# Contents

## Idea

A ‘structure type’ is a type of extra structure that can placed on finite sets, e.g. a coloring, or ordering.

## Definition

We define a structure type to be a faithful functor

$p:X\to \mathrm{core}\left(\mathrm{FinSet}\right)$p: X \to core(FinSet)

from some groupoid $X$ to core(FinSet), which is the groupoid of finite sets and bijections. Equivalently, we can think of it as a presheaf of sets on the groupoid of finite sets and bijections, or in other words a functor

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

These two points of view are related by the Grothendieck construction:

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

But since a groupoid is equivalent to its opposite, we can also think of a structure type as a functor

$\mathrm{core}\left(\mathrm{FinSet}\right)\to \mathrm{Set}$core(FinSet) \to Set

In this guise, a structure type is more commonly called a (combinatorial) species of structure, or species for short.

A structure type is a special case of a stuff type, so see stuff type for more information.

## References

Revised on May 29, 2012 22:04:00 by Andrew Stacey (129.241.15.200)