### 2-Structure Types

#### Posted by David Corfield

Structure types (aka species) are functors $F: FinSet_0 \to Set,$ where $FinSet_0$ is the groupoid of finite sets and isomorphisms. For example, we could look at the $F$ which sends the $n$-element set to the set of its orderings, which has cardinality $n!$.

We talked before about the forgetful functor from PointedSet to Set, and how this is used to pullback functors to construct things like the action groupoid. We can, of course, also do the same thing in the case of structure types. Pulling back our $F$ above, we see sitting above the $n$ element set, the set of orderings of that set with morphisms between them corresponding to permutations down below.

Might we expect there to be a similar story one level up? Here we would be interested in 2-functors $F: FinGpd_0 \to Gpd,$ where $FinGpd_0$ is the 2-groupoid of finite groupoids, equivalences, and natural isomorphisms. Examples include the identity 2-functor and the terminal 2-functor.

Just as we can take a skeletal category for $FinSet_0$ by selecting one set of each finite cardinality, now instead of $FinGpd_0$ we can take the 2-groupoid with objects finite multisets of finite groups. For a single copy of a group $G$, we then have automorphisms of $G$ as 1-morphisms, and as 2-morphisms from $1_G$ to itself, elements of the centre of $G$ (see 3 here).

Does anything like this appear in combinatorics? One thing we won’t have are the nice series expansions of structure types. But we do still have straightforward equivalents for the sum and product operations. As for the composite of structure types $G$ and $H$, recall we define a structure $G \circ H$ by saying a $G \circ H$-structure on a set $S$ consists of a way of partitioning $S$ into disjoint parts, putting a $G$-structure on the set of parts, and putting an $H$-structure on each part. Similarly with 2-functors $G$ and $H$ we could partition a finite groupoid into a set of subgroupoids and put a $G$- structure on the set of parts, considered as a discrete groupoid, and an $H$-structure on each part.

We also talked about pulling back 2-functors along the 2-category classifier, from $PointCat^+ \to Cat$. In the case under consideration, we might consider pulling back these 2-structure types along $PointGpd^+ \to Gpd$.

## Re: 2-Structure Types

So we might look for an analogue of the structure type which assigns the set of orderings to a set. This is the one represented

$1 + X + X^2 + ...$

I think we could make a case for the 2-functor which assigns to a groupoid $G$ the underlying groupoid of the 2-group $AUT(G)$.