Is there a natural way to regard a valued quiver as a category?

Question

You can think of a quiver, a collection of vertices $Q_0$ and arrows $Q_1$, as a free category with objects $Q_0$ and morphisms generated by the morphisms of $Q_1$. More precisely this is a functor from $\mathsf{Quiv}$ to $\mathsf{Cat}$ that sends a quiver to the free category it generates. Then there is a similar functor for quivers with relations where your category is no longer free.

A valued quiver $Q$ is quiver where we equip to each arrow $r\colon i \to j$, a pair of positive integers $(d_{ij}^r, d_{ji}^r)$ such that for each vertex $i$ there exists a positive integer $f_i$ where $d_{ij}^rf_j = d_{ji}^rf_i$. See this paper by Joel Lemay for details. Is there a way to think of valued quivers as a category like we can for usual quivers or for quivers with relations? Is there a functor from the category of valued quivers to some other category besides $\mathsf{Cat}$ that has the same spirit as the functor $\mathsf{Quiv} \to \mathsf{Cat}$?

And then we could ask the same question for a species too, but I think an answer for a valued quiver should extend easily to species. Also see this other question of mine which, if it has an answer, will probably extend to an answer here too.

Answer 1

The following are some remarks that don't really answer your question but are something in between "an elegant(ish) way to do this and a history lesson why it isn't meaningful".

1) If I'm not mistaken valued quivers arise firstly in the representation theory of algebras over non-algebraically closed fields. Over an algebraically closed field every finite dimensional algebra is Morita equivalent to the quotient of a path algebra of a quiver by some admissible ideal. For perfect fields that are not algebraically closed this is no longer true but what one can say is that every finite dimensional algebra is Morita equivalent to the quotient of a path algebra (or tensor algebra) of a species by some admissible ideal. The species encodes the information $(\operatorname{End}(S_i), \operatorname{Hom}(S_i,S_j))$ where the $S_i$ are the simple $A$-modules.

It was then noted that certain information like when the path algebra of a species (not taking a quotient) is of finite representation type depends only on the collection of $\dim_{\operatorname{End}(S_i)}\operatorname{Hom(S_i,S_j)}$ and $\dim_{\operatorname{End}(S_j)}\operatorname{Hom}(S_i,S_j)$. This information is encoded in a valued quiver. Given a valued quiver there is in my opinion no particularly nice choice of the corresponding skew fields $\operatorname{End}(S_i)$ and the bimodules $\operatorname{Hom}(S_i,S_j)$ given only the data of a quiver. Such skew fields don't even necessarily exist.

2) What to me makes more sense is considering the analoguous question for species. You can view a species as a strict 2-functor from the category $Q$ to the bicategory $\operatorname{Skew}$ whose objects are $k$-skew fields and whose morphisms are bimodules $M$ (whose duals w.r.t the two sides are isomorphic). Then you can define the path algebra as the tensor algebra whose degree zero part is the product of the skew fields on the vertices and whose degree 1 part is the direct sum of the bimodules associated to the arrows. This tensor algebra behaves like the path algebra of a quiver (which is the $k$-linear analogue of the free category you ask about).

3) Recently, several people started thinking about generalising this definition to strict 2-functors which go from the category $Q$ to some more general bicategory. Fang Li was maybe the first to write this down and he uses arbitrary algebras instead of skew fields, otherwise the conditions are more or less the same. In particular, you obtain a tensor algebra in the same way that in many ways behaves like the path algebra of a quiver. A major contribution was the successful application of these concepts by Geiss, Leclerc and Schröer who only used the case where all the algebras are of the form $k[x]/(x^{n_i})$ for some $n_i\in \mathbb{N}$ to obtain statements about categorification of cluster algebras. They even start with the quiver and relations for the tensor algebra and only later realise it as the path algebra of a species. A bit of self-advertisement: I also have a paper on this entitled Pro-species of algebras where I explain this 2-functorial perspective. Another step in abstraction (which was very recently put on the arxiv by Mozgovoy) is to consider a functor from $Q$ to the bicategory whose objects are abelian categories and whose morphisms are (certain exact) functors between these abelian categories. In this case there is of course in general no free algebra which models this situation.

4) More self-advertisement: Together with Nan Gao, Sondre Kvamme, and Chrysostomos Psaroudakis (where we also independently studied Mozgovoy's setup) we are currently writing a paper explaining another categorical perspective on this. In this paper we replace the "free category" by a "free monad on an endofunctor". In the classical case of the quiver the underlying category would be the sum of $\operatorname{mod} k$ taking as many copies as they are vertices. Then the endofunctor $X$ is the "push-along" functor, i.e. $X((M_i)_i)=(\oplus_{j\to i} M_j)_i$. The generalisation to e.g. Mozgovoy's setup is then straightforward: Take the underlying category to be the sum of all the abelian categories on the vertices and $X((M_i)_i)=(\oplus_{a\colon j\to i} F_a(M_j))_i$ where $F_a$ denotes the functor on the arrow $a$. In this case, for example the category of representations is then equal to the Eilenberg-Moore category of this free monad. In fact much of the classical theory for quivers generalises to this abstract framework of just an endofunctor on an abelian category given certain finiteness properties (e.g. the functor $X$ is nilpotent, has a right and a left adjoint which are isomorphic, and this adjoint is also nilpotent).