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\begin{document}

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\section*{PROP}

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{model_structure_on_simplicial_props}{Model Structure on Simplicial PROPs}\dotfill \pageref*{model_structure_on_simplicial_props} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{bialgebras}{Bialgebras}\dotfill \pageref*{bialgebras} \linebreak
\noindent\hyperlink{endomorphism_prop}{Endomorphism PROP}\dotfill \pageref*{endomorphism_prop} \linebreak
\noindent\hyperlink{relation_to_polycategories}{Relation to Polycategories}\dotfill \pageref*{relation_to_polycategories} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A `PROP' --- an abbreviation of `products and permutations category' --- is a [[symmetric monoidal category]] generated by a single [[object]], used to describe a given sort of algebraic structure. One can think of PROPs as a variant of [[Lawvere theory|Lawvere theories]] suitable for non[[cartesian monoidal category|cartesian]] contexts. In this respect they are similar to [[operad|operads]]. However, they are more general, because they can be used to describe operations with many outputs as well as many inputs.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{PROP} is a strict [[symmetric monoidal category]] where every object is of the form

\begin{displaymath}
x^{\otimes n} = x \otimes x \otimes \cdots \otimes x
\end{displaymath}
for a single object $x$ and $n \ge 0$.

\end{defn}
There is also a notion of colored PROP akin to [[colored operads]]. One way to define a colored PROP is as a certain kind of [[symmetric monoidal category]] (see Remark 2.2.14 of \hyperlink{Yau}{Yau}):

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{colored PROP} with set of colors $\mathfrak{C}$ is a strict symmetric monoidal category $(P,\odot)$ whose [[monoid]] of objects is freely generated by $\mathfrak{C}$.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
A \textbf{morphism of PROPs} $\phi:Q\to T$ is just a strict symmetric monoidal functor that takes the generators for the objects of $Q$ to the generators of the objects of $T$. Equivalently, a morphism of PROPs is a pair of functions: one from the colors of $Q$ to the colors of $T$, $\phi_c\colon Col(Q)\to Col(T)$, and for each pair of finite list of colors $\vec{x}=\{x_1,\ldots,x_n\}$ and $\vec{y}=\{y_1,\ldots,y_m\}$ in $Col(Q)$, a function $\phi_m\colon Q(\vec{x},\vec{y})\to T(\phi_c(\vec{x}),\phi_c(\vec{y}))$. The two functions $\phi_c$ and $\phi_m$ are of course required to preserve compositions, units and symmetries.

\end{defn}
Thus there is a \textbf{category of PROPs}. For more on this category, as well as some its properties, see \hyperlink{HR1}{HR1}.

\begin{defn}
\label{}\hypertarget{}{}
Given a PROP $T$ and a symmetric monoidal category $C$, a [[symmetric monoidal functor]]

\begin{displaymath}
F : T \to C
\end{displaymath}
is called an \textbf{algebra} or \textbf{model} of $T$ in $C$. The category of algebras of $T$ in $C$, say $Alg(T,C)$, has

\begin{itemize}%
\item symmetric monoidal functors $F : T \to C$ as objects,
\item [[symmetric monoidal natural transformation|symmetric monoidal natural transformations]] as morphisms.

\end{itemize}
\end{defn}
Note that all of the above definitions can be [[enriched category|enriched]] over a [[symmetric monoidal category]] which yields the notion of enriched PROPs. For instance, we can have [[simplicially enriched category|simplicial]] and [[topologically enriched category|topological]] PROPs where the sets of morphisms are [[simplicial sets]] or [[topological spaces]].

\hypertarget{model_structure_on_simplicial_props}{}\subsection*{{Model Structure on Simplicial PROPs}}\label{model_structure_on_simplicial_props}

First, in the case that we are working only with simplicial PROPs with a fixed set of colors (or, in other words, PROPs whose free monoids of objects all have the same generators), we have the following theorem of \hyperlink{HR2}{HR2}:

\begin{prop}
\label{}\hypertarget{}{}
There is a cofibrantly generated model structure on the category of simplicial PROPs with a fixed set of colors in which a morphism of simplicial PROPs $\phi:Q\to T$ is a weak equivalence (resp. fibration) if for each simplicial set of morphisms the induced map $Q(\vec{x},\vec{y})\to T(\vec{x},\vec{y})$ is also a weak equivalence (resp. fibration).

\end{prop}
Recall now that simplicial PROPs admit an obvious forgetful functor to [[categories]] that factors through [[simplicial categories]]. Denote this functor by $\pi_0\colon sPROP\to Cat$. Using this notation, we have (again from \hyperlink{HR2}{HR2}):

\begin{prop}
\label{}\hypertarget{}{}
There is a cofibrantly generated model structure on the category of all simplicial PROPs where a morphism of PROPs $\phi\colon Q\to T$ is a weak equivalence (resp. fibration) if:

\begin{itemize}%
\item the induced morphism of mapping complexes $\phi_m\colon Q(\vec{x},\vec{y})\to T(\phi_c(\vec{x}),\phi_c(\vec{y}))$ is a weak equivalence (resp. [[Kan fibration]]) of simplicial sets, and


\item the functor $\pi_0\colon\pi_0Q\to \pi_0T$ is a weak equivalence (resp. [[isofibration]]) of categories.



\end{itemize}
\end{prop}
Note that the model structure on simplicial PROPs is \emph{not} the model structure on gets by lifting the model structure of [[(infinity,1)-operad|simplicial operads]] along the free forgetful adjunction between simplicial operads and simplicial PROPs.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{bialgebras}{}\subsubsection*{{Bialgebras}}\label{bialgebras}

A perhaps paradigmatic example is that there is a $Vect$-enriched prop whose algebras are bialgebras. It should be observed here that there is no $Vect$-enriched [[operad]] (or cooperad) whose algebras are bialgebras, so this is a genuine example that illustrates a gain in generality of props over operads.

See \hyperlink{Pirashvili}{Pirashvili} for some more details on this prop.

\hypertarget{endomorphism_prop}{}\subsubsection*{{Endomorphism PROP}}\label{endomorphism_prop}

Given a set of colors $\mathfrak{C}$ and a [[closed category|closed]] [[symmetric monoidal category]] $E$ with a chosen collection of objects $\mathbf{X}=\{X_c\}_{c\in\mathfrak{C}}$, there is an $E$-enriched PROP $End_{\mathbf{X}}$ with morphism $E$-mapping spaces $End_{\mathbf{X}}(\{X_i\},\{Y_j\})=E(\otimes_i X_i,\otimes_j Y_j)$.

\hypertarget{relation_to_polycategories}{}\subsection*{{Relation to Polycategories}}\label{relation_to_polycategories}

Every PROP defines a [[polycategory]]; see there for more. PROPs can compose along many objects at once, whereas polycategories compose along a single object.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[PRO]]
\item [[polycategory]]
\item [[properad]]
\item [[dioperad]]

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

\begin{itemize}%
\item Wee Liang Gan, \emph{Koszul Duality for Dioperads}, \href{http://arxiv.org/abs/math/0201074}{(arXiv:0201074v2)}.

\end{itemize}
\begin{itemize}%
\item [[Philip Hackney]] and [[Marcy Robertson]], \emph{On the Category of PROPs}, \href{http://arxiv.org/pdf/1207.2773v2.pdf}{arXiv:1207.2773v2}.


\item [[Philip Hackney]] and [[Marcy Robertson]], \emph{The Homotopy Theory of Simplicial PROPs}, \href{http://arxiv.org/pdf/1209.1087.pdf}{arXiv:1209.1087}.


\item [[Steve Lack]], \emph{\href{http://www.tac.mta.ca/tac/volumes/13/9/13-09abs.html}{Composing PROPs}}, [[TAC]] 13 (2004), No. 9, 147--163.


\item Teimuraz Pirashvili, \emph{On the PROP corresponding to bialgebras}, http://arxiv.org/abs/math/0110014. (\href{http://arxiv.org/abs/math/0110014}{link})


\item [[Donald Yau]], \emph{Higher dimensional algebras via colored PROPs}, \href{http://arxiv.org/pdf/0809.2161v1.pdf}{(arXiv:0809.2161v1)}.


\item M. Markl, S. Merkulov, S. Shadrin, \emph{Wheeled PROPs, graph complexes and the master equation}, J. Pure Appl. Algebra, 213(4):496–535, 2009, \href{https://arxiv.org/abs/math/0610683}{math.AT/0610683}



\end{itemize}
[[!redirects PROP]] [[!redirects PROPs]] [[!redirects prop]] [[!redirects props]]



\end{document}