August 8, 2007

Arrow-Theoretic Differential Theory, Part II

Posted by Urs Schreiber

Recently I was contacted by somebody who had thought truly long, hard and deep about higher morphisms of Lie $n$-algebras: homotopies and higher homotopies of maps of $L_\infty$-algebras.

He expressed concerns that the formulas for these higher homotopies which we give in the provisional Structure of Lie $n$-Algebras, while coming close, actually in general have to receive correction terms.

Correction terms, that is, which apparently nobody has managed to get a complete handle on. (Should you, dear reader, be the exception to this statement, please let me know.)

While I have to admit that at one point I did falsely believe that the formulas we give are correct in general (I am grateful to Danny Stevenson for a remark on this point), I was relieved to be able to point out that there are explicit warnings on p. 3 and on p. 12 pointing out that the formulas in fact do apply – but only for the case that the target Lie $n$-algebra has a rather special property.

In fact – and this is where it seems things become interesting – this special property is shared in particular by Lie $(n+1)$-algebras which are inner derivation Lie $(n+1)$-algebras $\mathrm{inn}(g_{(n)})$ of Lie $n$-algebras $g_{(n)} \,.$ As readers of this blog know, I have for a long time now assembled what I consider increasing evidence that inner derivation Lie $n$-algebras and inner automorphism Lie $(n+1)$-groups are quite important concepts – for various reasons, actually. See for instance the recent article with David Roberts on The inner automorphism 3-group of a strict 2-group for more details.

So now I am wondering: is it a coincidence that all attempts to explicitly define higher homotopies of Lie $n$-algebras so far have failed, while the only case that is understandable is that where the target Lie $n$-algebra is one of inner derivations? Or is this maybe trying to tell us something?

I am now going to argue that this is possibly supposed to be telling us something.

More precisely, I shall indicate that using what I called Arrow-Theoretic Differential Theory one finds what is actually a simple, obvious and natural explanation.

At least as far as I can see currently.

Morphisms and Tangent Vectors

There are many different ways to relate ($n$-)categories with spaces, one way or another.

The way to think of an ($n$-)category as a space for the purpose of our arrow-theoretic differential theory is the immediate generalization of the way orbifolds may be regarded as groupoids.

So we should be thinking of the space in question as being actually the space of objects of our category. The morphisms of the category, on the other hand, encode something like neighbourhood relations among the points in that space. Tangency relations, if you wish.

In the following precise sense (details for $n=1$ and $n=2$ and everything strict are available, and for $n \geq 3$ are expected to follow straightforwardly, though possibly tediously):

Every $n$-groupoid $C$ comes with an $n$-groupoid $T C \,,$ called, here, its tangent $n$-bundle, which is essentially defined to be the space of maps of the fat point $\mathbf{pt}$ into $C$.

This $n$-groupoid $T C$ has the following remarkable properties:

a) $T C$ is indeed an $n$-bundle over the space underlying $C$ in that there is a canonical projection functor $p : T C \to \mathrm{Obj}(C)$

Even more is true: $T C$ is a “deformation retract” of the space underlying $C$, in that we have an equivalence of $n$-groupoids $T C \simeq \mathrm{Obj}(C) \,.$ (I am grateful to David Roberts for emphasizing the importance of this fact, regarded this way, and to David Roberts and Jim Stasheff for discussion on this point.)

b) The $n$-bundle $T C$ fits into the short exaxt sequence $\mathrm{Mor}(C) \to T C \to C$ of $n$-groupoids.

c) There is a canonical monomorphism $\Gamma(T C) \hookrightarrow T_{\mathrm{Id}_C}(\mathrm{End}(C))$ of the space (an $(n-1)$-category, really) of sections of $T C$ into the categorical tangent space at the identity in the space of endomorphisms of $C$.

The latter is the space of categorical flows on $C$. This is naturally an $(n+1)$-group. And by this embedding the space of sections $\Gamma(T C)$, too, receives the structure of an $(n+1)$-group.

Endofunctors and Vector Fields

To appreciate the concept formation above, it is helpful to see how ordinary vector fields – and then their generalizations – arise from this point of view.

Definition. For $C$ any $n$-groupoid and $G_{(n)}$ any $n$-group, we say that an $n$-group homomorphism $v : G_{(n)} \to \Gamma(T C)$ is a $G_{(n)}$-flow on $C$.

We shall hence think of $\mathrm{Hom}(G_{(n)}, \Gamma (T C))$ as the space of $G_{(n)}$-vector fields on $C$.

Example. For $X$ a smooth manifold and $P_1(X)$ its path groupoid, ordinary vector fields on $X$ are smooth $\mathbb{R}$-flows on $P_1(X)$.

Hence the space of $\mathbb{R}$-vector fields on $P_1(X)$ is indeed the space of ordinary vector fields on $X$: $\mathrm{Hom}(\mathbb{R}, \Gamma T(P_1(X))) \simeq \Gamma(T X) \,.$ (As before, I should admit that I have a strict proof so far only for the inclusion of the right hand side into the left hand side. The converse is a little more technical.)

But noticing that for any Lie group $G$ we have $\mathrm{Hom}(\mathbb{R}, G) \simeq \mathrm{Lie}(G)$ we should be able to restate this as $\mathrm{Lie}( \Gamma T(P_1(X))) \simeq \Gamma(T X) \,.$ This is actually essentially just saying that vector fields on $X$ form the Lie algebra of diffeomorphisms on $X$ connected to the identity.

Maps of categorical tangent bundles

With this in hand, there is now a straightforward definition of categorical maps of vector fields.

Definition.

For $C$ and $D$ $n$-groupoids and with some $n$-group $G_{(n)}$ fixed, we say that a morphism of their tangent $n$-bundles $f : T C \to T D$ is a map of $G_{(n)}$-vector fields if $f$ respects $G_{(n)}$ flows on $C$ and $D$ in the following, essentially obvious sense:

Recall what the “arrow theory” behind these concepts actuallly looks like:

Under our embedding $\Gamma(T C) \hookrightarrow T_{\mathrm{Id}_C}(\mathrm{End}(C))$ a section of the categorical tangent bundle is a “categorical tangent” in $T_{\mathrm{Id}_C}(\mathrm{End}(C))$ to the identity map from $C$ to itself, hence a transformation $\array{ & {}^{\;\;}\nearrow \searrow^{\mathrm{Id}} \\ C &\;\;\Downarrow^{\mathrm{exp}(v)}& C \\ & {}_{\;\;\;\;\;\;}\searrow \nearrow_{\mathrm{Ad}_{\mathrm{exp}(v)}} } \,.$

As we precompose this with any map $x : \mathrm{point} \to C$ from the point $\mathrm{pt} = \{\bullet\}$ into $C$, we obtain an element in $T_x C \,,$ namely $\array{ & {}^{\;\;}\nearrow \searrow^{x} \\ C &\;\;\Downarrow^{}& C \\ & {}_{\;\;\;\;\;\,}\searrow \nearrow_{\mathrm{exp}(v)(x)} } \;\;\; := \;\;\; \array{ & & & {}^{\;\;}\nearrow \searrow^{\mathrm{Id}} \\ \mathrm{pt} & \stackrel{x}{\to}& C &\;\;\Downarrow^{\mathrm{exp}(v)}& C \\ & & & {}_{\;\;\;\;\;\;}\searrow \nearrow_{\mathrm{Ad}_{\mathrm{exp}(v)}} } \,.$

This just means picking out the value of the given section of $T C$ at the point $x \in \mathrm{Obj}(C)$.

This element of $T_x C$ may then be sent with with our map $f$ to $T_{f(x)} D$. And clearly, for $f$ to be a map of $G_{(n)}$-vector fields, we should require that this procedure takes $G_{(n)}$-flows to $G_{(n)}$-flows, i.e. $f \;\;\;\; : \;\;\;\; \array{ & & & {}^{\;\;}\nearrow \searrow^{\mathrm{Id}} \\ \mathrm{pt} & \stackrel{x}{\to}& C &\;\;\Downarrow^{\mathrm{exp}(v(t))}& C \\ & & & {}_{\;\;\;\;\;\;\;\;}\searrow \nearrow_{\mathrm{Ad}_{\mathrm{exp}(v(t))}} } \;\;\;\; \mapsto \;\;\;\; \array{ & & & {}^{\;\;}\nearrow \searrow^{\mathrm{Id}} \\ \mathrm{pt} & \stackrel{f(x)}{\to}& D &\;\;\Downarrow^{\mathrm{exp}(w(t))}& D \\ & & & {}_{\;\;\;\;\;\;\;\;}\searrow \nearrow_{\mathrm{Ad}_{\mathrm{exp}(w(t))}} }$ for a suitable $w$.

This last sentence is intentionally left a little vague. I am still thinking about how to formulate this best. There is an issue here to be dealt with whenever $C$ and $D$ have more than one object.

But, actually, my main point here, to which I will now finally come, is what happens in the case that $D$ has just a single object, i.e. in which $D = \Sigma H_{n}$ is the suspension of an $n$-group. In that case $T D = \mathrm{INN}_0(H_{(n)})$ is the inner automorphism $(n+1)$-group of $D$. Then our tangent vector map $f$ will take values in the inner automorphism $(n+1)$-group of $H_{(n)}$. Since that’s the tangent space to the single point of $D$. (Compare this to the notion of tangent stacks, courtesy of David Ben-Zvi.)

So looking at it from this point of view, we find that a connection $n$-form for given structure $n$-group $H_{(n)}$ is actually a map $\mathbf{A} : T P_n(X) \to T \Sigma H_{(n)}$ hence a map $\mathbf{A} : T P_n(X) \to \mathrm{INN}_0(H_{(n)}))$ hence should come, differentially, from a Lie $(n+1)$-algebra morphism $A^* : \mathrm{inn}(h_{(n)})^* \to \Omega^\bullet(X) \,.$ Since $\mathrm{inn}(h_{(n)})$ is trivializable, any such map will be homotopic to the trivial such map – but as discussed at great length at various places now, it the choice of trivializing homotopy which encodes the expected information. And second order homotopies of that encode gauge transformations, and so on.

My suggestion, therefore: maybe we are being told here we are not supposed to be looking at homotopies of $L_\infty$ morphisms unless the target is $\mathrm{inn}$ of something.

While, clearly, this requires further thinking, I thought I’d mention this observation here.

Posted at August 8, 2007 8:48 PM UTC

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Re: Arrow-Theoretic Differential Theory, Part II

I’m having to swallow a bit of pride here and say that even though I feel like I’m getting the gist of these discussions better, I feel woefully lost in the technical details. I’m trying to come up with a good prefatory reading list in order to understand everything that you’re doing here with what you call arrow-theoretic differential theory.

Now I imagine that I should be able to understand Isham’s series on “A New Approach to Quantizing Space-Time” and John Baez’s last few years of quantum gravity seminar notes, but are these good places to start?

Posted by: Creighton Hogg on August 10, 2007 10:34 PM | Permalink | Reply to this

Re: Arrow-Theoretic Differential Theory, Part II

I hear you Creighton. If we ask really nicely, I wonder if Urs would give a brief “Arrow Theory 101” course? It seems that following this guiding light has taken him pretty far. I can tell it is important, but still haven’t broken through yet.

Posted by: Eric on August 10, 2007 11:19 PM | Permalink | Reply to this

Re: Arrow-Theoretic Differential Theory, Part II

Sorry for the slow reply here. It’s not that I don’t care about replying to this. Quite the opposite: I keep waiting for a moment of leisure to answer this in the detail it deserves.

Generally, you should be well aware that what I wrote here is stuff that I dreamed up while thinking about the general problem of $n$-categorical quantum theory. This means there is not quite a “canonical reading list” with background material. In as far as this is any good at all, this is a theory in the making!

But, yes, it is true that Isham’s “arrow fields” are actually an example of what I would now call a “section of the tangent category”.

(As I describe in that entry, I was thinking about “flows on categories” while unaware of Isham’s work. When I then read it I realized that this is closely related.)

And also, quite generally in what I am doing here I feel strongly influenced by John Baez’s general way of using $n$-categories (ort at least that’s how it appears to my mind):

the point here is to take quite seriously the idea that if we have something $n$-dimensional, then it ought be modeled by something $n$-categorical.

For instance: if we want parallel surface transport, then we do want to be looking at 2-functors. This is what got me started on all this!

And this is also where all this is headed: I would like to understand, systematically and from first principles, how to deal with “extended $n$-dimensional QFT”.

So, I first want to consider $n$-categorical analogues of points. Then the differential geometry of these beasts. Then the theory of connections these may couple to. Then the quantization of that setup. And then finally the resulting quatum propagation $n$-functors.

In this arrow-theoretic differential theory I am trying to put into a coherent form a couple of ingredients which, by lots of trial and error, I happen to have found crucial for doing this exercise.

One of these ingredients is the curious appearance of those “inner automorphism $(n+1)$-groups”. One main point of the “aroow theoretic differential theory” which I am talking about here is to clarify why exactly these show up, by demonstrating that they are really nothing but tangent categories! And that there is a general “exterior differential” operation on arbitrary functors which sends each $n$-functor to an $(n+1)$-functor “with values in a tangent categoy”, roughly.

And on top of that, I am trying to indicate how all this actually follows from an idea which is at the very heart of synthetic geometry and of supergeometry: that a tangent vector is the image of an “infinitesimal interval”, mapped into our space.

That’s where the superstuff enters here. And I am still hoping to understand this even better. Superalgebra is apparently so fundamental to physics, that I think chances are good that it really is something quite fundamenmtal on a rather abstract level. I would like to see supergeometry arise from “arrow theory” not as an add-on done by hand, but as the most natural thing in the world, which arises all by itself.

I think the way tangent categories and supercategories are related, as I tried to discuss, indicates that this hope is not completely vain.

Posted by: Urs Schreiber on August 14, 2007 7:39 AM | Permalink | Reply to this

Re: Arrow-Theoretic Differential Theory, Part II

Hi Urs,
I was out of town for a week and there’s so many more posts made! This variation of Zeno may well have me beat.

In all seriousness though, I think your summary gives me a bit better feel for what to focus on and I appreciate it very much. I do find it elegant and natural to connect internal degrees of freedom to arrows in the configuration n-category.

Now, one thing that I keep wondering is how one defines a proper notion of smoothness for these arrows when considered as transitions of state. Nature may not be smooth at it’s heart, but it certainly looks like it is.

Posted by: Creighton Hogg on August 22, 2007 7:01 PM | Permalink | Reply to this

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