I am interested in understanding a certain phenomenon. I am hoping this sort of problem has been studied before, but I don't know the proper terminology and am having trouble finding answers. I am going to describe a baby case for a PROP, but the same basic question makes sense in many other contexts such as universal algebra, operads, etc.
Suppose that I have a symmetric monoidal category A which I understand fairly well, by which I mean I have at my disposal a nice (finite) presentation of it. For example A could be the universal PROP for commutative Frobenius algebras, so that symmetric monoidal functors from A into a another symmetric monoidal category C are the same as commutative Frobenius algebras in C. For example C could be the category of finite dimensional vector spaces of a field k. In that case such functors are the same as commutative Frobenius algebras over k.
Now I can ask: When is A faithfully represented in C?
This really could be interpreted in two ways. First we could ask if there is a single faithful representation of A in C, i.e. a faithful functor from A to C. This means that there is a single Frobenius algebra c in C which can detect morphisms in A. In other words if two expressions in the language of A give the same value for the single Frobenius algebra c, then they are in fact the same for any Forbenius algebra in any target category D, including A itself.
This is a lot to ask. It might be the case that there are no such faithful representations of A in C. Instead we could ask for a related property. Instead of asking for a single functor to be faithful, we could ask if somehow all functors jointly are faithful in the following sense:
Defintion: We will say that C weakly detects A if for all morphisms f and g of A, we have $f \neq g$ if and only if there exists some $c:A \to C$ (possibly dependent on f and g) such that $c(f) \neq c(g)$. [Does this go by another name?]
Let's carry on with the Frobenius algebra example. In this case A has another well-known incarnation. It is the symmetric monoidal category of 2D cobordisms. The objects are closed 1-manifolds and the morphisms are 2-dimensional cobordisms.
Now suppose that C is the category of real vector spaces. Using the classification of surfaces, it is possible (and not too hard) to construct a single Frobenius algebra which which does the job. Essentially we construct a Forbenius algebra which takes a bordism to $x^{2g -2}$ where x is an irrational number not a root of unity and g is the genus of the surface. So this C satisfies the strong faithful property.
However this example depends on working over the reals. It might be the case that over other fields there is no single Frobenius algebra that does the job. I don't know the answer, but I would like to know. In the comments below, Noah Snyder gives some reasons we might think this is the case for finite fields. Is there any example which fails the weak detection property?
I am more interested in understanding how we might address this problem abstractly. Are there any techniques for understanding this problem? In the above example I used the connection between A and the bordism category, then I used a (big?) topological theorem (the classification of surfaces) to deduce the detection. I'd like to avoid that if possible and so I am looking for other techniques. Of course I am mostly interested in some other more exotic A, not the Forbenius algebra example I used to motivate.
I suspect that this is probably hopeless, but hopefully I'll gain a better understanding by asking this question. Here is a related MO queastion.
