The n-Category Café

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 $\mathrm{AUT}(G)$.

the forgetful functor from PointedSet to Set, and how this is used to pullback functors to construct things like the action groupoid.

Now that I finally got this point you have all my attention.

But I am maybe not following this statement:

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.

Why is that? If the representation $$\rho :BG\to \mathrm{Set}$$ happens to be on a set with cardinality not a factorial, then there is no fiber of the ordering structure type sitting above it at all. So then the pullback $$\begin{array}{ccc}& & {\mathrm{FinSet}}_0\\ & & {\downarrow }^{\mathrm{orderings}}\\ BG& \stackrel{\rho }{\to }& \mathrm{Set}\end{array}$$ does not even exist. Am I misunderstanding something?

Concerning moving up the categorical ladder here, I just want to point out again that there is one puzzle left which you might have more success thinking about than I had so far:

we have seen that the action groupoid $V//G$ of a representation $\rho :BG\to \mathrm{Set}$ is the pullback of the universal $\mathrm{Set}$-bundle:

$$\begin{array}{ccc}V& \to & s^{-1}\mathrm{pt}\\ \downarrow & & \downarrow \\ V//G& \to & T_{\mathrm{pt}}\mathrm{Set}\\ \downarrow & & \downarrow \\ BG& \stackrel{\rho }{\to }& \mathrm{Set}\end{array}$$

But we also know that the left column here usefully continues on further down, where it starts inclreasing in categorical dimension

$$\begin{array}{ccc}V& \to & s^{-1}\mathrm{pt}\\ \downarrow & & \downarrow \\ V//G& \to & T_{\mathrm{pt}}\mathrm{Set}\\ \downarrow & & \downarrow \\ BG& \stackrel{\rho }{\to }& \mathrm{Set}\\ \downarrow \\ BEG\\ \downarrow \\ BBG.\end{array}$$

(The last item $BBG$ has the obvious interpretation if $G$ is abelian. If not, there should still be something playing its role, as we are discussion with Tim Porter. But feel free to ignore $BBG$ and let’s concentrate on $BEG$).

One idea was to use $$\begin{array}{ccc}V& \to & s^{-1}\mathrm{pt}\\ \downarrow & & \downarrow \\ V//G& \to & T_{\mathrm{pt}}\mathrm{Set}\\ \downarrow & & \downarrow \\ BG& \stackrel{\rho }{\to }& \mathrm{Set}\\ \downarrow & & {\downarrow }^{\mathrm{codisc}}\\ BEG& \to & \mathrm{Cat}\\ \downarrow \\ BBG\end{array},$$

where $\mathrm{codisc}$ sends each set to the codiscrete groupoid over it (one unique morphism per ordered pair of objects).

That works in some of the applications I have, but seems to be unsatisfactory for other reasons. Or I don’t fully understand it yet.

Yet another idea is this: concentrate on representations with values in $G\mathrm{Act}=G\mathrm{Set}$: sets equipped with $G$-action.

Then we can form $$\begin{array}{ccc}BG& \stackrel{\rho }{\to }& G\mathrm{Act}\\ \downarrow & & \downarrow \\ BEG& \to & \mathrm{Set}\downarrow G-\mathrm{Mod}\end{array}$$

Here $\mathrm{Set}\downarrow G$ is the category of sets over $G$ which we equip with the monoidal structure the way John taught us: $$\{a_g\}\times \{b_h\}=\{(a,b{)}_{g\cdot h}\}.$$

Then $\mathrm{Set}\downarrow G-\mathrm{Mod}$ has objects categories with a module structure over this, morphisms functors respecting that module structure and 2-morphisms not just what you might expect, but also transformations that may change the $G$-labels.

Then $$G\mathrm{Act}\to \mathrm{Set}\downarrow G-\mathrm{Mod}$$

sends each $G$-set $T$ to the category $\mathrm{Set}\downarrow T$ of sets over $T$, which is equipped with the obvious $\mathrm{Set}\downarrow G$-module structure induced from the cartesian product of sets and the action of $G$ on $T$.

This alos looks like it is pointing in some right directions. But I am really not sure about this at the moment.

If you have any thoughts on this, please let me know.

Some more candidates for 2-structure types: ordering the components of a groupoid and partially ordering hom sets compatible with composition.

Are there any restrictions on the classes of examples you are looking at?

Here is one, which happens to factor through finite groupoids:

since $\mathrm{FinGrpd}$ is closed, for any finite groupoid $C$ we get

$${\mathrm{Hom}}_{\mathrm{FinGrpd}}(C,-):\mathrm{FinGrpd}\to \mathrm{FinGrpd}$$

The analog at the level of sets is for any finite set $S$ the functor $${\mathrm{Hom}}_{\mathrm{FinSet}}(S,-):\mathrm{FinSet}\to \mathrm{FinSet}$$ which is, in words: “form the set of $S$-indexed subsets”.

David wrote:

Might we expect there to be a similar story one level up? Here we would be interested in 2-functors

$$F:{\mathrm{FinGpd}}_0\to \mathrm{Gpd}$$

Of course David knows there is such a story one level up. We get a 2-functor of the above sort whenever we have any type of ‘structure’ we can put on finite groupoids. Given a finite groupoid $X$, we define $F(X)$ to be the groupoid of all structures of this type on $X$.

Indeed, we can can run around taking our favorite structure types and trying to boost them up a level and get structure 2-types!

I’ll do one right now. Everyone reading this should do their own, and then David can collect them and write a paper on the subject.

My favorite structure type is being an $N$-colored finite set. This is the structure type

$$F:{\mathrm{FinSet}}_0\to \mathrm{Set}$$

given by

$$F(X)=N^X$$

for some set $N$m which we call the set of ‘colors’. An ‘$N$-coloring’ of the finite set $X$ is just a function from $X$ to the set of $N$.

Can we boost this example up a level? Sure! Take a groupoid $N$ and define the 2-structure type being an $N$-colored finite groupoid:

$$F:{\mathrm{FinGpd}}_0\to \mathrm{Gpd}$$

by the same formula:

$$F(X)=N^X$$

We can march on… but before we do, we should think a bit.

One problem, as David notes, is that the set of isomorphism (or equivalence) classes of finite groupoids is much less tractable than the set of isomorphism classes of finite sets. So, the ‘generating functions’ of structure 2-types are much harder to explicitly describe than for structure types.

We should also carefully ponder the yoga of properties, structure and stuff. We can equip finite sets with extra properties, structure, or stuff. Structure types are nice, but stuff types are better for many purposes. These are (weak) 2-functors

$$F:{\mathrm{FinSet}}_0\to \mathrm{Gpd}$$

Similarly, we can equip finite groupoids with properties, structure, stuff, or 2-stuff. Which of these corresponds to David’s ‘2-structure types’?

(The very question suggests that ‘2-structure types’ might turn out to be an awkward name.)

I know what I think the really good things are: the ‘2-stuff types’, that is, ways of equipping finite groupoids with extra 2-stuff. These are (weak) 3-functors

$$F:{\mathrm{FinGpd}}_0\to 2\mathrm{Gpd}$$

What’s an example? ‘Being a weak 2-group’ is a good warmup example. Making a finite groupoid into a 2-group is not just a matter of adding extra structure; we’re really adding extra stuff!

Why? Because the morphisms between weak 2-groups are functors between their underlying groupoids equipped with extra structure… and the 2-morphisms between these morphisms are natural transformations equipped with extra properties.

(This pattern is part of what I mean by ‘the yoga of properties, structure and stuff’.)

For a more full-fledged example, try ‘being the first of two finite groupoids’. Making a finite groupoid into the first of a pair of finite groupoids is not just a matter of adding extra structure or stuff: here we’re really adding extra 2-stuff!

Anyway, there’s a lot more to say, but this is doubtless already more than most people want to hear.