PROPs representations, free module analog

Question

Ordinary operad with one ouput can be obviously regarded as free module on itself. Is there are analogous construction for operad with many outputs (PROP)? This must be difficult question, but what is conceptual reason for that in the context of operads? Also, please point me ways of producing of representations of such structures. Some references and results beyond ordinary operads are welcome too.

Answer 1

If I understand your first question, you are looking for a pasting schemes for PROPs.

For analogy, in a (non symmetric) free operad $\mathcal{P}$, pasting schemes are planar rooted trees where internal nodes are labeled on the generators of $\mathcal{P}$. The arity of a tree is the number of its leaves and the composition $S \circ_i T$ of two trees $S$ and $T$ is the grafting of the root of $T$ at the $i$th leaf of $S$.

In a free PROP $\mathcal{R}$ generated by a set $G$ of generators, you can regard an element $g \in G$ with $p$ inputs and $q$ outputs as a node of a directed graph with $p$ incoming edges and $q$ outcoming edges. Since these edges do not connect two nodes but a node with nothing, let us call these legs. Moreover, the incoming (resp. outcoming) legs are bijectively labeled on $\lbrace1, \dots, p\rbrace$ (resp. $\lbrace1, \dots, q\rbrace$).

In $\mathcal{R}$, the horizontal composition $g \star h$ of two generators is simply the juxtaposition of their graphs with a natural renumbering of the legs of $h$. Besides, the vertical composition $g \circ h$, defined only when $g$ has as many inputs $r$ than outputs in $h$, consists in connecting the $i$-th incoming leg of $g$ with the $i$-th outcoming leg of $h$, for all $1 \leq i \leq r$.

Hence, you can deduce that pasting schemes of $\mathcal{R}$ are directed graphs labeled on $G$ with no directed cycle and such that incoming (resp. outcoming) legs are bijectively labeled on an initial segment of $\mathbb{N} \setminus \lbrace 0 \rbrace$.

For your second question, an algebra over a PROP is just a vector space (or a set, or any another adequate category) equipped with operations and cooperations.

For instance, consider the PROP $\mathcal{B}$ in the category of vector spaces generated by an element $\mu$ with two inputs and one output and an element $\Delta$ with one input and two outputs, submitted to the following relations: \begin{equation} \mu \circ (\mu \star I) = \mu \circ (I \star \mu), \end{equation} \begin{equation} (\Delta \star I) \circ \Delta = (I \star \Delta) \circ \Delta, \end{equation} and \begin{equation} \Delta \circ \mu = ((\mu \star \mu) \cdot 1324) \circ (\Delta \star \Delta), \end{equation} where $I$ is the unit and $\cdot$ is the action of the symmetric group on $\mathcal{B}$.

Now, algebras over $\mathcal{B}$ are (non unital) bialgebras, that are vectors spaces $V$ with two (co)associative operations $\mu : V \otimes V \to V$ and $\Delta : V \to V \otimes V$ such that, for any $x, y \in V$, \begin{equation} \Delta(x \; \mu \; y) = \Delta(x) . \Delta(y), \end{equation} where $.$ in the right hand side is the tensor wise product using $\mu$.

You can find more detail in the following paper:

Markl, Martin. Operads and PROPs. Handbook of algebra. Vol. 5, 87--140, Handb. Algebr., 5, Elsevier/North-Holland, Amsterdam, 2008.