nLab
structure type
Contents
Idea
A ‘structure type’ is a type of extra structure that can placed on finite set s, 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 set s and bijection s. 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

Last revised on May 29, 2012 at 22:04:00.
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