July 21, 2007

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:

Tangent Categories
(pdf, ps)

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 $\mathrm{INN}(G_2)$.

Posted at July 21, 2007 4:09 PM UTC

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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.

Posted by: Urs Schreiber on July 21, 2007 4:58 PM | Permalink | Reply to this

Re: Tangent Categories

Let $C$ be any category. It’s tangent space at any object $x\in Obj(C)$, 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 $T C :=\oplus_{x\in Obj(C)}T_x(C)\, .$

Would it be too much trouble to give an example (a Lie group? the (based) loop space of a manifold? …) where this reproduces the usual notion of “tangent bundle”?

Posted by: Jacques Distler on July 22, 2007 5:02 PM | Permalink | PGP Sig | Reply to this

Re: Tangent Categories

Would it be too much trouble to give an example

I did! This is what I wrote above:

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.

Posted by: Urs Schreiber on July 22, 2007 5:54 PM | Permalink | Reply to this

Re: Tangent Categories

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 $(TP_2 (X))_{\sim} = T X$, where $( \cdot )_\sim$ here denotes dividing out 2-isomorphisms.

Sorry. I must be really slow.

1. Paths in $X$ have a beginning and an endpoint. $T X|_p$ should depend only on the “source” point, $p$.
2. It seems to me that putting some condition on the tangent vectors to a path is tantamount to assuming the answer. That is, that there’s a vector space attached to every point $p\in X$, and that we throw away everything about your construction, of paths up to homotopy, except for a choice of vector in that vector space.
3. But, even for the simplest case, $X=S^1$, your construction, as stated, doesn’t seem to give the right answer. $T S^1=\mathbb{R}\times S^1$, whereas (unless I have misunderstood you) the space one gets by following your prescription isn’t even connected.
Posted by: Jacques Distler on July 23, 2007 2:14 AM | Permalink | PGP Sig | Reply to this

Re: Tangent Categories

Sorry. I must be really slow.

I am grateful that you are asking!

Paths in $X$ have a beginning and an endpoint. $T X_p$ should depend only on the “source” point, $p$.

Yes, or rather, on the “vicinity” of $p$.

It seems to me that putting some condition on the tangent vectors to a path is tantamount to assuming the answer.

I know what you mean. Maybe it’s not quite tantamount to assuming the answer, but it might look a little unsatisfactory, true. Maybe I can improve on that eventually. (The second description which I mentioned (below) may be more satisfactory from this point of view. I’ll think about it, maybe there is a way to combine these two descriptions to one that shares the advantages of both.)

I think it does. Let’s see:

objects are paths starting at $p$.

Two such paths $\gamma,\gamma'$ are connected by an isomorphism (identified) if there is a path $\gamma''$ conecting their endpoints such that $\gamma'' \circ \gamma$ and $\gamma'$ are connected by a homotopy fixing both endpoints and their tangents.

Posted by: Urs Schreiber on July 23, 2007 12:51 PM | Permalink | Reply to this

Re: Tangent Categories

I think it does.

Paths which start at $p$ and end at $p'$, but which, along the way, wind a different number of times around the circle, are not homotopic to each other. Ergo, the space of such paths is not connected.

Posted by: Jacques Distler on July 23, 2007 4:59 PM | Permalink | PGP Sig | Reply to this

Re: Tangent Categories

Paths which start at $p$ and end at $p'$, but which, along the way, wind a different number of times around the circle, are not homotopic to each other. Ergo, the space of such paths is not connected.

Yes. Therefore it is crucial that there is one more ingredient in the game here:

Two such paths $\gamma,\gamma'$ are connected by an isomorphism (identified) if there is a path $\gamma''$ connecting their endpoints such that $\gamma'' \circ \gamma$ and $\gamma'$ are connected by a homotopy fixing both endpoints and their tangents.

This is a consequence of the fact that morphisms $\gamma \to \gamma'$ in $T_p P_1(x)$ are triangles $\array{ && p_1 \\ & {}^\gamma\nearrow \\ p &{}^\sim \Downarrow& \downarrow^{\gamma''} \\ & {}_{\gamma'}\searrow \\ && p_2 }$ and not just bigons $\array{ & \nearrow \searrow^{\gamma} \\ p &\Downarrow^\sim& p' \\ & \searrow \nearrow^{\gamma'} } \,.$

Posted by: Urs Schreiber on July 23, 2007 5:14 PM | Permalink | Reply to this

Re: Tangent Categories

There is also the following alternative formulation of the space of vector fields $\Gamma(T X)$ in terms of this language of tangent categories:

Let $X$ be any manifold and $P_1(X)$ its path groupoid (object space is $X$, morphisms are thin homotopy classes of paths in $X$).

Then a categorical tangent to the identity map on $X$ $\mathbf{v} \in T_{\mathrm{Id}_{P_1(X)}}(\mathrm{Mor}(\mathrm{Cat}))$ is a “path field” on $X$: an smooth assignment of a path starting at $x$ for each $x \in X$.

(This is essentially what Chris Isham calls an “arrow field”.)

So one tangent vector in $T_{\mathrm{Id}_{P_1(X)}}(\mathrm{Mor}(\mathrm{Cat}))$ is a tangent vector in $T_x P_1(X)$ for each $x$ in $X$.

But the crucial difference is that on $T_{\mathrm{Id}_{P_1(X)}}(\mathrm{Mor}(\mathrm{Cat}))$ there is a monoidal structure: we may compose two path fields to yield another path field.

This implies that an ordinary vector field on $X$ (as opposed to a path field) is a smooth functor $v : \Sigma \mathbb{R} \to \Sigma T_{\mathrm{Id}_{P_1(X)}}(\mathrm{Mor}(\mathrm{Cat})) \,.$ as also described here.

(Well, I should admit that it is easy to see that every vector field on $X$ gives injectively rise to such a smooth functor and that this is I think it is also bijective, but haven’t written out the proof of the converse statement yet, which is a little more technical.)

Posted by: Urs Schreiber on July 22, 2007 6:13 PM | Permalink | Reply to this

Re: Tangent Categories

Sorry for this belated reply - medical problems have distracted me.

At the level of musings, your entries have inspired the thoughts:

1. for topological manifolds, there is the
Nash notion of a stable tangent fiber space defined as the dual of the stable normal fiber space which is the space of paths
that start in M imbedded in some high dim Euclidean space but immediately depart into the complement

2.old attempts to define tangentvectors/vectorfields on the path space of a smooth manifold M
in terms of

a. paths in TM over paths in M

b. maps I x I –> M such that…

Posted by: jim stasheff on July 30, 2007 12:02 AM | Permalink | Reply to this

Re: Tangent Categories

Dear Jim,

thanks for these comments! Both sound quite relevant for what I am thinking about. Do you maybe have more details on references concerning point 1?

And regarding point 2: are you just thinking of the standard way of identifying tangent vectors with classes of paths that induce the same differential for functions, or do you have something more subtle in mind here?

Posted by: Urs Schreiber on July 30, 2007 12:10 PM | Permalink | Reply to this

Re: Tangent Categories

Unfortunately I was vague precisely because
memory is failing me. I think both appeared in the mat/physics litt.
The picture I have in mind for the `tangent vector’ to a path
is a strip/ribbon I x path
with 0 x path as the original path
and the I component sort of representing a unit ?tangent vector

NOTE: NOT necessarily such that the tangent vector on M is given by I x P(t)
where P is the path
just that the strip is diffeo I x path

not sure this makes sense
but hopefully some reader will recall what is in the litt

and or set me straight

Posted by: jim stasheff on July 30, 2007 3:12 PM | Permalink | Reply to this
Weblog: The n-Category Café
Excerpt: The last session of Recent Developments in QFT in Leipzig was general discussion, which happened to be quite interesting for various reasons....
Tracked: July 22, 2007 7:37 PM

Tangent Stacks

Can anyone help me with this:

Question: What is the tangent stack to $[\mathrm{pt}/G]$ like?

I know the general definition of tangent stack, but I am not sure about exactly how to evaluate it for this (simple) example.

This is the definition:

For $M$ a differentiable stack with atlas $X \to M$, we hit the groupoid $X \times_M X \stackrel{\to}{\to} X$ with the tangent space functor $T$ to get $T(X \times_M X) \stackrel{\to}{\to} T X$ and then form the quotient $[T X / T(X \times_M X)] \,.$ That’s the tangent stack $T M$of $M$.

I would like to compare this to the notion of tangent category which I am talking about. For me, $[\mathrm{pt}/G]$ is the category $\Sigma G$ and its tangent category, the way defined above, is $T(\Sigma G) = G // G = \mathrm{INN}(G)$ the codiscrete groupoid over $G$.

That’s certainly not directly comparable to the tangent stack thing, for one just because it doesn’t involve vector spaces, but maybe one could compare if it is morally the same.

So, what’s the value of the tangent stack $T[\mathrm{pt}/G]$ over the point? Should be two vector spaces $V_0$ and $V_1$ with a map $V_1 \to V_0$. Which one is it?

Posted by: Urs Schreiber on July 26, 2007 11:02 AM | Permalink | Reply to this

Re: Tangent Stacks

Urs,

There are a couple of ways to answer this. First, one can ask about the tangent complex to pt/G. More generally for a quotient stack X/G (with X smooth) we can calculate its tangent complex as $\g_X \to T_X$, where $\g_X$ is the action Lie algebroid (ie the trivial bundle with fiber the Lie algebra) and the map is the action (aka anchor map). So in this case it’s just $\g \to 0$. This needs to be interpreted as a complex of vector bundles on the quotient, ie we consider $\g$ (and $0$) not as vector spaces but as representations of $G$.

As for the tangent stack, it’s not very interesting here I think, it’s just pt/G itself - there are no new tangent directions in the quotient. Another way to say this: we can naively define the tangent stack as Spec of $H^0$ of the symmetric algebra of the cotangent complex (dual to the above tangent complex), which in this case is Spec of $H^0$ of the symmetric algebra on $g^*$ in degree one, which is just $pt/G$.

However, one does get the interesting answer one feels like one deserves if you look at the derived cotangent stack (ie pass to the world of stacks over commutative dgas, rather than over commutative rings). In other words while there are no new degree zero cotangent directions in pt/G, there is a g* worth of degree 1 cotangent directions…. said another way, Spec of the symmetric algebra on the tangent complex defines a (differential) graded vector bundle over pt/G, which is just g* in degree 1 (again considered as the coadjoint representation of G so as to define a vector bundle on $pt/G$). Maybe that fits better with what you would like?

(Note I switched from tangent to cotangent to be a little careful with gradings: the world of derived stacks is not symmetric with respect to positive and negative gradings – of course one can pass to the super-world, and remember only $\Z/2$ gradings, if one so desires, and this distinction goes away - or perhaps another stable setting..)

More generally if we pass to higher (smooth) stacks we’ll get tangent complexes concentrated in an arbitrary range of negative degrees I believe, and derived cotangent stacks which are nonnegatively graded “vector bundles” over them.

Posted by: David Ben-Zvi on July 30, 2007 4:24 PM | Permalink | Reply to this

Re: Tangent Stacks

Hi David,

great, thanks, very useful.

(differential) graded vector bundle over $\mathrm{pt}/G$, which is just $g^*$ in degree 1

Ah, that sounds good. So, let me ask this:

as I have said, I am inclined to address the 2-group $\mathrm{INN}(G) = \mathrm{Codisc}(G) = (G \stackrel{\mathrm{Id}}{\to} G)$ as something like the tangent bundle to $\mathrm{pt}/G$.

Now, the Lie 2-algebra of this 2-group is given by the differential graded algebra which as an algebra is the free graded commutative algebra $\wedge^\bullet (s g^* \oplus ss g^*)$ with $s g^*$ denoting $g^*$ regarded as sitting in degree 1 and with $ss g^*$ the same thing but now regarded as sitting in degree 2 (or use negative gradings, if desired).

The differential on that is that of the Weyl algebra.

Do you see that full structure appear from your perspective?

Posted by: Urs Schreiber on July 30, 2007 5:32 PM | Permalink | Reply to this

Re: Tangent Categories

[Assume the following comment is set in magic ink: you cannot read. So don’t.]

There are at least four strands running around the Café here which presumeably belong tied together, but first need to be disentangled:

a) $n$-curvature

b) super and tangent categories

c) the tale of groupoidification

d) categorified/arrow theoretic quantum mechanics.

To indicate why, consider this approach to curvature 2-functors from the perspective of super and tangent categories:

Start with a 1-bundle with connection $\mathrm{tra} : \mathrm{tar} \to G\mathrm{Tor} \,.$ on target space.

Suppose for simplicity at the moment that it’s trivial $\mathrm{tra} : \mathrm{tar} \to \Sigma G \,.$

Now couple the fat point (the superpoint) $\mathbf{pt} = \{ \bullet \stackrel{\sim}{\to} \circ \}$ to that. Pulling everything back to the configuration space of that superpoint, we get $d\mathrm{tra} : T \mathrm{tar} \to T \Sigma G$ hence $d\mathrm{tra} : T \mathrm{tar} \to \mathrm{INN}(G) \,.$

Now use the fact that the tangent category $T C$ is actually a 2-bundle over the objects of $C$ $p : T C \to \mathrm{Obj}(C) \,.$ So we can actually think of $\mathrm{tra}$ as being the connection on a trivial $\mathrm{INN}(G)$-2-bundle on $\mathrm{Obj}(\mathrm{tar})$. In this sense the transport $\mathrm{tra}$ along $x \stackrel{\gamma}{\to} y$ sends the space of possible “Wilson line values strating at $x$” to the space of possible “Wilson line values starting at $y$”.

But since now $\mathrm{tra}$ this way acts on a 2-bundle, we can actually check if the transport along two different paths this way is connected by a transformation. And indeed, they are: this transformation is the value of the curvature 2-functor around the surfave enclosed by the two paths (if such exists).

Open exercise: formulate this kind of passage from the transport transgressed to the superparticle’s config space to the curvature 2-functor in a nice powerful abstract formulation.

But also notice that the passage to the fat point’s config space turns the fibers from looking like $G$ to looking like $\mathrm{INN}(G) = G // G$.

By the Tale, this means the fibers now look like $G$-representations!

So in some magical sense the curvature of the principal transport looks like a vector transport, somehow.

Something is going on. Exercise: figure it out.

Posted by: Urs Schreiber on July 26, 2007 3:19 PM | Permalink | Reply to this

Re: Tangent Categories

The magic ink did work. Before I call it quits today, I’ll share this, a neat (I think) refinement of the concept of $n$-curvature using the concept of tangent categories.

Curvature from categorical differentials of transport

Abstract. Using the concept of tangent categories, a notion of differential of functors can be defined. We indicate how for the case that the functor is the parallel transport functor of a bundle with connection, its differential is the corresponding curvature 2-functor.

The point is that the tangent category construction in fact gives, for each category $C$, a functor $T C : C^{\mathrm{op}} \to \mathrm{Cat}$ which sends any morphism $f : a \leftarrow b$ to the functor $T_a C \stackrel{T_f C}{\to} T_b C$ between the tangent categories over the source and target, which simply precomposes with $f$.

(I wouldn’t be surprised if everything I am doing here already runs under some standard name and is well known. Rather, I am surprised that nobody has told me so yet. So if you are an expert and shaking your head at my naïvety, please drop me a note and release me from my ignorance!)

Using this, one finds that for every functor between categories $F : C \to D$ the composition $T_* F : C \stackrel{F}{\to} D \stackrel{T D}{\to} \mathrm{Cat}$ has a unique extension to a 2-functor $d\mathrm{tra} : C_2 \to \mathrm{Cat}$ when $D$ is a groupoid. Here $C_2 = \mathrm{Codisc}(C)$ is the 2-category whose Hom-categories are the codiscrite groupoids over the Hom-sets of $C$.

Applying all this to the special case that $C$ is some category of paths in some space and that $D$ is some codomain where parallel transport takes values in, this reproduces all the stuff about curvature 2-transport which I was going on about.

In particular, notice that if we have parallel transport $\mathrm{tra} : P_1(X) \to \Sigma G$ in a trivial $G$-bundle (possibly the local trivialization of some nontrivial one) then the curvature 2-transport $d\mathrm{tra} := \mathrm{curv} : P_2(X) \to \mathrm{Cat}$ sends each point to a fiber of the form $T \Sigma G := \mathrm{INN}(G)$ over it.

A nice aspect of this concise formulation of 2-curvature is that it unifies the two concepts of morphisms into the curvature 2-functor which I have been emphasizing so much.

The point is that $\mathrm{tra}$ is trivializable in various ways (closely related to the fact that $\mathrm{INN}(G)$ is equivalent to the trivial category.) But the choice of trivialization encodes interesting information:

a) Stokes’ theorem is essentially the fact that the transport 1-functor is the component map of an isomorphism $d I \stackrel{\mathrm{tra}}{\to} \mathrm{curv} \,,$ where $I$, in the above simple example, is the trivial $G$-bundle with trivial connection.

b) A section of the original $G$-bundle, together with its covariant derivative under the given connection, is an equivalence $e : d J \stackrel{\sim}{\to} \mathrm{curv} \,,$ where $J$ is the trivial point-bundle.

If you think about it, that’s pretty neat, I’d say. It’s the concept of holography, really.

Posted by: Urs Schreiber on July 26, 2007 8:15 PM | Permalink | Reply to this

Re: Tangent Categories

In particular, notice that if we have parallel transport $tra:P_1(X) \rightarrow \Sigma G$ in a trivial G-bundle (possibly the local trivialization of some nontrivial one) then the curvature 2-transport…

What confuses me is that since $tra$ is a functor from one category to another, what does the word ‘curvature’ mean? I would think that the curvature of any functor must be zero… doesn’t curvature somehow measure the failure of functoriality?

Posted by: Bruce Bartlett on July 26, 2007 9:06 PM | Permalink | Reply to this

Re: Tangent Categories

What confuses me is that since $\mathrm{tra}$ is a functor from one category to another, what does the word ‘curvature’ mean? I would think that the curvature of any functor must be zero… doesn’t curvature somehow measure the failure of functoriality?

Curvature should measure the difference, in some way, of going from $a$ to be $b$ in two different ways.

In a codiscrete groupoid, like a pair groupoid of some space, there is only one direct way to go between any two points. So on such a groupoid any functor is necessarily flat. Non-flatness could be achieved by making it a mere pseudofunctor, i.e. weakening its respect for composition.

But think instead of parameterized paths, for instance. Then it’s clear that you can have a functor which is still non-flat in that it may take different values on different morphisms with the same endpoints.

But it all ties together nicely: notice that the curvature 2-functor of a given 1-functor is a functor on the codiscrete groupoid of the Hom sets of the original domain, so it’s 2-flat. And that’s the Bianchi identity.

Posted by: Urs Schreiber on July 26, 2007 9:24 PM | Permalink | Reply to this

Re: Tangent Categories

I have received another reply to my comment above by email. If there is one person wondering, there are probably more, so I’ll reproduce here some more clarifying words which I sent by email:

Actually it is easy to get confused about this point, mostly because some people use different terminology. For instance in some of Anders Kock’s papers he is talking about graph maps and calls the failure of these to extend to functors on the corresponding graph categories “curvature”.

But, even if not explicitly so, this in fact harmonizes with what I said: that’s because the graphs that Kock eventually actually uses are essentially those underlying pair groupoids, or rather his synthetic version of these.

Another issue which makes it easy to get confused is this:

to a large extent, smooth $n$-functors on $n$-paths are like pseudo-functors (1-functors with values in an $n$-category respecting composition only weakly) on the pair groupoid. (There is supposed to be a theorem here, but it still hasn’t been written out in detail.) If you draw a picture, you’ll immediately see how this works. For instance the compositor of the pseudofunctor is what the 2-functor assigns to surfaces.

This fact is good to keep in mind for translating back and forth between the way Kock and others think about curvature, and the “more general” (I’d say) way.

And the most useful application of this fact is its differential version: the reason that smooth $n$-functors with values in $n$-groups correspond to Lie algebroid morphisms from the tangent algebroid (which is the differential of the pair groupoid!) to the Lie $n$-algebra is that the latter really are the differentials of the pseudofunctors corresponding to the former.

Posted by: Urs Schreiber on July 27, 2007 11:08 AM | Permalink | Reply to this
Read the post Arrow-Theoretic Differential Theory
Weblog: The n-Category Café
Excerpt: We propose and study a notion of a tangent (n+1)-bundle to an arbitrary n-category. Despite its simplicity, this notion turns out to be useful, as we shall indicate.
Tracked: July 30, 2007 4:02 PM
Weblog: The n-Category Café
Excerpt: The concept of an "Adinkra" - a graph used to describe representations of N-extended d=1 supersymmetry algebras - remarkably resembles some categorical structures which appear in the context of supersymmetry.
Tracked: August 7, 2007 11:14 PM
Read the post Arrow-Theoretic Differential Theory, Part II
Weblog: The n-Category Café
Excerpt: A remark on maps of categorical vector fields, inner derivations and higher homotopies of L-infinity algebras.
Tracked: August 8, 2007 10:53 PM

Re: Tangent Categories

I recently wrote an email to urs asking about literature for referencing super-things, he wrote:
what I find a useful introduction to superalgebra is

Manin, Yuri I. Gauge field theory and complex geometry. Grundlehren
der Mathematischen Wissenschaften, 289. Springer-Verlag, Berlin, 1997.
ISBN 3-540-61378-1

now another question:
since there are many different quivalent formulations of what the tangent space at a point is, shouldn’t there similarly be several different constructions that you can categorify and you should get equivalent (n-isomorphic) objects? why did you choose that realization of a manifold? the 2 morphisms seem a little forced, and the initial definition of the tangent category of C at x seems very naive. what happens when you categorify the leibniz rule? (i don’t know the answer) this seems like it could be a real big lead on what you should get in the end. Also, why not use sheaves on your category or an appropriate generalization to model smooth functions on a manifold just like is done in algebraic geometry?
these may all have really simple answers and maybe it is what you are interested in that leads you to pick your particular formulation. i will be thinking more about this, but i really think this stuff is fascinating. thanks for your encouragement!

sean “the hairy grad student” tilson

Posted by: sean tilson on August 14, 2007 3:00 PM | Permalink | Reply to this

Re: Tangent Categories

the 2 morphisms seem a little forced

Let’s see.

First to clarify, in case I wasn’t being clear on this:

notice that there can be nontrivial 2-morphisms in $T C$ only if $C$ is a 2-category to start with. If $C$ is an $n$-category, then so is $T C$.

what made me finally think that $T C$ is a good idea is precisely that its (higher) morphisms are not “forced” at all.

In fact, in the early stages I defined these higher morphisms by hand, such that they’d do for me what it seemed I needed to get done.

Then, after a little thinking, it occurred to me that what I did by hand was a actually just spelling out a very natural basic condition:

the higher morphisms in $T C$ are just higher morphisms of $n$-functors – of $n$-functor that start at the “fat point”.

Such higher morphisms of $n$-functors in general have components given by lots of “cylinders” in the codomain.

$T C$ is defined to be that maximal sub $n$-category of all these higher morphisms of $n$-functors where these cylinders have the identity morphism over one of the two objects of the fat point.

I liked to phrase this in the following suggestive (I belive) way:

$T C$ is that maximal sub $n$-category of $\mathrm{Hom}_{n\mathrm{Cat}}(\mathbf{pt},C)$ which collapses to a 0-category when pulled back along the inclusion of the point into the fat point.

I am thinking of this condition as the “arrow-theoretic analogue” of “infinitesimality”:

a tangent vector is something which becomes invisible when I stop distinguishing between a point and a “fat point”.

Of course this last sentence by itself is just heuristsics, which you may or may not find a helpful guide for thinking along the lines that I am proposing here. The real proof of principle is in checking whether $T C$ the way I defined it usefully captures what it is supposed to in applications.

Posted by: Urs Schreiber on August 14, 2007 3:31 PM | Permalink | Reply to this

Re: Tangent Categories

you are right, after you have made the definition the morphisms are natural, and i understand that you want the object to be corepresentable just like the earlier topos theoretic constructions. it just seems like you picked your 2-morphism of the category you made out of a manifold are a little forced, i guess that is what i meant. now, the two morphisms you picked should contain data about tangent vectors but it seems a little forced to me, ie if i asked a random person on the street to make a category out of a manifold or even a 2 category (supposing they knew what i was asking) i dont think they would come up with your construction. Is this really the best category to associate with a manifold? maybe, i dont know, but it seems like it is tailored to make your notion of tangent category be the tangent space. It just seems like it would be a more robust construction or rather notion if it was a categorified version of one of the other constructions. i guess first it would be good two know what an n-vector space should be then we would know how to categorify R^n then we can begin to categorify manifolds in the best possible way, locally a and n-category. Then you can go ahead and think of tangent vectors as natural transformations of n-functors to an appropriately “small” category (the reals).

the reason i think that your definition is naive is that it seems like there should be a limit somewhere around, the infinitessimal arrow object is initial right? also how do you generalize this to get a jet bundle it seems like there are a couple of ways.

these are essentially my worries.
peace,
sean “the hairy grad student” tilson

Posted by: Sean Tilson on August 14, 2007 4:03 PM | Permalink | Reply to this

Re: Tangent Categories

Oh, now I see what you are referring to. You are commenting on this discussion.

Yes, that’s right, the way to get tangent vectors which I describe there is a little forced. Right.

But as I also said there, there is another way to talk about not single tangent vectors but about tangent vector fields. And that’s in fact quite nice:

vecor fields on $X$ are $\mathbb{R}$-flows on $P_1(X)$, namely smooth $\mathbb{R}$-families of sections of the tangent category of the path groupoid.

See maybe Bruce’s description of this.

I think this notion of vector field as a $G$-flow on a category is the notion which is good and useful. That first notion of single tangent vectors is indeed much more “forced”.

Posted by: Urs Schreiber on August 14, 2007 4:35 PM | Permalink | Reply to this

Re: Tangent Categories

I have thought a bit more about this and i am now convinced that your construction is way to big. it seems much more analagous to unbased paths in the category, that is what you call TC should be called PC, and T_xC is more like paths in C starting at x.
My complaint about your example was more that this is a construction of a manifold as a category in an odd way that you have to force in order to get your construction to be the tangent space, not to mention that i dont recall you dividing out by 2 equivalences in the original definition.

Also it does not seem to have a natural complement that would play the role of the normal space/category. your construction would have as a complement in some embedding of the category in a larger one just the set theoretic complement and it would not seem to encode any interesting structure.

Another complaint, your “tangent vectors” are in the category while in general tangent vectors are not contained in the manifold they are tangent to, but it is implicit in your construction.
Perhaps in the case of super categories or other special structures that you are interested in, but i dont think it is the right object for categories in general.

I have a different proposed construction that may be completely wrong, please note that i am still trying to work on it and it is very much in a preliminary state. the ides is to take limits of maps out of x. i think this is a good idea, and that it philosophically has a lot in common with the standard construction. while a “tangent vector at x” may not be contained in the category we are working with, it will be contained in some complete category containing it. so a complete category could play a role similar to R^n. also note that we would get a type of vector addition in that if we have f:x–>y and g:y–>z then lim(fg) would be related to lim(f) and we might be able to think of this as the “sum of lim(f) and lim(g)” as “tangent vectors to C at x”. also we could get a type of scalar multiplication by using automorphisms of the target. another thought is that we could use equalizer diagrams to play the role of the inner product and that this will give us a notion of some sort of orthogonality.

there may be lots of problems with this, and the examples may not be write when you apply this, but i really think you are including too many “tangent vectors” and that there needs to be a limit somewhere in order to play the role of infinitessimals.

sorry for all the errors in spelling and i hope you like my suggestion.
peace
sean

Posted by: sean tilson on August 15, 2007 3:34 PM | Permalink | Reply to this

Re: Tangent Categories

Great. Thanks for this comment! You really thought through what I am proposing.

In fact, I pretty much entirely agree with you assessments. But do want to offer one or two points in favor of my terminology.

First, I did indeed consider for a while writing $P C$ instead of what I now called $T C$. Because, just as you say, taken at face value the objects of $T C$ are not “infinitesimal” at all.

But then I decided to keep the $T C$ notation (and the way of thinking about it as “tangents”) for the following two reasons:

a) $T C$ does enjoy a couple of properties which we do want to associate with the notion of a tangent bundle over $\mathrm{Obj}(C)$:

i) it is a “deformation retract” of $\mathrm{Obj}(C)$ in that there is a (canonical) equivalence $T C \simeq \mathrm{Obj}(C) \,.$ (I should say, as I did elsewhwere already, that this fact, and its importance, was emphasized to me initially by David Roberts and Jim Stasheff.)

ii) accordingly, there is a canonical “0-section” $\mathrm{Obj}(C) \to T C \,.$

b) Still, it is true, as you note, that $T C$ is “large” (too large, you might say, but wait a second). But this is supposed to be a feature, not a bug.

Namely, the idea is that we don’t just find ordinary vector fields in $\Gamma(T C)$, but also exotic generalizations of these, like tangent vectors to stacks but also “odd” vector fields as in supergeometry.

More precisely, this notion is very important, I think, for correctly estimating what $T C$ is and what it is not (and we may argue about the best notation afterwards):

the thing is that $\Gamma(T C)$ has a monoidal structure, and that we hence look at images of different groups inside $\Gamma(T C)$.

So, in particular, ordinary vector fields come from smooth $\mathbb{R}$-families in $\Gamma(T C)$. That’s where the “infinitesimal” aspect comes in: it’s because smooth $\mathbb{R}$-reps are specified already by their differential at the origin.

But already here more exotic stuff may be encoded in $\Gamma(T C)$: for $C$ a one-point orbifold, we find that the $\mathbb{R}$-families in $\Gamma(T C)$ correspond to the Lie algebra of the orbifold group.

That looks good, because it more or less coincides with the notion of tangent stack for the same orbifold, regarded as a stack. See my discussion with David Ben-Zvi about this.

But the point now is that we may look for images of more unusual groups inside $\Gamma (T C)$. As I keep emphasizing in the threads on “supercategories”, there are a couple of indications that morphisms from $\mathbb{Z}_2$ and other abelian groups into $\Gamma(T C)$ encode “odd” or “super” vector fields. See in particular maybe what I wrote in Lazaroiu on $G$-flows on Categories recently.

Essentially, for $G$ abelian Lazaroiu describes exactly what I am talking about here, namely group homomorphisms $G \to \Gamma(T C)$.

And I think that by looking at such morphisms as “odd vector fields on the category” (or rather on its space of objects), clarifies the situation.

So I’d like to make two points here:

$\bullet$ Considering $T C$ and in particular group images in $\Gamma (T C)$ is a useful way to look at ordinary vector fields and their generalizations

$\bullet$ at the same time, I acknowledge that the notation “$T C$” may still feel like a bit of a stretch. For me, this feeling disappeared after a while and reversed to its opposite. I now feel that it is very suggestive notation.

However, notation is always a matter of taste, so I won’t object if you object against this notation.

But let’s try to agree on the facts. Because it seems something interesting is going on.

In closing, I would like to mention that I do actually see on the horizon a much deeper reason to regard $T C$ as indeed a “vector bundle”, even though it does not seem to consist of vectors a priori at all.

This is related to John Baez’s program, known around here as the Tale of Groupoidification of replacing the category of vector spaces with combinatorial categories of spans of groupoids.

As I just today argued in The canonical 1-Particle, Part II, for $C = \Sigma G$ a one-object groupoid, we see in some appolications the tangent category $T C = T_\bullet \Sigma G = \mathrm{INN}(G) = G // G$ arise in pretty much exactly that role in which the groupoid $G // G$ (the action groupoid of $G$ on itself by one-sided multiplication) would arise as a “combiunatorial replacement” for a vector space on which $G$ is linearly represented.

Not sure if that convinces anyone, but I thought I’d mention it. I have the feeling we might eventually find that $T C$ has a much better interpretation really as a vector bundle than it currently seems, due to reasoning such as in the Tale. But that bit is speculative, at the moment.

Posted by: Urs Schreiber on August 15, 2007 4:54 PM | Permalink | Reply to this

Re: Tangent Categories

a natural complement that would play the role of the normal space/category

Well, to get to categorical normal bundles we obviously need to consider sub-categories first.

So let $C_1 \hookrightarrow C_2$ be an inclusion of categories.

This induces an inclusion $T C_1 \hookrightarrow T C_2$ of the corresponding tangent categories.

Next we would want to divide out the right hand side by the left hand side (possibly on the level of sections).

I haven’t thought about this until now. But I don’t see a priori why one couldn’t make good sense of this once one does think about it.

Posted by: Urs Schreiber on August 15, 2007 5:12 PM | Permalink | Reply to this

Re: Tangent Categories

the idea is to take limits of maps out of $x$

Yes, very good. Let me try to formalize this.

Let $C$ be our category and $x \in \mathrm{Obj}(C)$. We want a map “out of $x$” for each such object.

So it seems we want an endomorphism $f : C \to C$ mapping each object to some other object.

Now, we want to look at these “infinitesimally”. How do we do this? Well, noticing that if we have $f$, we can also form $f \circ f$, ect, we should require that $f$ itself is already built from “infinitely many small endomorphisms”, like a limit of $f = g \circ g \circ g \circ \cdots \circ g$.

We can simply say this as follows: we demand there to be a smooth group homomorphism $\mathbb{R} \to \mathrm{Aut}(C) \,.$ This will be completely determined by its differential at the origin. And gives our “infinitesimal generator”.

More or less equivalently, we can look $\mathrm{Lie}(\mathrm{Aut}(C))$.

(Of course this presupposes appropriate smooth structure on $C$ to make sense in the first place.)

Now, I say: we should add to that one more bit of information: we don’t just want to map any object $x$ to any other, $y$. But we want to specify a “path” along we do this. So we want a morphism from $x$ to $y$.

This is accomplished by not just looking at automorphisms, $f : C \to C$, but by looking at automorphisms with specified transformations connecting them to the identity automorphism.

This way we arrive at the deman that we want in fact a smooth group homomorphism from $\mathbb{R}$ not just to $\mathrm{AUT}(C)$, but to $\mathrm{INN}(C)$, where, recall, $\mathrm{INN}(C) = T_{\mathrm{Id}_C}(\mathrm{Aut}(C)) \,.$ Finally, noticing that the monoidal structure on $\Gamma(T C)$ which I kept going on about arises from the embedding $\Gamma(T C) \hookrightarrow T_{\mathrm{Id}_C}(\mathrm{Aut}(C))$ we find that following the idea you seemed to be hinting at here one actually arrives at what I am proposing to do! :-)

Posted by: Urs Schreiber on August 15, 2007 5:39 PM | Permalink | Reply to this

Re: Tangent Categories

i really dont think you should look at endomorphisms of see and i dont think tangent vectors should be morphisms in C. C need not be complete! a tangent vector should be a morphism in a completion of C! my proposal is essentially that for all maps out of x we have an associated tangent vector, lim(f) where f:x–>y. and i think depending on who you look to for definitions of limits, the morphism is implicit, or rather included in the information of the limit. the idea is that these limits may be indecomposable in some sense and that tangent vectors should be like this. i think that we do not need to have tangent vectors in the category, i dont see why this is a bad idea? maybe it isnt, but i dont think you have addressed it.

now there is a lot of motivation you mention that i just dont know, or rather havent learned yet. I have a lot of reading to do, so if in your reply you could mention which of the last posts i need to read, because i dont read every post, even though i should. thanks for all the encouragement.

Posted by: sean tilson on August 20, 2007 6:12 PM | Permalink | Reply to this

Re: Tangent Categories

Hi Sean,

there is really nothing I would enjoy more than to further discuss this. But right now I am not sure what say in reply to your comment that I had not already said before.

Also, I am not really sure what you actually have in mind.

For instance when you say

a tangent vector should be a morphism in a completion of $C$!

I don’t quite know by which criterium you would or would not regard my $T C$ as a “completion” of $C$. There is a surjection $T C \to C$ so in this sense $T C$ is larger. But I don’t know if that’s what you have in mind. Probably not.

Posted by: Urs Schreiber on August 20, 2007 7:55 PM | Permalink | Reply to this
Read the post On Roberts and Ruzzi's Connections over Posets
Weblog: The n-Category Café
Excerpt: On J. Roberts and G. Ruzzi's concept of G-bundles with connection over posets and its relation to analogous notions discussed at length at the n-Cafe.
Tracked: August 16, 2007 11:14 AM
Read the post More on Tangent Categories
Weblog: The n-Category Café
Excerpt: More comments on the nature of tangent categories and their relation to the notion of shifted tangent bundles to differential graded spaces.
Tracked: August 21, 2007 10:46 AM
Read the post The G and the B
Weblog: The n-Category Café
Excerpt: How to get the bundle governing Generalized Complex Geometry from abstract nonsense and arrow-theoretic differential theory.
Tracked: August 25, 2007 9:00 PM
Read the post Arrow-Theoretic Differential Theory IV: Cotangents
Weblog: The n-Category Café
Excerpt: Cotangents and morphisms of Lie n-algebroids from arrow-theoretic differential theory.
Tracked: September 3, 2007 4:29 PM

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