*This is a typed up copy of my lecture notes from the combinatorics seminar at KTH, 2010-09-01. This is not a perfect copy of what was said at the seminar, rather a starting point from which the talk grew.*

*In some points, I’ve tried to fill in the most sketchy and un-articulated points with some simile of what I ended up actually saying.
*

Combinatorial species started out as a theory to deal with enumerative combinatorics, by providing a toolset & calculus for formal power series. (see Bergeron-Labelle-Leroux and Joyal)

As it turns out, not only is species useful for manipulating generating functions, btu it provides this with a categorical approach that may be transplanted into other areas.

For the benefit of the entire audience, I shall introduce some definitions.

**Definition**: A *category* C is a collection of *objects* and *arrows* with each arrow assigned a *source* and *target* object, such that