The n-Category Café

What confuses me here is that you envisage supercategories as generalizing supergroups. I don’t know what a supergroup is , but it doesn’t seem that a supercategory generalizes one. A supercategory is “a thing”, and not “a collection of things”… this is weird.

I haven’t talked about morphism of supercategories yet.

In fact, I think one should go back one step and first consider the concept of a category with a $G$-flow on it. Then morphisms of that. Then supercategories as categories with a $\mathbb{Z}_2$-flow on them (an “odd vector field”) and one additional condition.

I am thinking here along the lines of the disucssion in What is a Lie derivative, really?

So:

Definition. For $G$ any group, a $G$-flow on a category $C$ is an action $$R: \Sigma G \to \mathrm{Aut}(C)$$ together with a collection of natural transformations $$\array{ & \nearrow \searrow^{\mathrm{Id}} \\ C &\Downarrow& C \\ & \searrow \nearrow_{R_g} }$$ for all $g \in G$, respecting that action.

A morphism of two categories $C$ and $C'$ with a $G$-flow on them is a functor $$F : C \to C'$$ such that $$\array{ & \nearrow \searrow^{\mathrm{Id}} \\ C &\Downarrow& C \\ & \searrow \nearrow_{R_g} & \downarrow \\ && \downarrow^{F} \\ && C' } \;\;\;\; = \;\;\;\; \array{ C \\ \downarrow^F \\ \downarrow & \nearrow \searrow^{\mathrm{Id}} \\ C' &\Downarrow& C' \\ & \searrow \nearrow_{R'_g} } \,.$$

The point is that a $G$-flow is a $G$-action such that everything is connected to the identity by natural transformations. The components of these natural transformations are the “flow lines” of the flow. Manifestly so, if we consider flows on the path groupoid, as described in Isham on Arrow Fields. Here we have a generalization of this concept of a flow of a vector field, which also allows us to consider flows of such weird things like “odd” vector fields.

I was claiming that this is one nice way to understand the Cartan condition on a connection on the total space of a $G$-bundle

Noah Snyder once explained to me how the category of super vector spaces should be defined. As a monoidal abelian category, it is just the category of Z/2 graded vector spaces with the grading not imposing any condition on the morphisms. However, super vector spaces was equipped with an unusual commutator. This commutator had the magical effect that if you wrote down the diagrammatic definition of a commutative algebra object or a lie algebra object in the category of super vector spaces, you got skew commutative algebras and super Lie Algebras.

I’ll shoot Noah an e-mail and see if he’ll post an explanation or a reference.

While hanging around in Conil de la Frontera (from where I sent my last telegram to the $n$-café), I did think a little (just a little) more about supercategories and all that. Here are some comments.

One major guiding light in this business is that we need to reproduce the following fact:

Morphisms from the superpoint to any ordinary space form the (odd) tangent bundle of that space.

I had made some comments on this here before (in other threads), but now it feels like I am better understanding what’s going on.

First: what is the tangent bundle, really?

Let $C$ be any category. It’s tangent space at any object $x \in \mathrm{Obj}(C)$, which I’ll write $$T_x(C)$$ ought to be the category whose objects are morphisms in $C$ starting at $x$ and whose morphisms are commuting triangles.

The entire “tangent bundle” of $C$ is then the disjoint union of all these categories $$T C := \oplus_{x \in \mathrm{Obj}(C)} T_x C \,.$$

We want to find a notion of supercategories such that with $C$ regarded as a supercategory in the trivial way, and with $\mathbf{pt}$ the superpoint, we have $$T C = \mathrm{Hom}(\mathbf{pt},C) \,.$$

And we want this to be such that it generalizes seamlessly to $n$-categories.

I think it works like this:

Let $$\mathrm{pt} := \{\bullet\}$$ be the ordinary point, i.e. the category with a single object and a single morphism.

Then let $$\mathbf{pt} := \{ \bullet \to \circ\}$$ be the the category with two objects and one nontrivial morphism going between these. Think of $\bullet$ as the body and of $\circ$ as the soul of the point.

(I don’t like this terminology, nor much of the rest of the “super”-terminology, but it is standard and I am mentioning it in order to help follow the modelling process.)

(Later I’ll require this morphism to be an isomorphism. This will make $\mathbf{pt}$ a category with nontrivial $\mathbb{Z}_2$-flow.)

The canonical inclusion $$\array{ \{\bullet\} \\ \downarrow^\subset \\ \{\bullet \to \circ\} }$$ is crucial for everything to follow.

Thinking of the ordinary category $C$ as a supercategory in the trivial way means looking at the obvious inclusion $$\array{ C \\ \downarrow^= \\ C }$$

(You know, in the silly standard terminology we’d say that $C$ has no soul. Poor $C$.)

So, a morphism from the superpoint to $C$ is a morphism $$\mathbf{pt} \to C$$ which covers a morphism $$\mathrm{pt} \to C$$ in that $$\array{ \mathrm{pt} &\to& C \\ \downarrow^\subset && \downarrow^\subset \\ \mathbf{pt} &\to& C } \,.$$ Moreover, a 2-morphism between such morphisms is a 2-morphism $$\array{ & \nearrow \searrow \\ \mathbf{pt} &\Downarrow& C \\ & \searrow \nearrow }$$ which vanishes when pulled back to $\mathrm{pt}$: $$\array{ &&& \nearrow \searrow \\ \mathrm{pt} &\to& \mathbf{pt} &\Downarrow& C \\ &&& \searrow \nearrow } \;\; = \;\; \mathrm{pt} \to C \,.$$

This way we indeed get that $$T C = \mathrm{Hom}(\mathbf{pt},C) \,.$$

And it is obvious how to generalize this to $C$ an arbitrary $n$-category.

Here is how we reproduce the ordinary tangent bundle of an ordinary manifold $X$ this way:

we need to model $X$ as a category. Take it to be the 2-groupoid $$P_2(X)$$ whose objects are the points of $X$, whose morphisms are piecewise smooth parameterized paths in $X$ and whose 2-morphisms are thin homotopy classes of homotopies between all those paths whose tangent vectors at source and target coincide.

Then $$(T P_2(X))_\sim = T X \,,$$ where $(\cdot)_\sim$ here denotes dividing out 2-isomorphisms.

That’s also the motivating idea behind the concept of tangent space here: we take all morphisms emanating at a given object and then let the 2-morphisms between them identify all those which “have the same tangent” at that object.

All this happens to tie in nicely with the stuff about $\mathrm{INN}(G)$, see The Inner Automorphism 3-Group of a Strict 2-Group, and in fact answers at least in part the question I was posing in a comment to that:

Let $G$ be an ordinary group. Then the “tangent space” of the category $\Sigma G$ at the single object $\bullet$ is $$T_\bullet (\Sigma G) = T_{\mathrm{Id}_{\Sigma G}} (\mathrm{Hom}(\mathrm{Cat})) = \mathrm{INN}(G) \,.$$

The structure of $\mathrm{INN}(G)$ as a groupoid is manifest from its realization as $T_\bullet (\Sigma G)$, while the monoidal structure on it (the 2-group structure) is manifest from its realization as $T_{\mathrm{Id}_{\Sigma G}} (\mathrm{Hom}(\mathrm{Cat}))$.

For $G_2$ any strict 2-group, we have $$T_\bullet (\Sigma G_2) = \mathrm{INN}(G_2) \subset T_{\mathrm{Id}_{\Sigma G_2}} (\mathrm{Hom}(2\mathrm{Cat})) \,.$$

The idenitification $\mathrm{INN}(G_2) = T_\bullet (\Sigma G_2)$ is a quick way to encode all the pictures that David Roberts and I have in that paper. The inclusion $\mathrm{INN}(G_2) \subset T_{\mathrm{Id}_{\Sigma G_2}} (\mathrm{Hom}(2\mathrm{Cat}))$ is a quick way to understand the monoidal structure on $\mathrm{INN}(G_2)$.