nLab structure type

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

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

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

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

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

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

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

$$core(FinSet) \to Set$$

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

• John Baez, …