nLab stuff type

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

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

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

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

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.

For more on stuff types see:

• John Baez, Groupoidification

For the generalization of stuff types using $\infty$-groupoids, see:

• Imma Gálvez-Carrillo, Joachim Kock, and Andrew Tonks, Homotopy linear algebra, Proc. Roy. Soc. Edinburgh Sect. A, 148(2) (2018), 293–325. (arXiv)