Recall that a PROP is a symmetric monoidal category whose object-set is freely $\otimes$-generated by a single object $x\in P$. A *representation* or *algebra* for a prop $P$ in a symmetric monoidal category $C$ is a symmetric monoidal functor $f: P \to C$ — i.e. it is an object $f(x)\in C$ and some morphisms between tensor powers of $f(x)$ that satisfy all the relations in $P$. I am interested in the $\mathbb Q$-linear versions of all of these: a *$\mathbb Q$-linear prop $P$* is a $\mathbb Q$-linear symmetric monoidal category whose object-set is freely $\otimes$-generated by a single object $x\in P$, and a *representation* thereof in a $\mathbb Q$-linear category $C$ is a $\mathbb Q$-linear symmetric monoidal functor. I will henceforth leave implicit the word "$\mathbb Q$-linear". By $\mathrm{Vect}$ I mean the category of $\mathbb Q$-vector spaces.

A general question you can ask is the following. Suppose you are given a prop $P$, and some presentation of it in generators and relations. Suppose you write some expression in the generators — i.e. you pick some morphism in the prop. Suppose furthermore that for every representation $P \to \mathrm{Vect}$, this morphism evaluates to $0$. Does it follow that the morphism is $0$ in $P$? Put another way: *Does every prop have a faithful representation in $\mathrm{Vect}$?*

The answer, of course, is "NO!". An example: let $P$ be the prop generated by the relation that the braiding $x\otimes x \to x\otimes x$ is minus the identity. Then the only representation of $P$ in $\mathrm{Vect}$ is $x = 0$, and in particular the identity map $x \to x$ evaluates to $0$ in this representation. On the other hand, $P$ has a non-zero representation in the category of super vector spaces.

My question, then, is for (checkable) conditions on a prop $P$ to assure that it does, in fact, have faithful representations in $\mathrm{Vect}$.

For example, the prop that I happen to care about has a presentation in which it is generated by (at most) one morphism between any two objects, and the relations are all (homogeneous) linear and quadratic in the generators. I could imagine this to be the type of condition that might assure faithfulness of representations. I would like to know that my prop has a faithful $\mathrm{Vect}$-representation, because I can prove that in any $\mathrm{Vect}$-representation a certain morphism evaluates to $0$, by choosing a basis for the underlying vector space of the representation. Of course, this proof does not universalize, but maybe some other results assure me that the morphism is universally $0$.

Vect, that does not require linear structure. Let there be one generating object $A$ with a right dual, and morphisms $f:A\to A$ and $g: A \to A$ such that $f \circ g = \text{id}_A$. The right dual property ensures that for any representation $Z$ inVect, $Z(A)$ is finite-dimensional, and then $Z(f) \circ Z(g) = \text{id}_{Z(A)}$ implies $Z(f) = Z(g)^{-1}$. But $f \neq g^{-1}$ in the PRO. – Jamie Vicary Jan 15 at 20:30