Species

A species of structures is a rule $F$ which

1. Produces, for each finite set $U$, a finite set $F[U]$,

2. Produces, for each bijection $\sigma:U\to V$, a function

$$F[\sigma]:F[U]\to F[V].$$

The functions $F[\sigma]$ should further satisfy the following functorial properties:

1. For all bijections $\sigma:U\to V$ and $\tau:V\to W$,

$$F[\tau\circ\sigma] = F[\tau]\circ F[\sigma],$$

2. For the Identity Map $\mathop{\rm Id}_U:U\to U$,

$$F[\mathop{\rm Id}_U] = \mathop{\rm Id}_{F[U]}.$$

An element $\sigma\in F[U]$ is called an $F$-structure on $U$ (or a structure of species $F$ on $U$). The function $F[\sigma]$ is called the transport of $F$-structures along $\sigma$.

References

Bergeron, F.; Labelle, G.; and Leroux, P. Combinatorial Species and Tree-Like Structures. Cambridge, England: Cambridge University Press, p. 5, 1998.