nLab
structure type

A structure type is a categorified generating function. Whereas a generating function assigns a number to each natural number (or finite set), a structure type assigns a groupoid. In this way, a structure type specifies a category of finite structured sets.

If we think of the term $Z^n$ as “the set with $n$ elements”, then the coefficient is the groupoid of ways for the set to be enowed with the given structure. For example, the structure type “being a finite set” is

(1)$$E^Z:=\frac{1}{\overline{0!}}+\frac{1}{\overline{1!}}Z+\frac{1}{\overline{2!}}{Z}^2+\cdots +\frac{1}{\overline{n!}}{Z}^n+\cdots ,$$E^Z := \frac{1}{\overline{0!}} + \frac{1}{\overline{1!}}Z + \frac{1}{\overline{2!}}Z^2 + \cdots + \frac{1}{\overline{n!}}Z^n + \cdots,

where $+$ is disjoint union, $//$ is the weak quotient, $n!$ is the permutation group $S_n$, and $1$ is the one-element set (since there’s only one way to be finite).

The structure type “being a totally ordered even set” is

(2)$$\frac{1}{(1-Z{)}^2}:=\frac{0!}{\overline{0!}}+0Z+\frac{2!}{\overline{2!}}{Z}^2+0Z^3+\cdots ,$$\frac{1}{(1-Z)^2} := \frac{0!}{\overline{0!}} + 0Z + \frac{2!}{\overline{2!}}Z^2 + 0Z^3 + \cdots,

since there are $n!$ ways to order a set with $n$ elements and $0$ ways for an odd set to be even.

Structure types generalize to stuff types?.

See Baez’s Fall 2003 to Spring 2004 quantum gravity notes.