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