nLab
stuff type

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

p:Xcore(FinSet)p: X \to core(FinSet)

from some groupoid XX 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) opGpdF: core(FinSet)^{op} \to Gpd

The idea is to use the Grothendieck construction and define

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

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

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

core(FinSet)Gpdcore(FinSet) \to Gpd

If a stuff type

p:Xcore(FinSet)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) opSetF: 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) opGpdF: core(FinSet)^{op} \to Gpd

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

(1)E Z:=10!¯+11!¯Z+12!¯Z 2++1n!¯Z n+,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!n! is the permutation group S nS_n, and 11 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)11Z 2:=0!0!¯+0Z+2!2!¯Z 2+0Z 3+,\frac{1}{1-Z^2} := \frac{0!}{\overline{0!}} + 0Z + \frac{2!}{\overline{2!}}Z^2 + 0Z^3 + \cdots,

since there are n!n! ways to order a set with nn elements and 00 ways for an odd set to be even.

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 June 29, 2010 19:29:22 by John Baez (99.11.157.15)