### More on Tangent Categories

#### Posted by Urs Schreiber

I am the luckiest man in the world, having John Baez around here at ESI in Vienna to talk to – over breakfast, over lunch, and in between – about $n$-curvature and that stuf which I like to think of as tangent categories and arrow-theoretic differential theory.

At the same time, the dg-wizards like Dmitry Roytenberg are around, and using that bridge which connects $n$-categorical algebra with physicist’s BRST-BV formalism I can connect these two huge reservoirs of ideas and let the information flow botrh ways. That helps a lot.

Here are some insights which I gained by

a) trying to address questions John urged me to answer more properly, and by all the input I get from him;

b) comparing this to what the dg-BV people are doing by passing back and forth over that bridge.

More concretely, I shall

A) point out in detail how the notion of forming the “tangent category” which I was talking about coincides, in the appropriate sense and in the applicable cases, *exactly* with what people do who form the *shifted tangent bundle of a differential graded manifold*, thereby giving yet another way of making the relation of tangent categories to the ordinary notion of tangent spaces manifest.

B) indicate how one should correctly think of the $n$-category of curvature $(n+1)$-functors and how this relates to the fact that they are related to universal $n$-bundles.

C) remark on the relation of *tangent categories* with the Yoneda embedding

**Inner Derivations and the Shifted Tangent Bundle**

**Proposition** *Forming the Lie $(n+1)$-algebroid *
$\mathrm{inn}(g_{(n)})$
*of* inner derivations * a Lie $n$-algebroid $g_{(n)}$ is equivalent to forming the the * shifted tangent bundle
$T[1] X_{g_{(n)}}$
* of the differential graded manifold $X_{g_{(n)}}$ corresponding to $g_{(n)}$ under the duality between Lie $n$-algebroids and dG manifolds *.

Proof. It is clear that in both cases the new complex is the direct sum of the fomer one with a shifted copy of itself. The only thing to check then is if the differential that people think the shifted tangent bundle $T[1] X$ to come equipped with, for $X$ any differential graded manifold, coincides with the one demanded by the $\mathrm{inn}(\construction)$. The latter is defined (section 3.2.5 of this provisional article) to simply be the *mapping cone* of the identity on the original complex.

Compare this to how the differential on $T[1] X$ is declared (which is apparently nowhere to be found fully explicitly in the literature (?), but which is exactly what I’d expect it is, and Dmitry Roytenberg kindly confirmed this to me a few minutes ago):

With $\wedge^\bullet ( V^*, \delta)$ the original complex (locally, if we have a dG-manifold which is not a point), we form the new complex
$\wedge^\bullet ( V^* \oplus s V^*, d' := d + L_{\delta})
\,,$
where $s$ denotes the shift in degree, where we write
$d t$
for the image under this shift of any element $t \in V^*$ (this is what is called $\sigma$ in our notes), where we think of $d$ as a derivation on the new complex acting as (obviously)
$d : t \mapsto d t$
$d : d t \mapsto 0$
for all $t \in V^*$, and where - and that’s finally the crucial point, where $L_\delta$ acts as the original $\delta$ on $V^*$ and anti-commutes with $d$:
$L_\delta : t \mapsto \delta t$
$L_\delta : d t \mapsto - d( \delta t)
\,.$
Just stare at this for a second, noticing, if it helps, that what is called $d$ here is called $\sigma$ there and that what is called $\delta$ here is called $d$ there and what is called $d + L_\delta$ here is called $d'$ there, to see that this exactly coincides with the definition of the differential on the inner derivations. And recall, once more, that both constructions are nothing but that of forming the *mapping cone*

**Remark** Let me emphasize what this means:

I’ll concentrate on the case where we are dealing with one-object $n$-groupoids and hence their corresponding $n$-algebroids “over a point”, in order not to get distracted by important but boring technical issues irrelevant to the main point.

So, I said in arrow-theoretic differential theory that there is a natural notion of tangent $n$-groupoid $T G \to G$ for any $n$-group $G$. In the case where $G$ has just a single object this is just the tangent groupoid $T_\bullet G$ over that point. This inherits the structure of an $(n+1)$-group by a canonical embedding $T_\bullet G \hookrightarrow T_{\mathrm{Id}_G}(\mathrm{Aut}(G))$ and equipped with this $(n+1)$-group structure I address $T_\bullet G$ as (a slightly smaller sub-thing of) the inner automorphism $(n+1)$-group $\mathrm{INN}_0(G) \hookrightarrow \mathrm{INN}(G) \,.$ I emphasized a lot how vector fields on (the space of objects of) a category $C$ correspond to group homomorphisms into $\mathrm{INN}(C)$. In particular, if we are looking at smooth group homomorphisms from the additive group of real numbers, we find “ordinary” vector fields (and their orbifold-like generalizations, in fact). But this means that vector fields on the $n$-groupoid $G$ form the Lie $n$-algabra $\mathrm{inn}(g) := \mathrm{inn}(Lie(G)) := \mathrm{Lie}(\mathrm{INN}_0(G))$ of the tangent $n$-groupoid of $G$.

And the above proposition tells us that this notion of tangent space of an $n$-groupoid is perfectly consistent with the notion of (shifted) tangent spaces to the corresponding dual incarnations of the corresponding Lie $n$-algebroids.

**more on $n$-Curvature**

I am claiming that for $F : C \to D$ any $n$-functor, it makes good sense to regard its differential (section 3.2)
$\delta F : C \to n\mathrm{Cat}$
which sends each object $x$ in $C$ to the tangent category $T_{F(x)} D$,
as its *curvature*.

Now, each $T_{F(x)} D$ is (that’s one of the crucial properties of tangent categories) “*contractible*” in that it is (not equal to but) equivalent to the trivial $n$-category.

I had some kind of operational idea what this means, and that it makes good sense, but John very much urged me to come up with a good *non-evil* (i.e. intrinsic) statement which clarifies in one sentence what is going on, what this contractibility means and why we aren’t simply talking about the trivial $(n+1)$-functor, up to equivalence.

There is now some progress in this direction.

The right way to think of it is this:

**Observation.**
For
$F : C \to D$
any $n$-functor, its *differential*
$\delta F : C \to n\mathrm{Cat}$
really takes values in $n$-categories over $D$, hence
$\delta F : C \to n\mathrm{Cat} \downarrow D$
and hence we regard it accordingly as an object in $\mathrm{Hom}(C,\mathrm{Cat}\downarrow D$.

As such, it is not trivial. In fact, I think we’ll get a canonical equivalence between $n$-functors with values in $D$ and their coresponding curvatures.

Here is a good and useful way to think about this in the special case which we are mostly interested in, that where $D = \Sigma G_{(n)}$ is a one-object $n$-groupoid (set $n=1$ or $n=2$ to get something we understand already in detail). Then the curvature $(n+1)$-functor factors through $\Sigma \mathrm{INN}(G_{(n)})$, so it is helpful to think of it as $\delta F : C \to \Sigma \mathrm{INN}(G_{(n)}) \,.$ Now recall that $\mathrm{INN}(G_{(n)})$ “is” the universal $G_{(n)}$-bundle in that we have an exact sequence $G_{(n)} \to \mathrm{INN}(G_{(n)}) \to \Sigma G_{(n)} \,.$ This essentially says that $\mathrm{INN}(G_{(n)})$ is made of two copies of $G_{(n)}$, one of them “shifted”. In fact, $\mathrm{INN}(G_{(n)})$ is the “mapping cone of the identity on $G_{(n)}$”.

It is the “shifted copy” $\Sigma G_{(n)}$ of $G_{(n)}$ in $\mathrm{INN}(G_{(n)})$ which hosts the various components of the curvature of $F$. Hence if we want the transformations of $\delta F$ to be those which correspond to transformations of the original $F$, we should demand that they have no component as we pass them along the projection $\mathrm{INN}(G_{(n)}) \to \Sigma G_{(n)}$.

More formally, we take the admissable transformations of $\delta F$, which are themselves $k$-functors with values in $\mathrm{Cylinders}(\mathrm{INN}(G_{(n)}))$ to become trivial as we postcompose them with $\mathrm{Cylinders}(\mathrm{INN}(G_{(n)})) \to \mathrm{Cylinder}(\Sigma G_{(n)}) \,.$

This is the fancy “nonabelian” version of a rather obvious and simple statement at the level of the corresponding Lie $n$-algebras.

There, an $n$-connection is a morphism $\Omega^\bullet(X) \leftarrow (\mathrm{inn}(g_{(n)}))^*$ of the differential algebra duals of the corresponding Lie $n$-algebras. Here the graded commutative algebra underlying $(\mathrm{inn}(g_{(n)}))^*$ is $\wedge^\bullet ( \s g_{(n)}^* \oplus s s g_{(n)}^* )$ and our exact sequence $G_{(n)} \to \mathrm{INN}(G_{(n)}) \to \Sigma G_{(n)}$ translates into the corresponding exact sequence $\wedge^\bullet( s g_{(n)}^* ) \leftarrow \wedge^\bullet ( \s g_{(n)}^* \oplus s s g_{(n)}^* ) \leftarrow \wedge^\bullet ( s s g_{(n)}^* ) \,.$

Our above restriction that the morphisms between our $n$-connection should be “projected out” by the map $\mathrm{INN}(G_{(n)}) \to \Sigma G_{(n)}$ then translates into the requirement that any homotopy $\Omega^\bullet(X) \leftarrow (\mathrm{inn}(g_{(n)}))^*$ vanishes when precomposed with the injection $\wedge^\bullet ( \s g_{(n)}^* \oplus s s g_{(n)}^* ) \leftarrow \wedge^\bullet ( s s g_{(n)}^* ) \,.$ This is exactly what we use (without quite saying it in such a general way) in The second edge of the cube. So, for instance, the Cartan condition on an ordinary connection involves a homotopy of morphisms involving $\mathrm{inn}(g)$, which is precisely restricted to have no component in the “shifted part” of $\mathrm{inn}(g)$:

**“tangential” Yoneda**

When I wrote up some of the things in Arrow-theoretic differential theory, I called the corresponding pdf file tanyon.pdf. The reason is that I noticed a curious similarity of the notion of “tangent category” which I felt the need to consider, and the ordinary Yoneda embedding: the tangent category is something like a “tangential Yoneda embedding”.

But I didn’t quite know what to make of this, then. So I never mentioned it except for this hint in the title.

But now John said I might want to emphasize this point, since it might help people better follow what I am trying to talk about.

So here it is:

For any $n$-category $C$ and any object $x \in C$, I write
$T_x C$
for the *n-category* (Yoneda would just have an $(n-1)$-category here!) whose objects are morphisms in $C$ emanating at $x$, and whose morphisms are (higher dimensional) triangles between these.

(I give what I think is a very nicely abstract precise deifnition of what exactly $T_x C$ looks like in my notes.)

While over objects this is slightly different from standard Yoneda, morphisms and higher morphisms in $C$ give morphisms and higher morphismsm between the tangent categories $T_x C$ in just the same kind of way as in ordinary Yoneda embedding, such that we obtain an $n$-functor $T C : C^{\mathrm{op}} \to \mathrm{n}\mathrm{Cat} \,.$

Notice that the whole point of this, in a way, is that $T C$ is *not* an equivalent way to think of $C$. Rather, it is a puffed-up way to think of just the space $\mathrm{Obj}(C)$ of its objects!

That, and how, $T C$ is still something of interest is in part what my remark in the previous section was about.

## Re: More on Tangent Categories

I can take over while Urs is at that talk.

I’ve been struggling to understand what he’s been up to for the last few months — don’t be fooled into thinking I’ve understood all his recent blog entries! — and I feel I’m making progress.

As usual, understanding something difficult requires reworking it: making it mesh with the ideas one already knows and loves. So, let me say a word about tangent categories and the Yoneda embedding, which might help people like me understand what Urs is up to.

(“People like me” — e.g., me, but maybe other people who like the Yoneda embedding.)

Let’s take a very basic version of Urs’ tangent category construction, where it

mustalready be familiar to category theorists.In its crudest form — a form so contemptibly crude I’ve never even seen it mentioned — the Yoneda functor of a category $X$

$Y: X \to Set$

assigns to each object $x \in X$ the set of all arrows going into $x$ — that is, morphisms like this:

$f: y \to x.$

(To placate Bertrand Russell’s ghost, assume $X$ is small enough so that these arrows form a set instead of a proper class!)

Now, this should remind you of the tangent bundle of a manifold, which assigns to each point a set of arrows going out of that point — namely, tangent vectors!

Of course, one difference is that the arrows in the category case are not particularly ‘infinitesimal’. But we can live with that.

Another is that in the category case, the arrows are pointing

into$x$ instead ofout ofit. We can fix that. Just consider the set of arrows going out:$f: x \to y.$

Now we get a different sort of Yoneda functor, which is contravariant. It’s not better; it’s not worse; it’s just different. But since Urs like this one, let’s switch to working with this one:

$Y : X^{op} \to Set$

and forget about the first one I mentioned.

Now, the mere

setof arrows out of $x \in X$ is a big disorganized mess. So, what most category theorists do is gather these arrows in bunches, according to where they land!Then we get, not a mere set of arrows, but an $X$-indexed set:

$Y: X^{op} \to Set^X.$

This is the usual Yoneda embedding — or more precisely a backwards version of it, since most category theorists focus on arrows shooting into $x \in X$, not shooting out of it. That’s just a convention; I’ll do things the other way to please Urs, who apparently prefers to shoot arrows than to get shot by them.

But here’s the real point: Urs organizes his arrows in a somewhat different way. He organizes the arrows going out of $x \in X$ into a

category!Given two arrows$f_1: x \to y_1$

and

$f_2: x \to y_2$

he says a morphism from $f_1$ to $f_2$ is a guy

$g: y_1 \to y_2$

making the obvious triangle commute. I won’t draw the triangle here, but it just says that $f_2$ is $f_1$ followed by $g$.

So, Urs gets a souped-up Yoneda functor, which I’ll call

$T: X^{op} \to Cat.$

If one wanted, one could call this the ‘tangent category’ of $X$. Of course, you certainly might object to that terminology! After all, it seems a bit odd to use ‘

tangent category’ as the name for afunctor. But, it’s no worse than thinking of thetangent bundleof a manifold $X$ as afunction$T: X \to Vect$

that assigns a vector space $T_x$ to each $x$ point in the manifold.

This sort of function is sometimes called a ‘classifying map’, and we can recover the total space of tangent bundle from that: a big fat space $T X$ that’s just the union of all the spaces $T_x$, cleverly glued together with a nice topology. And, this total space comes with an obvious projection map

$p: T X \to X$

sending everybody in $T_x$ to the point $x$.

Indeed, we can categorify this idea! We can build a ‘total category’ from Urs’ souped-up Yoneda functor

$T: X^{op} \to Cat$

in more or less the same way. Grothendieck was the guy who first thought about this sort of stuff… and if I’m not getting too confused and tripping over myself (I have a feeling I am), this ‘total category’ $T X$ will be just the ‘category of arrows in $X$’, equipped with a functor

$p: T X \to X$

assigning to each arrow its target, and another

$q: T X \to X^{op}$

assigning to each arrow its source.

Note, this is where the non-infinitesimal nature of our arrows really matters! A tangent vector is an arrow so short that its tip and tail lie at the same point, so we don’t get two different projections in the tangent bundle. But now, in the categorified case, we do!

Anyway, this is just a smidgen of what Urs is doing — he’s been generalizing this idea in various different directions, especially to Lie $n$-groupoids and Lie $n$-algebroids and the like, and using it to understand higher gauge theory and stuff like that.

But, there seems to be some simple stuff at the beginning here, which surely must have been studied before. Does anyone have thoughts about it?