# Contents

## Idea

A ‘stuff type’ is a type of stuff that can placed on finite sets, e.g. ‘being a 2-colored finite set’, or ‘being the first of two finite sets’.

## Definition

To make this precise, we define a stuff type to be a functor

$p: X \to core(FinSet)$

from some groupoid $X$ to the groupoid of finite sets and bijections, which is the core of the category FinSet. Equivalently, we can think of a stuff type as a 2-functor (of a suitably weakened sort, namely a pseudofunctor)

$F: core(FinSet)^{op} \to Gpd$

The idea is to use the Grothendieck construction and define

$F(n) = p^{-1}(n)$

taking advantage of the fact that we may assume without loss of generality that $p$ is a fibration.

We can think of $F$ as a suitably weakened sort of presheaf of groupoids on $core(FinSet)$. But since a groupoid is equivalent to its opposite, we can also think of a stuff type as a functor

$core(FinSet) \to Gpd$

If a stuff type

$p: X \to core(FinSet)$

is faithful, we call it a structure type. Structure types are also called species, and they can be thought of as presheaves of sets

$F: core(FinSet)^{op} \to Set$

A stuff type can also be thought of as a categorified generating function. Whereas a generating function assigns a number to each natural number (or finite set), a stuff type assigns a groupoid. Namely, the stuff type

$F: core(FinSet)^{op} \to Gpd$

assigns to the finite set $n$ the groupoid $F(n)$. We can write $F$ as a power series where the coefficient of $Z^n$ is the groupoid $F(n)$. In these terms, the structure type ‘being a finite set’ is

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

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

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

## References

One advantage of stuff types over the more familiar structure types (i.e., species) is that they allow one to categorify the theory of Feynman diagrams:

• John Baez and James Dolan, From finite sets to Feynman diagrams, in Mathematics Unlimited - 2001 and Beyond, vol. 1, eds. Björn Engquist and Wilfried Schmid, Springer, Berlin, 2001, pp. 29-50. (arXiv)

• Jeffrey Morton, Categorified algebra and quantum mechanics, Theory and Applications of Categories, 16 (2006), 785–854. (arXiv)

• John Baez, Fall 2003 to Spring 2004 seminar notes.

Revised on January 5, 2015 10:31:56 by Urs Schreiber (127.0.0.1)