### Tangent Categories

#### Posted by Urs Schreiber

I am back from vacation. While being away, I had two seemingly unrelated things in mind:

- the desire to better understand the true nature of inner automorphism $n$-categories and hence $n$-curvature.

- the desire to realize the concepts of tangency and supergeometry in a purely and genuinely arrow-theoretic and $n$-categorical way.

Then, while stunned from the Spanish sun, it occured to me how things might fit together. At nightfall I managed to rise and escape my friends to an internet Café, from where I had sent this postcard from my dreams, reproduced below.

Today I wanted to write it all up cleanly. But I didn’t get as far as I wanted to. And now I have to run once again, to get my train to Leipzig, where I want to attend at least one day of Recent Developments in QFT in Leipzig.

But never shy of sharing unfinished thoughts, I can provide at least these five pages:

Abstract:An arrow-theoretic formulation of tangency is proposed. This gives rise to a notion of tangent $n$-bundle for any $n$-groupoid. Properties and examples are discussed.

The notion of tangent category and tangent bundle given in the following is just a simple variation of the familiar concept of comma categories, albeit generalized to $n$-categories. While very simple, it still seems to me that there is something interesting going on here. I present the concept in a slightly redundant fashion which is supposed to suggest to the inclined reader the more general picture which seems to be at work in the background. For more hints, see Supercategories.

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. Its 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 $T_\bullet (\Sigma G) = T_{\mathrm{Id}_{\Sigma G}} (\mathrm{Mor}(\mathrm{Cat})) = \mathrm{INN}(G) \,.$

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 $T_{\mathrm{Id}_{\Sigma G}} (\mathrm{Mor}(\mathrm{Cat}))$.

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

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 $\mathrm{INN}(G_2) \subset T_{\mathrm{Id}_{\Sigma G_2}} (\mathrm{Mor}(2\mathrm{Cat}))$ is a quick way to understand the monoidal structure on $\mathrm{INN}(G_2)$.

## Re: Tangent Categories

I have one more minute. Here are some more comments.

A) I haven’t tried to describe it in the text. But can you see the nice intuition which is appearing here? It’s this:

I was puzzled before by how similar often the passage from non-super to $n$-fold super is to $n$-fold categorification. So what’s going on?

Take the point. Then puff it up to an equivalent $n$-category. Think of this as fattening the point. Map this fat point into some space. It doesn’t quite look like a point there, since it is rather a big fluffy ball. But it wiggles and wobbles and being that flexible, it feels like just a single point after all.

But now map that fat point into the space and then pin it down at one place. Now it may still wiggle and wobble, but much less so. We have pinned it down. Suddenly all the wiggling and wobbling doesn’t quite make the full fat point collapse to an ordinary point. Rather, it somehow sweeps out the neighbourhood of the place where we pinned it down – this way we get a notion of tangent space to a point.

B) And, in fact, I think what I am calling the “tangent $n$-category” $T_x C$ should really be thought of as $\oplus_{p = 0}^n \Lambda^p T_x C$. Do you see this? I’ll try to say more about this later.

C) Notice that then the entire $n$-particle technology applies: an $n$-fold superpoint is like an $n$-particle with only “infinitesimal extension”.

D) Exercise: assuming we agree on B). Give the arrow theory of the wedge product of differential forms.