# Are valued-quiver species and combinatorial species related?

In this paper, Joel Lemay defines a species as a generalization of a valued quiver $$Q$$, where to each node $$i \in Q_0$$ we assign a division ring $$\mathbf{k}_i$$, and to each arrow $$(a \colon i \to j) \in Q_1$$ we assign a $$(\mathbf{k}_j, \mathbf{k}_i)$$-bimodule $$M_a$$ that satisfies a couple of conditions:

• $$\operatorname{Hom}_{\mathbf{k}_j}(M_a, \mathbf{k}_j) \simeq \operatorname{Hom}_{\mathbf{k}_i}(M_a, \mathbf{k}_i)$$ as $$(\mathbf{k}_i, \mathbf{k}_j)$$-bimodules.

• $$(\dim_{\mathbf{k}_j}M_a, \dim_{\mathbf{k}_i}M_a)$$ is equal to the pair of values on the edge $$a$$.

Now a combinatorial species is an endofunctor of the category of finite sets, where the morphisms are set bijections. Is this first sort of species related at all to a combinatorial species? or do these two types of objects just coincidentally (unfortunately?) have the same name? And if they're not the same, what is the origin of the valued-quiver species? The term for combinatorial species comes from Andre Joyal's "espèces de structure".

• I have the very strong impression that it is just a coincidence, but I would love to be proven wrong! – Ivan Di Liberti Dec 3 '18 at 23:40