \tikzfig{
        \node [vertex] (left) at (-0.5,0) {$\beta$};
        \node [vertex] (right) at (0.5,0) {$\alpha$};
        \draw [->-] (right) to (left);
        \draw [->-] (left) to (-1,1);
        \draw [->-] (left) to (-1,-1);
        \draw [->-] (right) to (1,-1);
        \draw [->-] (right) to (1,1);
    } ~ \right)
\end{align*}\normalsize

Note that all of these relations, and the cyclicity condition on $\beta$, can be given by linear combinations of diagrams embedded in a disc with a consistent labeling of inputs and outputs, using only the cyclic action, which comes from the rotational symmetry of the disc; this is why this dioperad is planar, see \cref{sec:planar}. In this note we establish the following theorem.

The properad $\scrY^{(n)}$ was defined in \cite{emprin2025properadic}, and one of its uses is to encode the data of an orientation on a space $X$ into the chains on the based loop space. This properad is defined by planar genus zero relations, that is, it satisfies
\[ \mathscr{Y}^{(n)}_{prop} = F \mathscr{Y}^{(n)} = F\Phi(\mathscr{Y}^{(n)}_{pl})\]
for a certain planar dioperad $\mathscr{Y}^{(n)}_{pl}$. An algebra $A$ over this planar dioperad has
\begin{itemize}
 \item A product $\mu \colon A \otimes A \to A$ of degree zero,
 \item A coproduct $\alpha \colon A \to A \otimes A$ of homological degree $(1-n)$,
 \item A cyclically symmetric rank three tensor $\beta \in A \otimes A \otimes A$ of homological degree $(2-2n)$,
\end{itemize}
satisfying the following relations.

\small\begin{align*}
 &\tikzfig{
 \node [vertex] (bot) at (0,-0.2) {$\mu$};
 \node [vertex] (top) at (-0.5,0.5) {$\mu$};
 \draw [->-] (-1,1) to (top);
 \draw [->-] (0,1) to (top);
 \draw [->-] (top) to (bot);
 \draw [->-] (1,1) to (bot);
 \draw [->-] (bot) to (0,-1);
 } ~ - ~ \tikzfig{
 \node [vertex] (bot) at (0,-0.2) {$\mu$};
 \node [vertex] (top) at (0.5,0.5) {$\mu$};
 \draw [->-] (-1,1) to (bot);
 \draw [->-] (0,1) to (top);
 \draw [->-] (top) to (bot);
 \draw [->-] (1,1) to (top);
 \draw [->-] (bot) to (0,-1);
 } ~ \quad ; \quad \tikzfig{
 \node [vertex] (top) at (0,0.5) {$\mu$};
 \node [vertex] (bot) at (0,-0.5) {$\alpha$};
 \draw [->-] (-1,1) to (top);
 \draw [->-] (top) to (bot);
 \draw [->-] (1,1) to (top);
 \draw [->-] (bot) to (-1,-1);
 \draw [->-] (bot) to (1,-1);
 } ~ - ~ \tikzfig{
 \node [vertex] (top) at (0.5,0) {$\alpha$};
 \node [vertex] (bot) at (-0.5,0) {$\mu$};
 \draw [->-] (-1,1) to (bot);
 \draw [->-] (top) to (bot);
 \draw [->-] (1,1) to (top);
 \draw [->-] (bot) to (-1,-1);
 \draw [->-] (top) to (1,-1);
 } ~ - ~ \tikzfig{
 \node [vertex] (top) at (-0.5,0) {$\alpha$};
 \node [vertex] (bot) at (0.5,0) {$\mu$};
 \draw [->-] (-1,1) to (top);
 \draw [->-] (top) to (bot);
 \draw [->-] (1,1) to (bot);
 \draw [->-] (top) to (-1,-1);
 \draw [->-] (bot) to (1,-1);
 } ~;\\
 &\tikzfig{
 \node [vertex] (left) at (-0.5,0) {$\alpha$};
 \node [vertex] (right) at (0.5,0) {$\mu$};
 \draw [->-] (0,1) to (left);
 \draw [->-] (0,-1) to (right);
 \draw [->-] (left) to (-1.2,0);
 \draw [->-] (left) to (right);
 \draw [->-] (right) to (1.2,0);
 } ~ - ~ \tikzfig{
 \node [vertex] (left) at (-0.5,0) {$\mu$};
 \node [vertex] (right) at (0.5,0) {$\alpha$};
 \draw [->-] (0,1) to (right);
 \draw [->-] (0,-1) to (left);
 \draw [->-] (left) to (-1.2,0);
 \draw [->-] (right) to (left);
 \draw [->-] (right) to (1.2,0);
 } ~ + (-1)^n \left(\tikzfig{
 \node [vertex] (left) at (-0.5,0) {$\alpha$};
 \node [vertex] (right) at (0.5,0) {$\mu$};
 \draw [->-] (0,1) to (right);
 \draw [->-] (0,-1) to (left);
 \draw [->-] (left) to (-1.2,0);
 \draw [->-] (left) to (right);
 \draw [->-] (right) to (1.2,0);
 } ~ - ~ \tikzfig{
 \node [vertex] (left) at (-0.5,0) {$\mu$};
 \node [vertex] (right) at (0.5,0) {$\alpha$};
 \draw [->-] (0,1) to (left);
 \draw [->-] (0,-1) to (right);
 \draw [->-] (left) to (-1.2,0);
 \draw [->-] (right) to (left);
 \draw [->-] (right) to (1.2,0);
 }\right) ~;
\end{align*}
\begin{align*}
 &\tikzfig{
 \node [vertex] (top) at (0,0.4) {$\mu$};
 \node [vertex] (bot) at (0,-0.6) {$\beta$};
 \draw [->-] (bot) to (top);
 \draw [->-] (bot) to (-1.2,-1);
 \draw [->-] (bot) to (1.2,-1);
 \draw [->-] (-1.2,0.4) to (top);
 \draw [->-] (top) to (0,1);
 } ~ - \qquad \tikzfig{
 \node [vertex] (top) at (0,0.1) {$\beta$};
 \node [vertex] (bot) at (-0.7,-0.5) {$\mu$};
 \draw [->-] (top) to (0,1);
 \draw [->-] (bot) to (-1.2,-1);
 \draw [->-] (top) to (1.2,-1);
 \draw [->-] (-1.2,0.4) to (bot);
 \draw [->-] (top) to (bot);
 } ~ + ~ \tikzfig{
 \node [vertex] (top) at (0,0.4) {$\alpha$};
 \node [vertex] (bot) at (0,-0.6) {$\alpha$};
 \draw [->-] (top) to (bot);
 \draw [->-] (bot) to (-1.2,-1);
 \draw [->-] (bot) to (1.2,-1);
 \draw [->-] (-1.2,0.4) to (top);
 \draw [->-] (top) to (0,1);
 } ~ - (-1)^{n-1} ~ \tikzfig{
 \node [vertex] (top) at (0,0.1) {$\alpha$};
 \node [vertex] (bot) at (-0.7,-0.5) {$\alpha$};
 \draw [->-] (top) to (0,1);
 \draw [->-] (bot) to (-1.2,-1);
 \draw [->-] (top) to (1.2,-1);
 \draw [->-] (-1.2,0.4) to (bot);
 \draw [->-] (bot) to (top);
 } ~; \\ \\
 &\tikzfig{
 \node [vertex] (top) at (0,0.5) {$\alpha$};
 \node [vertex] (bot) at (0,-0.5) {$\beta$};
 \draw [->-] (bot) to (top);
 \draw [->-] (top) to (-1,1);
 \draw [->-] (top) to (1,1);
 \draw [->-] (bot) to (-1,-1);
 \draw [->-] (bot) to (1,-1);
 } ~ - ~ \tikzfig{
 \node [vertex] (left) at (-0.5,0) {$\alpha$};
 \node [vertex] (right) at (0.5,0) {$\beta$};
 \draw [->-] (right) to (left);
 \draw [->-] (left) to (-1,1);
 \draw [->-] (left) to (-1,-1);
 \draw [->-] (right) to (1,-1);
 \draw [->-] (right) to (1,1);
 } ~ + (-1)^n \left( ~ \tikzfig{
 \node [vertex] (top) at (0,0.5) {$\beta$};
 \node [vertex] (bot) at (0,-0.5) {$\alpha$};
 \draw [->-] (top) to (bot);
 \draw [->-] (top) to (-1,1);
 \draw [->-] (top) to (1,1);
 \draw [->-] (bot) to (-1,-1);
 \draw [->-] (bot) to (1,-1);
 } ~ - ~ \tikzfig{
 \node [vertex] (left) at (-0.5,0) {$\beta$};
 \node [vertex] (right) at (0.5,0) {$\alpha$};
 \draw [->-] (right) to (left);
 \draw [->-] (left) to (-1,1);
 \draw [->-] (left) to (-1,-1);
 \draw [->-] (right) to (1,-1);
 \draw [->-] (right) to (1,1);
 } ~ \right)
\end{align*}\normalsize

Note that all of these relations, and the cyclicity condition on $\beta$, can be given by linear combinations of diagrams embedded in a disc with a consistent labeling of inputs and outputs, using only the cyclic action, which comes from the rotational symmetry of the disc; this is why this dioperad is planar, see \cref{sec:planar}. In this note we establish the following theorem.

If $\mathscr{Q}$ is a quadratic dioperad that is Koszul (as a dioperad), it is not true in general that $F\mathscr{Q}$ will be Koszul as a properad; this is notably the case with non-commutative Frobenius algebras \cite{merkulov2009deformation} and $\scrV^{(n)}$-algebras \cite{poirier2019koszuality}. There is, however, a class of dioperads for which dioperadic Koszulity implies properadic Koszulity; this happens for a quadratic dioperad $\mathscr{Q}$ when the Koszul dual coproperad of $F\mathscr{Q}$ is in fact a codioperad, that is, its decomposition map does not generate any higher genus terms; in this case, one says that the properad is \emph{contractible}.

It was proven in \cite{emprin2025properadic} that the dioperad $\mathscr{Y}^{(n)}$ is contractible; the other component that allows us to establish Koszulity of the properad $\scrY^{(n)}_{prop}$ is a geometric interpretation of the bar complexes associated to (the Koszul dual of) the planar dioperad $\scrY^{(n)}_{pl}$. This interpretation is in the same spirit as the description of the ``PROP of open-closed marked surfaces'' of \cite{kontsevich2025pre} using moduli spaces of meromorphic Strebel differentials. In \cref{sec:planar} we will recall this theory, in the specific case of differentials on $\mathbb{CP}^1$ with a higher-order pole at infinity, or equivalently, polynomial quadratic differentials on $\CC$. The moduli spaces of such objects has a natural non-compact cell complex structure that is dual to the cell complex structure of the \emph{assocoipahedra} of \cite{poirier2018combinatorics}, and gives an alternative proof of their compatibility, established in that reference; by the results of \cite{poirier2019koszuality} this implies Koszulity of the dioperad $\mathscr{V}^{(n)}$.

As a consequence of \cref{thm:main}, the map
\[ \scrY^{(n)}_\infty = \Omega (\scrY^{(n)}_{prop})^{\antish} \to \scrY^{(n)}_{prop} \]
is a quasi-isomorphism of dg properads and therefore gives a cofibrant resolution of the properad $\scrY^{(n)}_{prop}$; the results on formality in \cite{emprin2025properadic} can then be seen as completely analogous to the formality of $A_\infty$-algebras, but in the properadic setting of $\scrY^{(n)}_{prop}$-algebras. Together with the explicit formulas for homotopy transfer of \cite{hoffbeck2021properadic}, we have as a consequence of our main theorem.
\begin{corollary}
 The homotopy transferred structures for $\scrY^{(n)}_{prop}$-algebras can be computed by formulas only involving sums of planar trees.
\end{corollary}

Therefore, for any $\mathbf{C}$ as above, the subspace
\[ \mathrm{ClovQ}_\mathbf{C} \subset Q^\mathrm{Str}_\mathrm{reg}(k;i_1,\dots,i_k) \]
of regularized Strebel differentials that are cloven by $\mathbf{C}$ is a union of cells, each of which labeled by a directed tree that has at least $r$ bivalent vertices. This union of cells is coclosed, in the sense that the if $a$ is in $\mathrm{ClovQ}_\mathbf{C}$ and is on the boundary of $b$, then so is $b$. So we can write
\[ \mathrm{Clov}_\mathbf{C} \subset Z_{(k;i_1,\dots,i_k)} \]
for the corresponding union of cells in the assocoipahedron, which will be a cell subcomplex. We now give the main proposition of this section.
\begin{proposition}\label{prop:contractible}
 For any collection of pairwise distinct equivalence classes
 \[ \mathbf{C} = \{C_1,\dots,C_r\}, \]
 the space $\mathrm{Clov}_\mathbf{C}$ is contractible if and only if there are $r$ cuts $\gamma_1,\dots,\gamma_r$, one in each equivalence class, that are pairwise disjoint and divide the complex plane into regions such that each contains at least one of the $k$ output directions. In every other case, $\mathrm{Clov}_\mathbf{C}$ is empty.
\end{proposition}
\begin{proof}
 Let us start by proving the contrapositive of the last statement. If $\mathrm{Clov}_\mathbf{C}$ is not empty, $\mathrm{ClovQ}_\mathbf{C}$ must contain at least one point; taking the $r$ saddle cuts gives us the desired representatives $\gamma_1,\dots,\gamma_r$.

\begin{proposition}\label{prop:barcomplexKoszul}
 The complex $(\mathbf{B}^*_{pl} (\scrY^{(n)}_{pl})^!)_{(k;i_1,\dots,i_k)}$ has cohomology concentrated in degree zero.
\end{proposition}
\begin{proof}
 For ease of notation, let us write
 \[ Y^* = (\mathbf{B}^*_{pl} (\mathscr{Y}^{(n)}_{pl})^!)_{(k;i_1,\dots,i_k)}, \quad Z^* = (\mathbf{B}^*_{pl} (\mathscr{V}^{(n)}_{pl})^!)_{(k;i_1,\dots,i_k)} \cong C^*(Z_{(k;i_1,\dots,i_k)},\ZZ) \]
 By definition, we have an exact sequence
 \[ Y^{*-k+1} \hookrightarrow Z^* \twoheadrightarrow R^* \]
 and the result then follows from the induced long exact sequence in cohomology, together with the vanishing results above, and contractibility of $Z^*$.
\end{proof}

\subsection{Dioperadic and properadic Koszulity}
The contractibility of these planar bar complexes immediately implies dioperadic Koszulity.
\begin{corollary}\label{cor:diopKoszul}
 The dioperad $\mathscr{Y}^{(n)} = \Phi(\scrY^{(n)}_{pl})$ is a Koszul dioperad.
\end{corollary}
\begin{proof}
 From \cref{prop:KoszulDiop,prop:barcomplexKoszul} it follows that $(\scrY^{(n)})^! \cong \Phi((\scrY^{(n)}_{pl})^!)$ is a Koszul dioperad; since $\scrY^{(n)}$ is finitely generated, we conclude that $\scrY^{(n)} \cong (\scrY^{(n)})^{!!}$ is also a Koszul dioperad.
\end{proof}

The functor $F$ \emph{does not} intertwine dioperadic and properadic Koszul duality. That is, for an arbitrary finitely-generated quadratic dioperad $\mathscr{Q}$, the properads
\[ F(\mathscr{Q}^!) \quad \text{and} \quad (F \mathscr{Q})^! \]
need not be isomorphic, even if $\mathscr{Q}$ is Koszul; in other words, the Koszul dual properad $(F \mathscr{Q})^!$ might have nontrivial higher-genus relations, or dually, the decomposition maps of the Koszul dual coproperad $(F \mathscr{Q})^{\antish}$ is not a codioperad, that is, if its decomposition map generates higher-genus graphs from genus-zero ones.
\begin{definition}
 A quadratic dioperad $\mathscr{Q}$ is called \emph{contractible} when $(F \mathscr{Q})^{\antish}$ is a codioperad.
\end{definition}
If $\mathscr{Q}$ is a finitely-generated quadratic dioperad, then it is contractible if and only if the canonical map $F(\mathscr{Q}^!) \to (F \mathscr{Q})^!$ is an isomorphism of properads. Contractibility means that one can deduce Koszulity of the properad from Koszulity of the dioperad.
\begin{proposition}
 If a quadratic dioperad $\mathscr{Q}$ is Koszul (as a dioperad) and contractible, then the properad $F\mathscr{Q}$ is Koszul (as a properad).
\end{proposition}
In the case the proposition above holds, one says that the properad $F\mathscr{Q}$ is \emph{Koszul contractible}. Together with \cite[Thm.1.23]{emprin2025properadic}, which shows that $\scrY^{(n)}$ is a contractible dioperad, and \cref{cor:diopKoszul}, we deduce the main theorem of this paper.
\begin{theorem*}(\cref{thm:main})
 The properad $\scrY^{(n)}_{\mathrm{prop}} = F(\scrY^{(n)})$ is Koszul contractible, and therefore a Koszul properad.
\end{theorem*}

It was proven in \cite{emprin2025properadic} that the dioperad $\mathscr{Y}^{(n)}$ is contractible; the other component that allows us to establish Koszulity of the properad $\scrY^{(n)}_{prop}$ is a geometric interpretation of the bar complexes associated to (the Koszul dual of) the planar dioperad $\scrY^{(n)}_{pl}$. This interpretation is in the same spirit as the description of the ``PROP of open-closed marked surfaces'' of \cite{kontsevich2025pre} using moduli spaces of meromorphic Strebel differentials. In \cref{sec:planar} we will recall this theory, in the specific case of differentials on $\mathbb{CP}^1$ with a higher-order pole at infinity, or equivalently, polynomial quadratic differentials on $\CC$. The moduli spaces of such objects has a natural non-compact cell complex structure that is dual to the cell complex structure of the \emph{assocoipahedra} of \cite{poirier2018combinatorics}, and gives an alternative proof of their compatibility, established in that reference; by the results of \cite{poirier2019koszuality} this implies Koszulity of the dioperad $\mathscr{V}^{(n)}$.

In the $\mathscr{Y}^{(n)}$ case, one needs to remove a certain subcomplex of \emph{cloven Strebel differentials}; these are quadratic differentials that split the complex plane into regions in a specific way. Using geometric arguments, we can control the topology of this subcomplex, concluding that it is always homologous to a bouquet of $(k-2)$-spheres, where $k$ is the number of outputs we are looking at. Together with a comparison between the bar complexes associated to $\mathscr{V}^{(n)}$ and $\mathscr{Y}^{(n)}$, vanishing of this homology outside of that specific degree implies vanishing of higher cohomology of the relevant bar complex, establishing our main theorem.

As a consequence of \cref{thm:main}, the map
\[ \scrY^{(n)}_\infty = \Omega (\scrY^{(n)}_{prop})^{\antish} \to \scrY^{(n)}_{prop} \]
is a quasi-isomorphism of dg properads and therefore gives a cofibrant resolution of the properad $\scrY^{(n)}_{prop}$; the results on formality in \cite{emprin2025properadic} can then be seen as completely analogous to the formality of $A_\infty$-algebras, but in the properadic setting of $\scrY^{(n)}_{prop}$-algebras. Together with the explicit formulas for homotopy transfer of \cite{hoffbeck2021properadic}, we have as a consequence of our main theorem.
\begin{corollary}
    The homotopy transferred structures for $\scrY^{(n)}_{prop}$-algebras can be computed by formulas only involving sums of planar trees.
\end{corollary}

\subsection{Relation to other known results}
We would like to point out that there are three other closely related dioperads for which Koszulity has been studied. The first one is the dioperad $\scrV^{(n)}$ of Poirier and Tradler, following earlier work of Tradler and Zeinalian \cite{tradler2007infinity}; algebras over this dioperad are pre-Calabi--Yau algebras, in the formulation of \cite{kontsevich2025pre}. We cite results of \cite{poirier2019koszuality}.
\begin{theorem*}
    The dioperad $\scrV^{(n)}$ is Koszul (as a dioperad) but not contractible.
\end{theorem*}

Another one is the dioperad of balanced infinitesimal bialgebras $\mathrm{BIB}^\lambda$ used by Quesney in \cite{quesney2024balanced}; the algebras that this dioperad encodes are associative analogues of Lie bialgebras, and have a coproduct of degree $\lambda$. It was proved in that paper that this dioperad is contractible, but using deformation theory techniques, Merkulov showed in \cite{merkulov2025complex} the following result.
\begin{theorem*}
    The properad $F(\mathrm{BIB}^\lambda)$ is not Koszul, so as a consequence the dioperad $\mathrm{BIB}^\lambda$ cannot be Koszul.
\end{theorem*}

Finally, the third dioperad that I would like to mention is the dioperad $\mathrm{DPois}$ encoding the double Poisson algebras of van den Bergh; this dioperad was studied by Leray in \cite{leray2019protoperadsI,leray2020protoperadsII}, who showed the following result, see also \cite{leray2025precalabiyau} for a more recent discussion.
\begin{theorem*}
    The dioperad $\mathrm{DPois}$ is Koszul contractible.
\end{theorem*}

The relationship between $\scrY^(n)$ and these three other dioperads was established in \cite{leray2025precalabiyau,quesney2024balanced,emprin2025properadic}. Their Koszul dual dioperads are related by inclusions
\[ \mathrm{DPois}^! \overset{n=2}\hookrightarrow (\mathrm{BIB}^{1-n})^! \hookrightarrow (\scrY^{(n)})^! \hookrightarrow (\scrV^{(n)})^! \]
where the superscript on the first arrow means that this map only exists for that value of $n$.\footnote{There are shifted versions of double Poisson algebras, but the results we cite are only phrased for the unshifted case.} We note that the first and third maps do not respect weight-grading, and do not come from taking Koszul dual maps.

Dually, we have quotients of codioperads
\[ \mathrm{DPois}^{\antish} \overset{n=2}\twoheadleftarrow (\mathrm{BIB}^{1-n})^{\antish} \twoheadleftarrow (\scrY^{(n)})^{\antish} \twoheadleftarrow (\scrV^{(n)})^{\antish} \]
and taking cobar, quotients of dg codioperads
\[ \mathrm{DPois}_\infty \overset{n=2}\twoheadleftarrow \Omega (\mathrm{BIB}^{1-n})^{\antish} \twoheadleftarrow \scrY^{(n)}_\infty \twoheadleftarrow \scrV^{(n)}_\infty = \mathrm{pCY}, \]
so the notion of a  $\scrY^{(n)}_\infty$-algebra is extremely close to the notion of a $\scrV^{(n)}_\infty$ or pre-CY algebra; in fact, it is just a pre-CY algebra with vanishing copairing. When $n \ge 1$, every pre-CY algebra that is concentrated in non-negative homological degree must have vanishing copairing by degree reasons, and will be a $\scrV^{(n)}_\infty$-algebra. We summarize the results cited above in a table for convenience.
\begin{center}
    \begin{tabular}{|c||c|c|c|c|}
    \hline
                            & $\mathrm{DPois}$ & $\mathrm{BIB}^{1-n}$  & $\scrY^{(n)}$ & $\scrV^{(n)}$ \\
    \hline\hline
    Dioperad is Koszul?     & Yes              & No                    & Yes           & Yes \\
    \hline
    Properad is Koszul?     & Yes              & No                    & Yes           & Unknown \\
    \hline
    \end{tabular}
\end{center}

\noindent \textbf{Acknowledgments:} I would like to thank C.~Emprin, S.~Merkulov and B.~Vallette for helpful discussions, and Uppsala University for the great working environment. This work was supported by the Knut and Alice Wallenberg foundation.

It is recalled that \cite{emprin2025properadic} proves the dioperad $\mathscr{Y}^{(n)}$ is contractible; to establish Koszulity (and hence Koszul contractibility) for the properad $\scrY^{(n)}_{prop}$, the other ingredient is “a geometric interpretation of the bar complexes associated to (the Koszul dual of) the planar dioperad $\scrY^{(n)}_{pl}$.” The introduction outlines that one studies moduli spaces of meromorphic Strebel differentials (here specialized to $\mathbb{CP}^1$ with a higher-order pole at infinity / polynomial quadratic differentials on $\CC$), which carry “a natural non-compact cell complex structure” related to assocoipahedra; by \cite{poirier2019koszuality} this type of input yields Koszulity in the $\mathscr{V}^{(n)}$ setting.

For $\mathscr{Y}^{(n)}$, the sketch says one must “remove a certain subcomplex of \emph{cloven Strebel differentials}.” One then uses “geometric arguments” to control its topology, concluding this subcomplex “is always homologous to a bouquet of $(k-2)$-spheres, where $k$ is the number of outputs.” Finally, “together with a comparison between the bar complexes associated to $\mathscr{V}^{(n)}$ and $\mathscr{Y}^{(n)}$,” the fact that this homology vanishes outside that degree implies “vanishing of higher cohomology of the relevant bar complex,” which is said to “establish[] our main theorem” (i.e. Theorem~\ref{thm:main}).

It is recalled that \cite{emprin2025properadic} proves the dioperad $\mathscr{Y}^{(n)}$ is contractible; to establish Koszulity (and hence Koszul contractibility) for the properad $\scrY^{(n)}_{prop}$, the other ingredient is “a geometric interpretation of the bar complexes associated to (the Koszul dual of) the planar dioperad $\scrY^{(n)}_{pl}$.” The introduction outlines that one studies moduli spaces of meromorphic Strebel differentials (here specialized to $\mathbb{CP}^1$ with a higher-order pole at infinity / polynomial quadratic differentials on $\CC$), which carry “a natural non-compact cell complex structure” related to assocoipahedra; by \cite{poirier2019koszuality} this type of input yields Koszulity in the $\mathscr{V}^{(n)}$ setting.

For $\mathscr{Y}^{(n)}$, the sketch says one must “remove a certain subcomplex of \emph{cloven Strebel differentials}.” One then uses “geometric arguments” to control its topology, concluding this subcomplex “is always homologous to a bouquet of $(k-2)$-spheres, where $k$ is the number of outputs.” Finally, “together with a comparison between the bar complexes associated to $\mathscr{V}^{(n)}$ and $\mathscr{Y}^{(n)}$,” the fact that this homology vanishes outside that degree implies “vanishing of higher cohomology of the relevant bar complex,” which is said to establish the main theorem.

Question: Let \(\mathscr{Y}^{(n)}\) be the quadratic dioperad associated to the planar dioperad \(\mathscr{Y}^{(n)}_{pl}\), and let \(\mathscr{Y}^{(n)}_{\mathrm{prop}} = F(\mathscr{Y}^{(n)})\) be the properad obtained from it by the functor \(F\). Here, a properad of the form \(F\mathscr{Q}\) is called \emph{Koszul contractible} when the quadratic dioperad \(\mathscr{Q}\) is both Koszul and contractible; for finitely generated quadratic dioperads, contractibility is equivalent to the canonical map \(F(\mathscr{Q}^!) \to (F\mathscr{Q})^!\) being an isomorphism of properads. Which statement holds for \(\mathscr{Y}^{(n)}_{\mathrm{prop}}\)?

The properad \(\mathscr{Y}^{(n)}_{\mathrm{prop}}\) is Koszul contractible, and therefore it is a Koszul properad.

Choices:
B. The properad \(\mathscr{Y}^{(n)}_{\mathrm{prop}}\) is a Koszul properad, but it is not Koszul contractible because the canonical map \(F((\mathscr{Y}^{(n)})^!) \to (F\mathscr{Y}^{(n)})^!\) need not be an isomorphism of properads.
C. The properad \(\mathscr{Y}^{(n)}_{\mathrm{prop}}\) is a Koszul properad.
D. The properad \(\mathscr{Y}^{(n)}_{\mathrm{prop}}\) is contractible in the sense that the canonical map \(F((\mathscr{Y}^{(n)})^!) \to (F\mathscr{Y}^{(n)})^!\) is an isomorphism of properads, but it is not Koszul contractible because Koszulity of \(\mathscr{Y}^{(n)}\) as a dioperad does not imply Koszulity of \(F(\mathscr{Y}^{(n)})\) as a properad.
E. The properad \(\mathscr{Y}^{(n)}_{\mathrm{prop}}\) is Koszul contractible only after restricting to the genus-zero part; equivalently, the canonical map \(F((\mathscr{Y}^{(n)})^!) \to (F\mathscr{Y}^{(n)})^!\) is an isomorphism merely as a dioperad, not as a properad.

comparison-plus-vanishing needed to upgrade contractibility to properadic Koszul contractibility

drops the contractibility/Koszul-contractible assertion while keeping the properadic Koszul conclusion

proposition that Koszul + contractible dioperad implies \(F\mathscr Q\) is Koszul as a properad

full properadic isomorphism versus genus-zero/dioperadic-only control