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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".

Mike Pierce
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In the paper you linked, the author says that species were introduced as a generalization of quivers in 1973 by Gabriel. Combinatorial species were introduced by Andre Joyal in 1981 in Une theorie combinatoire des series formelles. If you look at the bibliography of this paper, there is no reference to Gabriel. Then presumably, the inventors of either theory were not basing it on the other's work at all.

Joe Moeller
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