More on Tangent Categories

August 21, 2007

Posted by Urs Schreiber

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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

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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.

Posted at August 21, 2007 9:48 AM UTC

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Re: More on Tangent Categories

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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 must already 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 of out of it. 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 set of 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 a functor. But, it’s no worse than thinking of the tangent bundle of a manifold $X$ as a function

$T: X \to Vect$

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?

Posted by: John Baez on August 21, 2007 1:32 PM | Permalink | Reply to this

Re: More on Tangent Categories

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Thanks for this comment. I’ll have to run to get something to eat before the next talk starts, but here is a quick remark:

it is pretty important, I think, that the projection $T X \to X$ which you mention, with $X$ any $n$ -category, has $\mathrm{Mor}(X)$ – regarded as an $n$ -category with only identity $n$ -morphisms, as its kernel, i.e. this $\mathrm{Mor}(X) \to T X \to X$ is an “exact” sequence!

That may look like a trivial statement, but notice that when we take $X$ to be a one-object $n$ -groupoid, $X := \Sigma G_{{n}}$ , then this sequence reads $G_{(n)} \to G_{(n)} // G_{(n)} \to \Sigma G_{(n)}$ and “is” the universal $G_{(n)}$ -bundle. This vividly shows that something quite interesting is going on with these tangent $n$ -categories.

Posted by: Urs Schreiber on August 21, 2007 1:53 PM | Permalink | Reply to this

Re: More on Tangent Categories

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The ‘contemptibly crude’ Yoneda functor that John starts with, $Y: X \to Set$ sending $x \in X$ to $\{ arrows into x \}$ , is usually called the Cayley embedding of the category $X$ . As the name suggests, it’s full and faithful (and injective on objects, if you care, which you probably shouldn’t). A corollary of this is that any small category $X$ is equivalent to a subcategory of Set.

In particular, if you do this for a group $X$ (viewed as a one-object category) then you get Cayley’s result that every group is a subgroup of a permutation group.

In the ‘total category’ stuff towards the end of the post, John wonders whether he’s tripping over himself. No, it looks right to me, except that I haven’t checked all the “op”s. It’s a standard example of a $Cat$ -valued functor and its corresponding fibration.

Posted by: Tom Leinster on August 21, 2007 3:28 PM | Permalink | Reply to this

Re: More on Tangent Categories

Looks like I may soon get some answers to questions I posed on the way to a (sadly neglected) categorified wreath product.

Posted by: David Corfield on August 21, 2007 3:43 PM | Permalink | Reply to this

Re: More on Tangent Categories

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It’s a standard example […]

Do you have a reference where this standard example is considered? Maybe also for higher $n$ ?

Posted by: Urs Schreiber on August 21, 2007 3:52 PM | Permalink | Reply to this

Re: More on Tangent Categories

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First of all, oops: I mis-stated the properties of the Cayley embedding. It’s faithful and injective on objects, but not usually full. Nevertheless, the conclusion stands: every small category is equivalent (in fact, isomorphic) to a subcategory of $Set$ .

I can’t name you a reference off the top of my head, but if I was looking for it I’d try looking in introductory texts on fibrations. There’s one online by Thomas Streicher, and there’s also the Handbook of Categorical Algebra by Borceux.

The only people I know who’ve written about higher-dimensional Cayley embeddings are Robin Houston and myself. (Of course, lots of people have thought about higher-dimensional Yoneda embedddings, which are very closely related.) My contribution is in Chapter III of this paper. However, this contains some mistakes, as pointed out by Robin Houston, who has further interesting thoughts.

Posted by: Tom Leinster on August 21, 2007 5:57 PM | Permalink | Reply to this

Re: More on Tangent Categories

Tom wrote:

In the “total category” stuff towards the end of the post, John wonders whether he’s tripping over himself. No, it looks right to me, except that I haven’t checked all the “op”s.

Good. I was afraid I was “tripping op” in some manner that couldn’t be fixed by merely sticking in or removing an “op” here and there.

Posted by: John Baez on August 22, 2007 12:31 PM | Permalink | Reply to this

Re: More on Tangent Categories

I haven’t quite followed the details so can’t say anything useful, but since you’re asking I think for similar-sounding constructions I was wondering about the relation between what you discuss and derived loop spaces. The reason I mention this is that the derived loop space of a variety is exactly the shifted tangent bundle T[-1] which (up to grading conventions) appears above, but the definition of a derived loop space sounds very close to what you discuss (except of course we have arrows going out of and then back into x). Also distributions on the derived loop space are Hochschild cochains of functions on the space, which are a BV algebra (or homotopical version thereof). The derived loops into BG (for a group G) are the stack G/G (nothing funny derived there). For a general stack the derived loop space is a combination of the shifted tangent complex and of the inertia, both things that seem to play a role in the world of these posts.. (I learned about this stuff from Toen’s great survey article, and it’s also explained here.)

One thing I find unsatisfying with the BV quantization story of AKSZ etc is that, if I understand correctly, by considering only dg manifolds you don’t really get gauge theories. e.g. the AKSZ Chern-Simonsy model, based on the BV algebras Urs described, doesn’t see “large” gauge transformations, only the Lie algebra (again I may have totally missed the point). That’s one reason I prefer to work with DG (or derived) stacks, which have both dg directions and automorphisms of objects. The problem is I think the difference between stable and unstable homotopical things — stably both positive and negative directions, simplicial and cosimplicial, dg and stacky, etc all meld together into one happy continuum, but for applications in TFT I think we need not dg manifolds but dg stacks.. aaaanyway.

Posted by: David Ben-Zvi on August 21, 2007 4:17 PM | Permalink | Reply to this

Re: More on Tangent Categories

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The derived loops into $B G$ (for a group $G$ ) are the stack $G/G$

Thanks for saying this! This seems to be the kind of statement I was looking for when we talked about this last time, though then I had not emphasized the shift sufficiently, I guess.

So, what you just said, clearly is, when passing from stacks to the groupoids presenting them, just the statement I had emphasized so much: $T \Sigma G = G // G$ in my funny notation.

In words: the category whose objects are the morphisms starting at the single point of $\Sigma G$ (and hence, necessarily, also ending there) and whose morphisms are commuting triangles of these is nothing but the action groupoid of $G$ acting on itself from one side.

So, possibly then, it might be useful to think of “derived path spaces” here, more generally. Is anything like that considered anywhere?

One thing I find unsatisfying with the BV quantization story of AKSZ etc is that, if I understand correctly, by considering only dg manifolds you don’t really get gauge theories. e.g. the AKSZ Chern-Simonsy model, based on the BV algebras Urs described, doesn’t see “large” gauge transformations

Yes, exactly. That’s why I would like to identify the right “structure Lie $n$ -algebra” which appears here and then pass to the Lie $n$ -group integrating it.

In fact, I think this comment applies to the entire BV approach (but I might be missing something): by its very construction (as far as I perceive it) it is based on passing from the Lie $n$ -groupoid of physical configurations to just the underlying Lie $n$ -algebroid.

So, can you point me to anything like a “large” discussion of Chern-Simons using those stacky dg methods you mention?

Posted by: Urs Schreiber on August 21, 2007 4:43 PM | Permalink | Reply to this

Re: More on Tangent Categories

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One thing I find unsatisfying with the BV quantization story […]

That’s why I would like to identify the right “structure Lie n-algebra” which appears here and then pass to the Lie $n$ -group integrating it.

By the way, this really strikes a nerve:

after my talk I was asked:

“Why on earth do you want to understand a 3-functor underlying Chern-Simons?”

and

“Why on earth do you want to determnine a 3-group underlying Chern-Simons?”

I did try to explain why. But independent of how well I did, I found it remarkable that in the other talks in the conference one could and can see people struggling with pretty much exactly these issues – just without addressing them as such.

For instance Alberto Cattaneo a few hours ago talked about how to quantize certain Courant-algebroid sigma models on manifolds with boundary using AKSZ-BV. He described how he was trying to assign amplitudes of sorts to little squares of surfaces, trying to get a double groupoid. But, lo and behold, he finds that composition in the double groupoid works only up to certain homotopies, unless one tries to fake it.

In other words, he is seeing the need for a 3-functor! But doesn’t say so.

Posted by: Urs Schreiber on August 21, 2007 5:03 PM | Permalink | Reply to this

Re: More on Tangent Categories

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David, I’m a little confused, but do have a head-cold and haven’t thought about $T[-1]$ for a while. You say that $T[-1]$ is the derived loop space – is that what you mean? I was thinking that $O_\Delta\otimes O_\Delta$ (as on object in the derived category $D(X\times X)$ ) was something like functions on the derived loop space and that it is dual to the internal hom $Hom(O_\Delta,O_\Delta)$ which when pushed down to $D(X)$ is the universal enveloping algebra of the Lie algebra $T[-1]$ . In what (intuitive?) sense is $T[-1]$ the derived loop space?

Posted by: Simon Willerton on August 22, 2007 12:06 PM | Permalink | Reply to this

Re: More on Tangent Categories

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Simon, Yes that’s right, this is part of the same story about the Atiyah class etc that you explain (and apply) beautifully with Roberts. The derived loop space of a smooth variety is Spec of the Hochschild chain complex, i.e. $O_\Delta\otimes O_\Delta$ , which is calculated by the Koszul resolution to be the symmetric algebra of $T^*[1]$ , i.e. the total space of $T[-1]$ . (The same is true in a singular setting if we take the tangent complex instead.) The derived loop space is a (homotopical version of a) bundle of groups over the base, and its Lie algebra is $T[-1]$ (considered as a Lie algebroid or $L_\infty$ algebroid), its enveloping algebra is the Hochschild cochains (which are distributions on the derived loop space), etc. (We explain a little of this in our paper, but didn’t give a derived loop space interpretation of the full Markarian-type story, though I think that’s a natural context for it..)

The way I see it it’s confusing to restrict to smooth schemes, because linear and nonlinear things get mixed up. If we work with stacks we see some things more clearly: if we look at BG, then $T[-1]$ is just the Lie algebra $\g$ with its adjoint action of G, but the derived loop space is $G/G$ , the adjoint quotient - $T[-1]$ is just its linearization. Also in this setting one needs to distinguish big and small loops – ie the enveloping algebra construction really sees the formal group of the loop space, not the big loops.

That relates to my comment to Urs about Chern-Simons, which really needs the full loop space. It also relates to the question Urs is asking about negative vs positive cohomological directions – ie the stacky vs the dg directions.

Actually one gets a topological field theory out of the derived loop space which looks like a categorified form of string topology, I’m sorting out the details now with Nadler and John Francis.

Posted by: David Ben-Zvi on August 22, 2007 3:12 PM | Permalink | Reply to this

Re: More on Tangent Categories

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David B-Z(*),

Ah, you say $T[-1]$ is the Lie algebra of the loop space, that’s what I was trying to get at; it’s more an infinitesimal version of the loop space (not that I really understand what that means).

What Justin Roberts and I were doing was precisely trying to mimic (perturbative) Chern-Simons theory in this context, formally using $T[-1]$ as the Lie algebra of the gauge group and the derived category $D(X)$ as its representation category. Of course, what we wanted to say was that in the case that $X$ is symplectic there is an extended 3d TQFT with $D(X)$ as the category associated to the circle. However, at the time we (or at least I) couldn’t see how to associate this functorially to a circle – what have $D(X)$ and $T[-1]$ got to do with maps of a circle to $X$ ?

As you say, it is instructive to consider more stacky things, and at the opposite end of the spectrum to smooth spaces you have the example of a finite group over a point (corresponding to Dijkgraaf-Witten theory). From this perspective there is the same simple, lovely, monadic categorical reason for why the group algebra of finite group (with the adjoint action) is a Hopf algebra in the category of representations and for why $\pi_* {\Ext}(O_\Delta,O_\Delta)$ , the universal enveloping algebra of $T[-1]$ , is a Hopf algebra in $D(X)$ .

(*) We have three Davids on this thread – can we disambiguate (as Wikipedia would have us say) by trying some other forms of David? Anyone of you prefer Dave or Davey or Davie or Dai…?

Posted by: Simon Willerton on August 30, 2007 12:32 PM | Permalink | Reply to this

Re: More on Tangent Categories

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Re: *, I’m happy to go as BZ for disambiguation purposes (and to save me from being called Davey..)

Simon: I agree in general for a stack T[-1] is the Lie algebra of the derived loop space - my point was that in the case of a scheme it actually is the loop space, in the sense that when we consider T[-1] as a (derived) space rather than as a vector bundle (complex) (i.e. pass to its “total space”) we get the loop space. More precisely, we know that given a vector bundle on a variety we can construct a variety mapping affinely to the original (namely the total space) by taking Spec of the symmetric algebra of the dual. In the case of T[-1], Spec of Sym T^*[1] is precisely Spec of the Hochschild homology, which is the derived loop space. Here this is just an odd nilpotent space, but it still has the formal structure of a loop space (and things like string topology operations). This is one advantage of the world of derived schemes, that such a space (Spec of a negatively graded dga for example) makes sense.

I guess the issue is the same when we consider $\R^{0|1}$ as a Lie algebra or as a group. Once you have stackiness then the shifted tangent and the derived loop space differ since the loop space gets honest points not just (odd) infinitesimals.

Is there a sequel in the works for your very enlightening paper with Justin Roberts? (I realized Roberts needs disambiguation here too..)

Posted by: David Ben-Zvi on August 30, 2007 2:52 PM | Permalink | Reply to this

Re: More on Tangent Categories

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David BZ,

Thanks, whilst I’m not completely on top of derived geometry [understatement], what you’re saying certainly seems to make sense.

Re a sequel: there was supposed to be a sequence of three papers. The second paper involved perturbative calculations for the TQFT, calculating S-matrices and lots of fun stuff; the third paper was on the full TQFT, and we’d done some bits and pieces with Justin Sawon (*) on that. However, the first paper took over 5 years to come out, and it’s not clear what the status of the other two is. There’s a lot of the second paper written, and I think we have a lot better idea of what’s going on in the full TQFT now for the third paper. However, I don’t quite know whether they’ll reach the light of day or not.

(*) So we can’t disambiguate Roberts by just calling him Justin.

Posted by: Simon Willerton on September 4, 2007 1:59 PM | Permalink | Reply to this

Re: More on Tangent Categories

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The second paper involved perturbative calculations for the TQFT

Is this still referring to extended Chern-Simons?

calculating S-matrices and lots of fun stuff; the third paper was on the full TQFT

Wow! Sounds impressive. Are you anywhere close to constructing CS as a 3-functor already?

Posted by: Urs Schreiber on September 4, 2007 2:10 PM | Permalink | Reply to this

Re: More on Tangent Categories

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These ideas are very pretty. The idea of a tangent space reminds me a lot of an arrow category.

One obstacle in trying to “discretize” the idea is that for a “diamond complex”, there would be no tangent spaces and hence no tangent bundle because for edges

$e_1:x\to y_1$

and

$e_2:x\to y_2,$

there is no edge

$e:y_1\to y_2$ $.

Probably not significant, but thought I would throw it out there.

Posted by: Eric on December 17, 2008 4:40 AM | Permalink | Reply to this

Re: More on Tangent Categories

Any candidates for a cotangent n-category?

Posted by: David Corfield on August 21, 2007 3:11 PM | Permalink | Reply to this

Re: More on Tangent Categories

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Hmm - contangent vectors eat vectors and spit out numbers. In the categorical context, you’d need something that likes to eat arrows and spit out… what?

Urs is doing his stuff for a darn good reason; if he bumps into a darn good reason to get a categorical version of the cotangent bundle, let’s hope he notices that and reports back!

Posted by: John Baez on August 22, 2007 12:27 PM | Permalink | Reply to this

Re: More on Tangent Categories

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if he bumps into a darn good reason to get a categorical version of the cotangent bundle

I was thinking about this a lot, lately. I feel there is indeed a good motivation to think of cotangent categories. However, I still feel puzzled about what it all really is about, so that’s why I haven’t really talked much about it.

Indeed, I had a couple of paragraphs already written in reply to David’s question yesterday, when I realized that none of these thoughts were anywhere near the point where communicating them would actually lead to a decrease of intellectual entropy.

Originally, I thought that something like categorical cotangent bundles would be helpful to trace down why differential graded algebras are so omnipresent in physics.

But, we are understanding that all these differential graded algebras are really just the Koszul dual incarnation of the Lie version of $n$ -groupoids of physical configurations and/or states.

So the question is, for me: are all things cotangent best understood as just a more or less weird alternative point of view on just Lie $n$ -groupoids (and their tangent $n$ -groupoids)?

Or should one maybe try to understand that this “weird change of viewpoint” is actually not as weird. Maybe by relating it to a notion of cotangent categories.

I really don’t know at the moment. But maybe the vague remarks I just made trigger something in somebody.

If so, please trigger me back by dropping a comment here! :-)

Posted by: Urs Schreiber on August 22, 2007 12:43 PM | Permalink | Reply to this

Re: More on Tangent Categories

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What if we just tried a completely naive approach and tried to mimic $T^* X = Hom(T X, \mathbb{R})$ ?

So, starting with the special case of $\Sigma G$ for some group $G$ ,

$T^* \Sigma G = Hom(T \Sigma G, ?) = Hom(G//G, ?).$

Then we might fall back on that old workhorse ? = Vect.

Posted by: David Corfield on August 22, 2007 1:22 PM | Permalink | Reply to this

Re: More on Tangent Categories

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I wouldn’t want to use $\mathrm{Vect}$ here, that would somehow go against the grain of the picture which is emerging here, which is purely “combinatorial”.

But, I’d say the $n$ -category dual to the $n$ -category $C$ is $C^* := \mathrm{Hom}_{n\mathrm{Cat}}(C^{\mathrm{op}}, (n-1)\mathrm{Cat}) \,,$ i.e. that of all pre $n$ -sheaves on $C$ .

It might actually be reasonable to think $T^* C := (T C)^*$ this way.

Posted by: Urs Schreiber on August 22, 2007 1:40 PM | Permalink | Reply to this

Re: More on Tangent Categories

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So,

$(T \Sigma G)^* = (G // G)^* = Hom_{Cat} ((G//G)^{op}, Set).$

Oh, does 2 show up a lot in duality theorems as it’s $(-1)-Cat$ ?

Peter Johnstone mentions its many guises as a ‘schizophrenic’ object (which we were considering renaming ‘Janusian’ or ‘Janus-faced’ after Tom Leinster’s valid objection). Can Set be seen similarly as Janusian, under different guises?

Speaking of renaming things, perhaps we should claim ‘dual’ for the sense you’ve just given, and call ‘opposite’ what’s sometimes called ‘dual’.

Posted by: David Corfield on August 22, 2007 2:32 PM | Permalink | Reply to this

Re: More on Tangent Categories

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Your ‘dual’ Lawvere calls ‘concrete duality’ for a particular choice of dualizing object. ‘Opposite’ he calls ‘formal duality’.

Then, we read, “a large part of the study of mathematics … may be considered as the detailed study of the extent to which formal duality and concrete duality into a favorite $V$ correspond or fail to correspond.”

Posted by: David Corfield on August 22, 2007 3:48 PM | Permalink | Reply to this

Re: More on Tangent Categories

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Your ‘dual’ Lawvere calls ‘concrete duality’ for a particular choice of dualizing object. ‘Opposite’ he calls ‘formal duality’.

Interesting. I vaguely rememeber you having talked about this before, but I completely forget the details.

(But if you fix the link you wanted to provide, I’ll try to follow it ;-)

What does “opposite” refer to right at the moment? Just reversal of arrows?

Are there a couple of easily accessible concrete examples where we could see this relation between formal and concrete duality in action, usefully?

Posted by: Urs Schreiber on August 22, 2007 4:05 PM | Permalink | Reply to this

Re: More on Tangent Categories

I fixed the link. It was just to a post where I quoted from Lawvere and Rosebrugh’s ‘Sets for Mathematics’.

What happens to opposites in higher categories? Do you just reverse everything?

I can’t think of a situation where the Hom from one category to another is equivalent to the opposite.

Posted by: David Corfield on August 22, 2007 8:15 PM | Permalink | Reply to this

Re: More on Tangent Categories

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I can’t think of a situation where the Hom from one category to another is equivalent to the opposite.

Here’s a toy 2-enriched example: let X be a sup-lattice (in a topos, $Set$ if you prefer), and let 2 denote the subobject classifier. Then the opposite $X^{op}$ is sup-lattice-isomorphic to hom( $X$ , 2), where this hom is the poset of sup-preserving maps. (It also realizes an equivalence between the category of sup-lattices and its opposite, of course.)

Posted by: Todd Trimble on August 22, 2007 8:54 PM | Permalink | Reply to this

Re: More on Tangent Categories

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Oh, does $\mathbf{2}$ show up a lot in duality theorems as it’s $(−1)$ −Cat?

Ah, very interesting remark. I hadn’t thought about this.

Right, by the definition I just proposed, we find that for $S$ any set (0-category) the “dual set” is $S^* = \mathrm{Hom}_{\mathrm{Set}}(S, \{0,1\}) \,.$ But that’s just the set of subsets of $S$ !

Fun.

Posted by: Urs Schreiber on August 22, 2007 4:09 PM | Permalink | Reply to this

Re: More on Tangent Categories

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David writes:

Oh, does 2 show up a lot in duality theorems as it’s $(-1)-Cat$ ? Peter Johnstone mentions its many guises as a ‘schizophrenic’ object (which we were considering renaming ‘Janusian’ or ‘Janus-faced’ after Tom Leinster’s valid objection). Can Set be seen similarly as Janusian, under different guises?

Here’s one 2-categorical duality in which $Set$ appears as an <ahem> ambimorphic or Janusian object: between the 2-category of Cauchy complete categories on one side, and the 2-category of presheaf toposes and bicontinuous functors on the other.

Sweeping aside any set-theoretic subtleties, we have a 2-categorical Galois connection

$\frac{A \stackrel{bicont}{\to} Set^{C}}{C \to Bicont[A, Set]}$

where $A$ is complete and cocomplete. For $A = Set^C$ , $Bicont[Set^C, Set]$ gives the Cauchy completion $\hat{C}$ of $C$ . And for such $A$ , we have $A \simeq Set^{Bicont[A, Set]}$ .

A more interesting example is given by Gabriel-Ulmer duality. Rather than attempt to explain this, let me just refer to a nice paper which encapsulates a broad class of such dualities:

C. Centazzo and E.M. Vitale, A duality relative to a limit doctrine, TAC vol. 10 no. 20 (2002).

Posted by: Todd Trimble on August 22, 2007 7:23 PM | Permalink | Reply to this

Re: More on Tangent Categories

Thanks. I wonder if there’s a category which is more U(1)-ish to provide a categorified Pontryagin duality.

Posted by: David Corfield on August 22, 2007 8:27 PM | Permalink | Reply to this

Re: More on Tangent Categories

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I wonder if there’s a category which is more $U(1)$ -ish

John offered us the category of $U(1)$ -phased sets $\mathrm{Set}\downarrow U(1) \,.$

Since we were talking about tangent categories, it might be of interest to notice that groupoids over the tangent category $T \Sigma U(1) = T_\bullet \Sigma U(1) = \mathrm{INN}(U(1)) = U(1) // U(1)$ of $U(1)$ , play a very similar role.

And it might be remarkable that such groupoids appear automatically as we take sections of differentials of $U(1)$ -valued functors. In this sense:

suppose $F : C \to \Sigma U(1)$ is a functor on some category $C$ which sends each object to the single object of $U(1)$ and each morphism to an element in $U(1)$ .

Then, by the operation which I denote $\delta$ and argued to be the right notion of differential applied to functors (section 3.2), we find that $\delta F : C \to \mathrm{Grpd}$ is a functor which sends each object to the groupoid $U(1) // U(1)$ , i.e. $\delta F : ( x \stackrel{\gamma}{\to} y) \mapsto (U(1)//U(1)) \stackrel{F(\gamma)^*}{\leftarrow} (U(1)//U(1)) \,.$ A section of this functor, to be thought of literally (therefore the name) as a section of the $U(1)$ -bundle with connection represented by $F$ is a morphism into $F$ , hence another groupoid-valued functor $E : C \to \mathrm{Grpd}$ with a transformation $e : E \to \delta F \,.$ By the magic of tangent categories (and this is at the heart of the reason for looking at them in this context here) such a transformation is entirely determined, already, by its values over objects (over morphisms it will then automatically assign the corresponding “covariant derivative” ).

But over any object $x$ , such a section is nothing but a $U(1)// U(1)$ -phased groupoid: $e(x) : E(X) \to U(1)// U(1) \,.$ To potentially see that this is potentially a good thing, concentrate for a moment on the case that these groupoids $E(x)$ all have no nontrivial vertex groups, such that they are equivalent to just a set of isomorphism classes of objects. In this case a section as above is a function on $\mathrm{Obj}(C)$ with values in $U(1)$ -phased sets.

According to some general expectations floating around the $n$ -Café, this is a good way to think of a $U(1)$ -valued function.

I am hoping that the general nonsense of tangent categories combined with the general nonsense on curvature of transport which I am working on will, this way, actually neatly make contact with the Tale – and help free the world from vector spaces ;-)

Anyway, I just thought I might mention this in the present context.

Posted by: Urs Schreiber on August 22, 2007 9:04 PM | Permalink | Reply to this

Re: More on Tangent Categories

Hi David,
So this might be completely off base, but wouldn’t it be more inline with the philosophy here to parameterize the cotangent bundle over some kind of target object, like the one Isham uses in his recent topos papers to take the place of the real numbers?

I think what I’m grasping at is how would one do “classical” diff geo based upon a topos other than Set? Surely someone has done that before.

Posted by: Creighton Hogg on August 22, 2007 6:36 PM | Permalink | Reply to this

Re: More on Tangent Categories

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I’d love to know more about how one chooses a “favorite $V$ ” for forming a “concrete” dual.

$n$ -Cat can’t be the only answer. Why $U(1)$ for dual locally compact abelian groups? Why $\mathbb{R}$ for tangent/cotangent spaces?

Posted by: David Corfield on August 23, 2007 8:33 AM | Permalink | Reply to this

Re: More on Tangent Categories

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David,

I thought a little more about cotangent categories now. Possibly there is something interesting going on here.

But let me propose this point of view, which might nicely “explain” why, how and if we make a choice of “value category”.

One of the big lessons – for me at least – of this business of tangent categories is that they allow us to think of vector fields as images of groups in the space of sections of the tangent category.

The important example to keep in mind is: $X$ a smooth space, $P_1(X)$ its smooth path groupoid, then smooth group homomorphisms $\mathbb{R} \to \Gamma(T P_1(X))$ are ordinary vector fields on $X$ (the flow line of of a given vector field $v$ over a parameter length $t$ starting at $x$ is the value of the section over $x$ ).

The important point of this is that it is useful to replace $\mathbb{R}$ by other groups here, to get other notions of vector fields.

For instance, by simply looking at $\mathbb{Z}_2$ -flows on categories, one finds precisely all the structure which characterizes categories of supersymmetric D-branes, on which an “odd vector field” allows us to “flow” from any object to its superpartner.

The lesson I seem to be learning from this is: we should not try to specify our “value object” before hand. But let it be arbitrary.

Here is what I mean:

what is a differential 1-form, really?

I mean, there are subtle subtleties here, which we need to sort out if we want to have a chance of understanding cotangent categories:

Is a 1-form something that sends vectors to numbers?

Or is a 1-form something that sends vector fields to functions?

That’s a huge difference, when it comes to arrow-theory.

But maybe we have a hint now. If a vector field is to be thought of as a group homomorphism

$\mathbb{R} \to \Gamma(T P_1(X))$ then a covector field is probably best thought of as a group homomorphism groung the other way round $\Gamma(T P_1(X)) \to \mathbb{R} \,.$

If you think about it, that actually makes good sense. I believe any such group homomorphism from the sections of the tangent category of the path groupoid to the real numbers comes from having a 1-form on $X$ , choosing a point of $x$ and then sending each section of the tangent bundle to the result of integrating the 1-form over the given morphism emanating at $x$ , and exponentiating.

(Hope I am making sense here.)

You see what I am getting at: if we are looking at $\mathbb{R}$ -vectors coming from $\mathbb{R}$ -flows, then the dual thing should take values, accordingly, in $\mathbb{R}$ .

Butr we don’t want to restrict attention to $\mathbb{R}$ . Maybe there are also interesting $G$ -flows on our category, for $G$ some other group. Then we would think of these as giving “ $G$ -vector fields”. Their duals would be gandgets that spit out elements in $G$ .

Posted by: Urs Schreiber on August 23, 2007 6:16 PM | Permalink | Reply to this

Re: More on Tangent Categories

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I wrote:

a covector field is probably best thought of as a group homomorphism groung the other way round $\Gamma(T P_1(X)) \to \mathbb{R}$

As you probably noticed, that is in fact not a good idea. The quickest way to see that it is is a bad idea is that it implies that pairing a vector field $\mathbb{R} \to \Gamma(T P_1(X))$ would yield an automorphism of the additive group of real numbers $\mathbb{R} \to \Gamma(T P_1(X)) \to \mathbb{R} \,,$ which is not at all what we are after.

So I need more thinking here. But I still do believe that this idea of not specifying the valuation object, but allowing all possible ones is good.

So here is another remark I should make:

Let’s go back to the idea that the dual of a category $C$ is $C^* := \mathrm{Hom}(C^\mathrm{op},0\mathrm{Cat}) \,.$ Then let’s internalize this into smooth spaces. So the category $C$ is now assumed to have a smooth space of objects, of morphisms and source, target and composition being smooth maps. Moreover, the category of sets gets replaced by the category of smooth space. Let me call that $S^\infty$ . And possibly, might just want to think of ordinary smooth manifolds for the moment.

Anyway, then we get that the dual category is $\mathrm{Hom}(C^{\mathrm{op}},S^\infty) \,,$ whose objects are smooth functors, etc.

Now, we know that such a smooth functor on $C^{\mathrm{op}}$ with values in smooth spaces is just a smooth $\mathrm{Diff}$ -bundle with connection over the objects of $C$ !

So, in particular, such a functor is determined by something like a $\mathrm{Diff}$ -valued 1-form (an ordinary 1-form when $C$ happens to be the path groupoid of some manifold).

Hence $C^*$ contains all possible connectin 1-forms on $\mathrm{Obj}(C)$ , in a way, taking values in all possible “Lie algebras”.

Or something like that.

Posted by: Urs Schreiber on August 24, 2007 8:59 AM | Permalink | Reply to this

Re: More on Tangent Categories

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I know this may be restrictive, but what does

$Hom_{Cat}(T P_1 (X), \Sigma \mathbb{R})$

give you?

Posted by: David Corfield on August 25, 2007 9:54 AM | Permalink | Reply to this

Re: More on Tangent Categories

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And does

$Hom_{Cat}(Hom_{Cat}(T P_1 (X), \Sigma \mathbb{R}), \Sigma \mathbb{R})$

give you something equivalent to

$T P_1 (X)?$

Posted by: David Corfield on August 25, 2007 10:04 AM | Permalink | Reply to this

Re: More on Tangent Categories

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That can’t be right. Maybe I meant one of those Baez-Crans 2-vector spaces $\mathbb{R} \to \mathbb{R}$ rather than $\Sigma \mathbb{R}$ . But even then…

Posted by: David Corfield on August 25, 2007 12:59 PM | Permalink | Reply to this

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By the way, thinking of shifted cotangent spaces highlights another general aspect, which at some point needs to be better understood:

A Lie $n$ -algebra is the same thing as an $L_\inft$ -algebra structure concentrated in degree $1 \leq d \leq n$ (in some conventions at least. For other conventions, shift this and all the numbers to follow by the desired amount and multiply by the desired signs. But do it globall! :-)

We can literally think of the vector space in degree $k$ here as some tangent space to the space of $k$ -morphisms of a one-object Lie $n$ -groupoid.

The fact that there is just one object (one “0-morphism”) is hence reflaced in the fact that there is nothing in degree 0.

Indeed, $L_{\infty}$ -algebras concentrated in degree $0 \leq d \leq n$ correspond to Lie $n$ -algebroids.

(Even though this is, as far as I am concerned and aware, not really well discussed anywhere in the literature. When it comes to the many-object version, people usually pass to the dual differential graded algebra. I shall have to make a remark on that later.)

But now, it turns out that an $L_{\infty}$ structure on the graded vector space $V$ is exactly the same as a degree -1 codifferential $D : S^c (V) \to S^c(V)$ on the cofree co-commutative coalgebra $C^c(V)$ over $V$ .

Remarkably, from that point of view there is nothing which would stop one to consider negative degrees in $V$ !

While this is a rather boring statement on the level of (co)differential (co)algebras – indeed people around me here in Vienna routinely consider such negativelygraded structures (when you hear them say “N-manifolds”, then they are explciitly restriction to dg-manifolds of Non Negative degree. That’s what this N is for!), this becomes a rather puzzling step as we pass the bridge back to the world of Lie $n$ -groupoids:

A negatively-graded $L_\infty$ -aklgebra seems to demand to be interpreted as the Lie $n$ -algebroid of a Lie $n$ -groupoid which has things like a (-3)-morphism!

What is a (-3)-morphism, though??

Is there a generalization of $\infty$ -categories to $\pm \infty$ -categories?

And this is related to David’s question about cotangent categories:

given a Lie $n$ -algebroid which is represented by an $L_{-infty}$ -algebra concentrated in up to degree $n$ , we may look at the corresponding dg-manifold and form its cotangent bundle. That is automatically a negatively graded dg-manifold!

So what’s going on here?

One option is: nothing interesting is going on. One indication for this is that it seems that in most applications, people who consider the cotangent bundle of a dg-manifold always shift its degree afterwwards back to a positive valued. So maybe this is an indication that it is no good to think of negative degrees in this context in the first place.

But, while a hint, this is certainly not a satisfactory answer. Does anyone have any insight into this issue?

Posted by: Urs Schreiber on August 22, 2007 1:13 PM | Permalink | Reply to this

Re:

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Urs,

The way I understand your question it is the same concern I was talking about in my response yesterday (and also in here). Namely derived (or dg) spaces correspond to cosimplicial spaces, or on the level of functions to simplicial (or dg) rings, in homological degrees.. the way these usually arise is from taking degenerate intersections, like the self-intersection of the diagonal in $X\times X$ , which is the definition of the derived loop space – on the level of functions these are Tors. In mirror symmetry these come up from the whole theory of the virtual fundamental class.

On the other hand we can look in the other direction, and that’s where stacks live - they’re simplicial spaces. They usually arise from taking bad quotients - on the level of functions these are Homs or Exts.

This is clearly seen (as you mention) by looking at tangent complexes - the derived direction contributes POSITIVE degrees to the tangent complex, while the stacky direction contributes NEGATIVE degrees — eg the tangent to pt/G is g in degree -1.

In physics this is usually blurred by looking at Z/2 graded things, but geometrically these two are very differently. Also one could decide to work in a stable setting — after all spectra can be thought of as a way of joining simplicial sets and cosimplicial sets, or positive and negative, in one seamless stable whole. The tangent complex, ie the $L_\infty$ algebroid we assign to a derived stack - is a stable object – it’s an element of the derived category, and we can shift it up and down with impunity. In this sense it doesn’t care about stacky vs dg directions, about connective vs coconnective, etc. And indeed the theory of $L_\infty$ algebras is stable — but rational homotopy theory OF SPACES (rather than spectra) cares about connectivity, and so only positively graded $L_\infty$ algebras arise (well I’m sure I’ve gotten my signs wrong repeatedly but you get my point).

Posted by: David Ben-Zvi on August 22, 2007 3:33 PM | Permalink | Reply to this

positive and negative categorical directions

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The way I understand your question it is the same concern I was talking about

Ah, okay. I guess I didn’t understand what you meant be “stacky directions”.

But now I am beginning to get your point, I think.

Would it be right to think of it this way (roughly):

- the “positive directions” of order $k$ are those in which we have $k$ -equivalences.

- whereas the “negative directions” of order $k$ are those where we have divided out by $k$ -equivalences.

If true (i.e. if I understand that correctly), that actually would beautifully match with something I was about to say concerning BV-BRST quantization:

- there we have positively graded ghosts of degree $k$ corresponding to order- $k$ gauge transformations (i.e. equivalences) and

- negatively (-1)-graded ghosts corresponding to each “constraint which we divide out”

and

- negatively (-2)-graded ghosts for each Noether relation between these relations, etc.

Well, take this last statement with a grain of salt, it is only just materializing in me (and might go away soon – or it might stay :-)

Posted by: Urs Schreiber on August 22, 2007 3:56 PM | Permalink | Reply to this

Re: positive and negative categorical directions

Sounds at least roughly right to me.
The first example of derived stack
that ever made sense to me (and probably the only) is what you get by taking Hamiltonian reduction (which is basically what BRST/BV
is doing). The process has two parts,
a sub and a quotient: setting the moment map value and quotienting out by symmetries. The latter leads to stackiness if there are stabilizer, while the former leads to dg-ness if we’re not at a regular value of the moment map, ie if the constraints we are enforcing have some kind of redundancy. But you get this correct answer by taking the appropriate homological-cohomological complex: other than the issue of “big” vs “small” stabilizers, the BRST complex is calculating the reduced space as a derived stack..

(this comes up nicely in geometric Langlands:

Q: what is the space of flat G-bundles (with no singularities) on the Riemann sphere?

A: more interesting than what one might first think, but can be found by the above
thinking)

Posted by: David Ben-Zvi on August 22, 2007 4:09 PM | Permalink | Reply to this

Re: positive and negative categorical directions

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The process has two parts,a sub and a quotient […] The latter leads to stackiness if there are stabilizer, while the former leads to dg-ness

That’s awesome, thanks.

Posted by: Urs Schreiber on August 22, 2007 4:17 PM | Permalink | Reply to this

Re: positive and negative categorical directions

Apologies for not responding sooner.
For me the point of BV (or BFV in the Hamiltonian case), is that it’s dg all the way and no need for stacks. this also addresses that issue of why negative degrees as well as positive. Both the BV construction and the BFV construction combine two well known pieces of homological algebra: easiest to see in the case of a constrained Hamiltonian:

the BRST part (really good old Cartan-Chevalley-Eilenberg complex for
the symms acting on e.g. the symplectic manifold and the Koszul (in the regular)
or Koszul-Tate (in the general case) resolution for the constraint surface = 0-locus of the constraints.

Posted by: jim stasheff on September 8, 2007 2:10 AM | Permalink | Reply to this

Re: positive and negative categorical directions

Hi,

First I should say I’m a big fan of the BRST and BV stories, ever since I read your (Stasheff’s) papers on them. But one (and I think maybe the only) disadvantage is that they don’t
fully replace the need for stacks, at least as far as I understand.

More precisely, I believe you only see the Lie algebra of the group you’re quotienting by, and not the group. For example if your group is disconnected you can’t see other components, and you can’t see pi_1. More generally you’ll only see the rational homotopy type of the group. So the BRST picture is a kind of “linearization” of the quotient in the stack language. I may be missing something however.

Maybe for typical (connected) groups that one runs into this is not a problem. But in general there’s a big difference I think between having stacks (whose points
over a ring are a topological space, usually a K(pi,1) ) and purely dg objects, which only see rational homotopy theory.

As far as gauge theories (eg Chern-Simons) are concerned it seems to me that the BV approach doesn’t handle large gauge transformations, only small ones (and what’s generated by them). But I would be glad to learn otherwise!
Thanks!
David

Posted by: David Ben-Zvi on September 8, 2007 3:06 AM | Permalink | Reply to this

Re: positive and negative categorical directions

I agree, and thanks for the complement on my papers. I don’t speak `stacks’ fluently and must admit, even in rational homotopy theory, stacks do seem to provide a natural
setting, cf. my work with Schlessinger where there is indeed a group acting on the coarse moduli space (which is a variety)
with a quotient which is not.

Posted by: jim stasheff on September 8, 2007 2:47 PM | Permalink | Reply to this

Re: More on Tangent Categories

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The tangent category functor, as Urs calls it, is essentially the globular version of the functor $Dec^1$ defined by Illusie. It takes a simplicial object, and moves all the simplices down a dimension (so $Dec^1X_n = X_{n+1}$ ) and removes the first (or last, depending on convention) face and degeneracy operator in each dimension. This removed face operator gives $Dec^1X$ a map to $X$ , and also by a lot of composition with other face operators to $X_0$ (we say $Dec^1X$ is an augmented simplicial object). Also, the extra degeneracy operator provides a homotopy equivalence between $Dec^1X$ and the constant simplicial object on $X_0$ . $Dec^1X$ also behaves very much like a total path space of $X$ - in other words some sort of properly topologised union of the path spaces for each point.

Posted by: David Roberts on August 22, 2007 7:00 AM | Permalink | Reply to this

Re: More on Tangent Categories

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The tangent category functor, as Urs calls it, is essentially the globular version of the functor $\mathrm{Dec}^1$ defined by Illusie.

Ah, great. I was hoping it would turn out to be something known in other contexts and other languages. If that tangent construction is as useful and fundamental as it seems to be, it should have surfaced in different guises already.

So I am glad to have found out by now that

- as we pass from Lie $n$ -groupoids to their $n$ -algebroids, the categorical tangent bundle just produces what is known as the shifted tangent bundle of the underlying dg-manifold

- apparently this can be understood, as David Ben-Zvi remarked as some construction known in derived algebraic topology

- now you are adding another facet to that!

But I am having trouble finding the work you seem to be referring to. Could you provide a reference?

Posted by: Urs Schreiber on August 22, 2007 12:53 PM | Permalink | Reply to this

Re: More on Tangent Categories

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Duskin refers to it in his AMS Memoir (section(0.14) - treats the basics, really. Illusie used it to talk about the cotangent complex of a thing - I can’t remember what the thing is, but the cotangent complex he defines contains a whole lot of homotopy information.

The review here of the book should point you in the right direction.

More properly, $Dec^1$ is called the decalage functor (with a accent in there somewhere), and it forms a cotriple of the category of simplicial objects - as does your ‘tangent’ functor.

Posted by: David Roberts on August 23, 2007 5:54 AM | Permalink | Reply to this

Re: More on Tangent Categories

David,
That link seems to require membership.
Can you supply the bib data instead?

jim

Posted by: jim stasheff on August 30, 2007 3:03 PM | Permalink | Reply to this

Re: More on Tangent Categories

Sorry - the books are Illusie’s “Complexe cotangent et deformations” I and II, LNM 239 and 283. I’ve never looked at them, but thay’re the usual reference people give

Posted by: David Roberts on August 31, 2007 3:22 AM | Permalink | Reply to this

Re: More on Tangent Categories

Yes, the usual reference but I’ve never seen
a page reference for a particular result.

In the algebraic geometric context, see papers of Mike Schlessinger.

Posted by: jim stasheff on August 31, 2007 2:27 PM | Permalink | Reply to this

Re: More on Tangent Categories

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More properly, $Dec^1$ is called the decalage functor (with a accent in there somewhere), and it forms a cotriple of the category of simplicial objects

Ah! I’d heard people use the word decalage, but didn’t know what it meant until your post. So it’s the comonad I was using all along here, when I was describing what was so special about the bar construction, but I didn’t have a name for it. Now I do. Thanks!

Posted by: Todd Trimble on August 31, 2007 4:54 AM | Permalink | Reply to this

Re: More on Tangent Categories

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So it’s the comonad I was using all along here

That’s interesting. I need to re-read your entry.

So, is the claim that where I write $\mathrm{Mor}(C) \to T C \to C$ for $C$ any category, passing this to the simplicial world yields something well known, where the term in the middle would be addressed as $\mathrm{Dec}^1(N(C))$ or the like?

Posted by: Urs Schreiber on August 31, 2007 11:32 AM | Permalink | Reply to this

Re: More on Tangent Categories

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John Baez was suggesting in Vienna several times to me that forming $\mathrm{INN}$ and $T(\cdot)$ and the like is similar to forming resolutions and related to the bar construction.

Let me see if I remember this simple example correctly, which might illustrate what’s going on, if I understand correctly.

For $G$ any group, we would like to construct the universal $G$ -bundle.

The bar-construction way is this, as far as I recall (thanks to John for walking me through this on a blackboard at the Schrödinger institute):

we look at the $G$ -set $\{\bullet\}$ with a single object. Using the forgetful functor

$G\mathrm{Set} \to \mathrm{Set}$ and the free construction adjoint to that $\mathrm{Set} \to G\mathrm{Set}$ we have a mondad acting on $\{\bullet\}$ . Applying the bar construction to that, we get a simplicial set whoe vertices are the elements of $G$ , which has unique edges between any two vertices, and so on.

So, unless I am mixed up, this simplicial set is nothing but the nerve of $\mathrm{INN}(G) = G // G = \mathrm{Codisc}(G)$ , the codiscrete groupoid over $G$ . And indeed, that’s the right answer.

So, that leaves me with the vague impression that we are talking about a commuting square of sorts, where the horizontal arrows correspond to passing from categories to simplicial sets, and where the vertical arrow on the left is $T(\cdot)$ and where the vertical arrow on the right is the bar construction, or maybe $\mathrm{Dec}^1$ or the like.

Is that getting anywhere close to a true statement??

Posted by: Urs Schreiber on August 31, 2007 11:46 AM | Permalink | Reply to this

Re: More on Tangent Categories

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Urs asks:

Is that getting anywhere close to a true statement??

Yes, absolutely! Interesting – it also reminds me of a question you asked over here. I’ll give the connection toward the end of this comment.

So you (and David Roberts and I) are suggesting there’s a square (commuting up to isomorphism)


Cat ---> [Delta^{op}, Set]
 |    N           |       
 | T              | Dec^1 := [1 + (-), Set]
 V    N           V
Cat ---> [Delta^{op}, Set]

where $T$ is your tangent category functor, $N$ is the nerve, and $1 + (-)$ is the ordinal sum which sticks a new element at the beginning. (To bring this in line with comments on the bar construction, I’ll assume we’re dealing with augmented simplicial sets, and things seem to work out best if we assume that the nerve of a category $C$ ,

$... C_1 \stackrel{\to}{\to} C_0 \to C_{-1},$

is augmented by its set of path components: $C_{-1} = \pi_0(C)$ .) Let me try to make this precise.

As I understand it, the tangent category is defined to be

$T C = \sum_{c \in Ob(C)} c/C,$

the sum of all the co-slice categories; alternatively, $T C$ is the following pullback in $Cat$ :


TC --> C^2 = Mor(C)
 |      |
 |      | dom         (1)
 V   i  V
C_0 --> C

where $i: C_0 \to C$ is the inclusion of the discrete category of objects. This is an exact analogue of the sum of path spaces, construed as a pullback in $Top$ :


PX --> X^I
 |      |
 |      | eval_0     (2)
 V   i  V
|X| --> X

where $|X|$ is the set of points of $X$ with the discrete topology.

The nerve, being a right adjoint, preserves the pullback (1); in fact, the $n$ -dimensional simplices $[n] \to T C$ in dimension $n \geq 0$ , where $[n]$ is the ordinal $0 \leq 1 \leq ... \leq n$ , are precisely commutative squares in $Cat$ of the form

 

[n] --> C^2
 |       |
 |!      | dom
 V       V
[0] ---> C;

equivalently, commutative squares of the form


[n] --> 2 x [n]
 |  i_0   |
 |!       |        (3)
 V        V
[0] ----> C

The pushout in $Cat$ of the legs out of $[n]$ in (3) is the cone $[1 + n]$ , and all of this is functorial in $[n]$ . In other words, $Cat([-], T C) \cong Cat([1 + (-)], C)$ . In other words, the square at the top of this post commutes (up to isomorphism).

[It may be felt that I’ve made an essentially trivial calculation look hard, but the point was to do it reasonably carefully and functorially and with minimal pain to myself!]

Let me connect this up with some other stuff that you and I have said. You mentioned a construction of the (total space of the) classifying bundle of a group as the nerve of the codiscrete groupoid attached to $G$ . I think another name for $Codisc(G) = G //G$ is the translation groupoid of $G$ , and I think when I first learned about this from Chuck Weibel (a long time ago now), he used $T G$ to denote the translation groupoid – an amusing coincidence with your notation for the tangent category!

So now that this has all come out, here’s another way to think of the tangent category, this time in terms of the bar construction. A category $C$ can be considered as a monad $C_1: C_0$ -|-> $C_0$ in the bicategory of spans, acting by pullback $(-) \times_{C_0} C_1$ . This monad acts in a tautological way on the span

$1 \stackrel{!}{\leftarrow} C_0 \stackrel{id}{\to} C_0$

(which plays a role analogous to the single-element $G$ -set), which I’ll call $C_0$ . The bar construction applied to this monad-algebra pair $(C_1, C_0)$ is precisely the nerve of the tangent category!

So when you asked here whether a category arises as the nerve of cartesian monad, my answer in hindsight should have been: yes, if it’s a tangent category!

By the way, as David Roberts pointed out, decalage is equipped with a comonad structure. So far you’ve mentioned the projection $T C \to C$ , which corresponds to the counit of decalage. Do you have any use for a comultiplication $T C \to T T C$ ?

Posted by: Todd Trimble on September 1, 2007 2:37 AM | Permalink | Reply to this

Re: More on Tangent Categories

$MathML-enabled post (click for more details).$

And now that I think of it, it’s obvious that the tangent category comonad is right adjoint to something which could reasonably be called the cone monad acting on $Cat$ , whose value at $C$ is defined to be the pushout in $Cat$ of the diagram


   C ---> 2 x C
   |  i_0
   |!
   V
pi_0(C)

In fact, I suspect that the right POV on a lot of this is through closed model categories: the tangent and cone category constructions are certain fundamental fibrations and cofibrations one can attach to a category $C$ , adjoint to one another.

If that tangent construction is as useful and fundamental as it seems to be, it should have surfaced in different guises already.

Damn straight!

Posted by: Todd Trimble on September 1, 2007 3:17 AM | Permalink | Reply to this

Re: More on Tangent Categories

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Todd,

many thanks indeed for these insightful comments! Where would I be without the Café and contributors like you?

Right now I am on weekend and busy with other things, but I’ll get back to you on this at least on Monday.

Posted by: Urs Schreiber on September 1, 2007 5:23 PM | Permalink | Reply to this

Re: More on Tangent Categories

How about the COtangent version? since I’m more familair with that in it’s deformation theory atavar.

Posted by: jim stasheff on September 2, 2007 1:27 AM | Permalink | Reply to this

Re: More on Tangent Categories

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I haven’t thought about what the comultiplication would be good for, cause in my own cogitation I was contrasting the construction $TC$ with $INN_0(G)$ , in particular, the equivalent to $INN_0$ for internal simplicial groups, to get a sort of large-scale view of what Urs and I did. In that case we need to sit in simplicial groups because of course $INN_0$ takes one from $n$ -groups to $(n+1)$ -groups. Then one gets a monad on $sGrp$ .

In Todd’s comment the first square commutes for real, in the case where one throws away the augmentation that comes for free with $Dec^1$ - I was being slack and talking about $Dec_+^1$ - the positive part - without telling you so.

Posted by: David Roberts on September 3, 2007 5:32 AM | Permalink | Reply to this

Re: More on Tangent Categories

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augmentation

I have to admit that I am not entirely following the simplicial discussion here.

Could somebody explain to me what augmentation means, and maybe repeat again in detail how $\mathrm{Dec}^1$ is defined?

Posted by: Urs Schreiber on September 3, 2007 8:35 AM | Permalink | Reply to this

Re: More on Tangent Categories

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I’m sure by now you’ve figured all this out, but just for the record:

A simplicial object in a category $C$ (topologists’ convention) is a functor

$\Delta_{+}^{op} \to C$

where here I am using $\Delta_{+}$ to denote the category of non-empty ordinals [topologists usually write $\Delta$ for this category]. The tail end of face operators here looks like $... C_1 \stackrel{\to}{\to} C_0.$

A simplicial object (algebraists’ convention) is a functor $\Delta^{op} \to C$ where this time $\Delta$ is the category of finite ordinals (including the empty one). The tail end of face operators this time looks like $... C_1 \stackrel{\to}{\to} C_0 \to C_{-1}$ where $C_{-1}$ is the value of the empty ordinal (of dimension $-1$ ). The last map $C_0 \to C_{-1}$ is often called the augmentation, and topologists would call these things augmented simplicial objects.

For technical reasons (often centering around applications of the bar construction), I prefer augmented simplicial objects whenever possible. (For example, without the empty ordinal, the monoidal category $\Delta$ lacks a monoidal unit!) There are two canonical choices of augmentation at opposite ends of the spectrum, corresponding to, respectively, the left and right adjoints to the evident pullback along inclusion $i: \Delta_{+} \to \Delta$ :
$Set^{\Delta^{op}} \stackrel{Set^{i^{op}}}{\to} Set^{\Delta_{+}^{op}}.$ The first is to augment by the canonical projection $C_0 \to \pi_0(C)$ to the object of path components. The second is to augment by the unique map $C_0 \to 1$ where $1$ is terminal.

For either brand of simplicial object, $Dec^1$ (as we recently learned from David Roberts) is the functor $C^{\Delta^{op}} \to C^{\Delta^{op}}$ which precomposes a simplicial object $X: \Delta^{op} \to C$ with $[n] \mapsto [1 + n]: \Delta^{op} \to \Delta^{op}$ : $Dec^1(X) \stackrel{def}{=} X \circ ([1 + -]),$ but please note it makes a difference where you put the ‘1’ – we are putting it at the beginning to get a better match with coslice categories, as opposed to slice categories. (Finally, let me emphasize again that you need the empty ordinal to be able to say that $\Delta$ is the walking monoid, which is the key observation underlying the bar construction.)

Posted by: Todd Trimble on September 3, 2007 6:45 PM | Permalink | Reply to this

Re: More on Tangent Categories

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Thanks, Todd.

I have a question: would you be interested in turning some of the things we talked about concerning tangent categories and décalage into a nice coherent discussion?

I am planning to create a file devoted to a coherent and comprehensive discussion of this topic, and I would be enjoy sharing this with you and David Roberts. Would you be interested?

Posted by: Urs Schreiber on September 4, 2007 12:34 PM | Permalink | Reply to this

Re: More on Tangent Categories

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Sure, I’d be glad to participate – thanks for the invitation!

Posted by: Todd Trimble on September 4, 2007 1:22 PM | Permalink | Reply to this

Re: More on Tangent Categories

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Sure, I’d be glad to participate – thanks for the invitation!

Nice, thanks.

This is how far I got today:

Tangent Categories

The TeX source will already have reached you by email by now.

Please note that nothing in that file is meant as being unnegotiable. Quite the opposite. I am hoping you will feel free to improve and modify all the things that require improvement and modification. But one needs to start somewhere.

And in particular: the file is clearly completely incomplete. There is lots of stuff which I know about that needs to be included eventually. And certainly there is much more stuff which I don’t even know about, but which does need to be included.

In closing here, I want to address one important aspect of all of this, which we should discuss lest we’ll run into misunderstandings:

Tangent categories play two different important roles, which a priori seem to be rather unrelated:

a) The tangent $n$ -category $T C$ is a puffed up version of the space of objects $C_0$ . For $C$ an $n$ -groupoid, the canonical inclusion $C_0 \to T C$ is an equivalence. The canonical sequence $\mathrm{Mor}(C) \to T C \to C$ is the $n$ -groupoid incarnation of the universal $C$ -bundle.

But, b),

at the same time, $T C$ does know about the tangency relations on $C_0$ induced by $\mathrm{Mor}(C)$ : for $C$ an $n$ -groupoid, $G$ -flows $\Gamma_G(T C) := \{G \to \Gamma(T C) \subset T_{\mathrm{Id}_C}(\mathrm{End}(C))\}$ do provide a generalization of the concept of vector fields on $C_0$ in that for $G = \mathbb{R}$ and with everything taken to be smooth we have that sections $\Gamma_\mathbb{R}(T C) \simeq \Gamma(\mathrm{Lie}(C))$ do coincide with the sections of the Lie $n$ -algebroid associated with $C$ .

I think that the apparent dichotomy – universal $C$ spaces on one hand, differentials on $C$ on the other – is resolved by noticing that $T C$ is actually to be regarded as the

universal $C$ -bundle equipped with the universal $C$ -connection

This remark currently features in the introduction of our file. I would be lying if I claimed to have fully exhibited or even fully understood this last statement. But I do believe that evidence is accumulating that this is what is going on. Personally, I regard as particularly useful in this regard the observation which I talked about in this recent comment to John’s entry on Higher Gauge theory and Elliptic Cohomology.

Since that is part of a moderately long comment monologue, I cannot quite tell to which extent I am making sense there to anyone else. But I do think that’s important and at the heart of what “tangent categories” are about.

I’d be delighted to hear your (critical) comments on this. But also, if none of this seems to make any sense at the moment and just appears as a distraction to you, please feel free to ignore it. While it is meant as an attempt to put these tangent category constructions into what I consider a useful larger perspective, we can discuss their features just as well without worrying about these particular ideas of mine.

Posted by: Urs Schreiber on September 4, 2007 8:13 PM | Permalink | Reply to this

Re: More on Tangent Categories

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Well it’s fascinating stuff, Urs, and it’s clear I’ll learn a lot. I’m a little diffident to expose here the true depths of my ignorance about this material, but you’ll probably be getting a ton of dumb questions privately.

This phrase in particular fascinates me:

universal C-bundle equipped with the universal C-connection

especially since it seems to touch on the question I was trying to ask over here. In reply Jim Stasheff gave some references to universal bundles with connection, and it motivates me to seek a connection (ha ha) with what you’re doing here.

Posted by: Todd Trimble on September 4, 2007 9:20 PM | Permalink | Reply to this

Re: More on Tangent Categories

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Yes, those universal connections on the universal bundle have always mystified me. As Jim says in one of the comments you pointed to, one finds in the literature some constructions where $B G$ is turned into a smooth space by some mean or other, but nothing really close to a conceptual understanding.

But now I am thinking that the problem is that we shouldn’t pass to spaces if we can stay in the world of groupoids.

The point being (concentrating on $n=1$ for the moment), that the groupoid $\mathrm{INN}(G) = T \Sigma G = G // G = \mathrm{codisc}(G) = \cdots$ is the universal $G$ -bundle. Or rather, the sequence it sits in $G \to T \Sigma G \to \Sigma G$ (which is just a special case of the general sequence $\mathrm{Mor}(C) \to T C \to C$ which every tangent category sits in) is.

For one, under taking nerves and their realizations, this sequence does map to the universal $G$ -bundle regarded as a space $\array{ G &\stackrel{|\cdot|}{\mapsto}& G \\ \downarrow && \downarrow \\ T G &\stackrel{|\cdot|}{\mapsto}& E G \\ \downarrow && \downarrow \\ \Sigma G &\stackrel{|\cdot|}{\mapsto}& B G } \,.$

But I think it gets better than that. Once we realize that we don’t need to pass to spaces by taking realizations of nerves. Everything can be discussed entirely in the world of groupoids. And in that world, many things become much simpler.

(For a quick discussion of $G$ -bundles in terms of groupoids, see the last section of my article with David Roberts. )

You might recall that at one point I expressed my own personal puzzlement about the fact that I was caliming $\mathrm{INN}(G)$ to be important for two different reasons in the theory of $G$ -bundles:

- on the one hand as the universal $G$ -bundle itself

- on the other as the right codomain for the functor which encodes the connections.

It’s these two aspects of tangent categories we run into all the time. Once one gets used to one of these two points of views, it tends to be irritating to see somebody else (often oneself ;-) use tangent categories in the other sense, which seems to be so different.

But then I noticed: no, it’s not different. We are just being blind. $\mathrm{INN}(G)$ is not “just” the universal $G$ -bundle. It also has a universal $G$ -connection.

Which one you ask? The identity!

A $G$ -connection on a space $X$ is really a 2-functor $\mathrm{curv} : \Pi_2(X) \to \Sigma \mathrm{INN}(G) \,,$ where $\Pi_2(X)$ is the fundamental 2-groupoid of $X$ (morphisms are parameterized paths, 2-morphisms are homotopy classes of cobounding surfaces).

But $\Sigma \mathrm{INN}(G)$ is itself something like a model for the fundamental 2-groupoid of $\mathrm{INN}(G)$ . I think. So it would be nice if we had a canonical $G$ -connection on it in that there were a canonical 2-functor $\Sigma \mathrm{INN}(G) \to \Sigma \mathrm{INN}(G) \,.$ But of course there is. The identity: $\mathrm{curv}_{\mathrm{universal}} := \mathrm{Id} : \Sigma \mathrm{INN}(G) \to \Sigma \mathrm{INN}(G) \,.$

While I have to ask you to keep in mind that this is what I consider a rather fresh insight of mine, I do think it makes good sense. I believe this is best seen in the differential picture, as described in that comment of mine which I already mentioned:

see, Jim Stasheff kept emphasizing to me for a long time that the qDGCA $(\wedge^\bullet( s g^* \oplus s s g^*), d_{\mathrm{inn}(g)})$ Koszul dual to the Lie 2-algebra corresponding to $\mathrm{INN}(G)$ , which is nothing but the well-known Weil algebra, plays the role of differential forms on the universal $G$ -bundle.

But that’s right what we need for talking about universal connections! Something like differential forms on the universal $G$ -bundle.

Now, if $P$ is the total space of some $G$ -bundle, then $\Omega^\bullet(P)$ is, clearly, the space of differential forms on it and – now comes the point – a connection on $P$ is a qDGCA morphism $\Omega^\bullet(P) \stackrel{A}{\leftarrow} (\wedge^\bullet( s g^* \oplus s s g^*), d_{\mathrm{inn}(g)} ) \,.$ Here we have $\mathrm{INN}(G)$ on the right playing its role as differentials to $G$ , while before we had it in its role as the universal $G$ -bundle itself.

Now simply take these two statements and join them to the obvious conclusion: if $A$ as a above is a connection on the arbitrary $G$ -bundle $P$ , and if $(\wedge^\bullet( s g^* \oplus s s g^*), d_{\mathrm{inn}(g)} )$ itself is the algebra of differential forms on the universal $G$ -bundle, then, clearly, a connection on the universal $G$ -bundle ought to be a morphism $(\wedge^\bullet( s g^* \oplus s s g^*), d_{\mathrm{inn}(g)} ) \leftarrow (\wedge^\bullet( s g^* \oplus s s g^*), d_{\mathrm{inn}(g)} ) \,.$ Which one? The canonical one: the identity!

So there are two ways to regard the statement that a $G$ -connections is a morphism not to $G$ but to $\mathrm{INN}(G)$ (the statement I kept going on about in the context of “ $n$ -curvature):

- either we realize that passing from $G$ to $\mathrm{INN}(G)$ is the right thing to do for differential reasons of various sorts

- or we realize that this can be equivalently read as pulling back the universal $G$ -connection from the universal $G$ -bundle

I mean, read $\Omega^\bullet(P) \stackrel{A}{\leftarrow} (\wedge^\bullet( s g^* \oplus s s g^*), d_{\mathrm{inn}(g)} )$ as $\Omega^\bullet(P) \stackrel{A}{\leftarrow} (\wedge^\bullet( s g^* \oplus s s g^*), d_{\mathrm{inn}(g)} ) \stackrel{\mathrm{Id}}{\leftarrow} (\wedge^\bullet( s g^* \oplus s s g^*), d_{\mathrm{inn}(g)} )$

I need to sleep over all this and formulate it in a better way. Currently it’s just a mix of factoids and some intuition. But it seems to make good sense. Best of all, if I simply trust myself and assume this is the way things are, this explains a couple of previously somewhat mystifying aspects: it unifies

- the concept of the universal bundle

- the concept of tangents and differentials

- the concept of $n$ -curvature

all in one single notion.

Okay, before I get more elevated about something I haven’t fully worked out, I better stop right here. :-)

Posted by: Urs Schreiber on September 4, 2007 10:00 PM | Permalink | Reply to this

Re: More on Tangent Categories

$MathML-enabled post (click for more details).$

Sign me up for the tangent category ride!

On a different note:

One thing which has inrigued me recently was an exhortation to check that on geometric realisation, our universal $|G_2|$ -bundle is locally trivial (I am doing this nonetheless). Back in the day, people were very careful to check that various universal bundle constructions were indeed principal bundles by the defintions of the day, and under precisely which conditions.

It turns out that for a topological group $G$ , it is enough that it is locally contractible at the identity. (Recall that this means essentially there is a neighbourhood of the identity that is a deformation retract of the point $\{id\}$ ). However, if a groupoid approach is what we are after, then such things (and dealing with issues of cofibrancy and homotopy colimits) worry us less. This does mean that the nonabelian cohomology we define might not be representable by a honest space, but it should be representable by a groupoid, and we might need to use a category of fractions. But if we are honest with ourselves, the groupoid of cocycles is much better than the pointed set, anyway.

Posted by: David Roberts on September 5, 2007 1:54 AM | Permalink | Reply to this

Re: More on Tangent Categories

$MathML-enabled post (click for more details).$

Sign me up for the tangent category ride!

Very well.

One thing which has inrigued me recently was an exhortation to check that on geometric realisation, our universal $|G_2|$ -bundle is locally trivial.

[…]

It turns out that for a topological group $G$ , it is enough that it is locally contractible at the identity.

Ah, good.

Is there a way to state this and see this entirely on the groupoid level?

I am wondering:

when dealing with principal Lie groupoid bundles, one needs to assume (as far as I can see) that these groupoids are what is called “locally trivial” in the sense of Lie groupoids. If I recall correctly right now (should look this up again), this means that $\mathrm{Mor}(G) \stackrel{(s,t)}{\to} \mathrm{Obj}(G) \times\mathrm{Obj}(G)$ is a surjective submersion. This in turn is equivalent to $G$ looking locally like a pair groupoid extended by a group, i.e. with composition like $(x \stackrel{g_1}{\to} y \stackrel{g_2}{\to} z) = (x \stackrel{g_1 g_2}{\to} z) \,.$ It seems to me like this should be related to what you just said.

Posted by: Urs Schreiber on September 5, 2007 1:12 PM | Permalink | Reply to this

Re: More on Tangent Categories

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The condition

It turns out that for a topological group $G$ , it is enough that it is locally contractible at the identity.

which ensures that the universal $G$ -bundle is locally trivial, is one way of ensuring the resulting simplicial space (the nerve of $\Sigma G$ ) underlying $BG$ is “cofibrant enough”. Something else I have been working generalises this condition to general topological categories. The identity map

$C_0 \stackrel{id}{\rightarrow} C_1,$

can be thought of as a map over $C_0\times C_0$ , where the spaces involved are over-spaces by the diagonal and $(s,t):C_1 \to C_0 \times C_0$ respectively. Then we demand that $id$ is a fibrewise closed cofibration.

I don’t know how this relates to the condition you mentioned about Lie groupoids, but if $(s,t)$ is a surjective submersion, then we can pull it back along the diagonal $G_0 \to G_0 \times G_0$ and the resulting manifold $\Delta^*G_1$ should hopefully be a closed submanifold of $G_1$ . The inclusion of a closed submanifold is certainly a closed cofibration. It would be a good idea to figure this connection out.

Posted by: David Roberts on September 6, 2007 6:19 AM | Permalink | Reply to this

Re: More on Tangent Categories

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I wrote

I need to formulate it in a better way. Currently it’s just a mix of factoids and some intuition. But it seems to make good sense.

I think I made some progress. The key is to note that because for $C = \Sigma G$ a one-object groupoid, $T C$ is itself a 2-group, namely $T C = \mathrm{INN}(G) \,.$ we may iterate the tangent category construction to obtain $\array{ \mathrm{Mor}(C) &\to& T C &\to& C \\ \downarrow && \downarrow &\Downarrow^{\simeq}& \downarrow \\ T C &\to& T \Sigma T C &\to & \Sigma T C \\ \downarrow &\Downarrow^\simeq& \downarrow \\ \Sigma C &\to& \Sigma T C } \,,$ and that $\Sigma T C$ plays the role of the fundamental 2-groupoid of $E G$ whereas $T \Sigma T C$ plays the role of the Atiyah groupoid $E G \times_G E G$ of $E G$ pulled back to the fundamental 2-groupoid, or rather its version $EG \times EG$ before we divide out by $G$ .

We find from this that a splitting of the Atiyah projection $T \Sigma T C / G \to \Sigma T C$ is a 2-functor $\Sigma T C \to \Sigma T C$ and the canonical one, $\mathrm{curv}_{\mathrm{universal}} := \mathrm{Id} : \Sigma T C \to \Sigma T C$ is the universal connection on $E G$ in a sense which I describe in more detail in the new version of

Tangent Categories.

Notice that it is the top-right or equivalently bottom-left morphism $C \to \Sigma T C$ i.e. $\Sigma G \to \Sigma T \Sigma G$ i.e. $\Sigma G \to \Sigma \mathrm{INN}(G)$ which is to be regarded as the integrated version of the differential morphism $(\wedge^\bullet (s g^*), d_g) \leftarrow (\wedge^\bullet (s g^* \oplus s s g^*), d_{\mathrm{inn}(g)})$ that I kept mentioning. In the new notes I describe the integral analog of the differential construction in that comment which I keep pointing to.

I’ll try to post something summing this up, so far.

Posted by: Urs Schreiber on September 5, 2007 12:51 PM | Permalink | Reply to this

Re: More on Tangent Categories

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Recall the diagram from above. Then consider this:

Here

- $f$ is the classifying map of a $G$ -bundle, expressed in the world of groupoids, where $Y \to X$ is a good cover of base space $X$ and $Y^{[2]}$ the corresponding groupoid.

You may usefully think of $Y^{[2]}$ and $f$ as constituting an anfunctor $X \to \Sigma G \,,$ with $X$ regarded as a discrete category.

- $T C = \mathrm{INN}(G)$ is the universal $G$ -bundle in the world of groupoids.

The picture explains why and how the $n$ -connection is something that is locally expressed in terms of its $(n+1)$ -curvature $\mathrm{tra}$ taking values in $\Sigma \mathrm{INN}(G_{(n)})$ , while the transition data (the nonabelian cocycle) is restricted to be in $\Sigma G_{(n)}$ .

What I denoted $\mathrm{curv}_{EG}$ is the canonical section of $T \Sigma T C \to \Sigma T C$ , which I think is the morphism which we would want to regard as the universal $G$ -connection on $EG$ , in the world of groupoids.

So by refining the “classifying map” to the map $(\mathrm{curv},f)$ we pull back the universal $G$ -bundle together with its universal connection to $X$ .

Posted by: Urs Schreiber on September 5, 2007 7:32 PM | Permalink | Reply to this

Re: More on Tangent Categories

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I wrote:

Then consider this:

I am preparing some pdf slides which are supposed to better illuminate this business about the universal $G_{(n)}$ - $n$ -bundles, their universal connections and the relation to the Lie $n$ -algebra cohomology.

In the pdf

String and Chern-Simons $n$ -Bundles

you can find something like a diagram-movie with subtitles which is supposed to walk you through that big diagram controlling $G_{(n)}$ -bundles with connection, and explain all its parts stepwise.

It starts somewhere in the second half (may change, this is under construction). You find it directly by following the hyperlink to $n$ -bundles with $g_{(n)}$ -connection provided in the table of contents.

Posted by: Urs Schreiber on September 11, 2007 8:09 PM | Permalink | Reply to this

Re: More on Tangent Categories

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Concerning diagram (1)

$\array{ T C &\to & C^2 = \mathrm{Mor}(C) \\ \downarrow && \downarrow^{\mathrm{dom}} \\ C_0 &\to & C }$ you are thinking of $\mathrm{Mor}(C)$ as the category whose objects are morphisms in $C$ and whose morphisms are commuting squares (if I understand you correctly).

That’s what I denoted $\mathrm{Hom}_{\mathrm{Cat}}(\{\bullet \to \circ \}, C) \,,$ where $2_1 = \{\bullet \to \circ\}$ is the category with one nontrivial morphism.

I had two motivations for addressing this construction as a “tangent” construction.

The first was this appearance of $2_1$ in the above pullback. If we think of $2_1$ as being a “puffed up” version of the point $\{\bullet\}$ , then the above pullback says that $T C$ is the category of images of the fat point in $C$ which looks just like the objects of $C$ as soon as we stop distinguishing the fat point from the point.

I enjoyed regarding this as analogous to the basic setup of synthetic differential geometry.

(The second was that $G$ -flows in $\Gamma (T C)$ really do reproduce vector fields, hence things one would ordinarily address as “tangent”).

Is that $2_1 = \{\bullet \to \circ\}$ the “2” appearing in your third diagram $\array{ [n] &\stackrel{i_0}{\to} & 2 \times [n] \\ \downarrow^! && \downarrow^{\mathrm{dom}} \\ [0] &\to & C }$ ?

I hope it is, because then I think I understand what you are saying! :-)

The pushout in $\mathrm{Cat}$ of the legs out of $[n]$ in (3) is the cone $[1+n]$ ,

Ah, is that the very definition of $[1 + n]$ ?

I think another name for $\mathrm{Codisc}(G) = G // G$ is the translation groupoid of $G$

That may be. $G // G$ is meant to be read as “the action groupoid of the one-sided $G$ -action on itself” (or, less precisely, as the “weak quotient of $G$ by $G$ of course”) and I guess that may also well be thought of as a translation groupoid.

The bar construction applied to this monad-algebra pair $(C_1,C_0)$ is precisely the nerve of the tangent category!

Interesting. I am following your constructions here to 80% (while admiring them 100%). I’ll need to think about this a little more.

Do you have any use for a comultiplication $T C \to T T C$ ?

I am not sure. This might be related to something vaguely resembling “higher exterior powers” of the tangent bundle. But I really don’t know yet.

Posted by: Urs Schreiber on September 3, 2007 8:32 AM | Permalink | Reply to this

Re: More on Tangent Categories

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I wrote:

That’s what I denoted $\mathrm{Hom}_{\mathrm{Cat}}(\{\bullet \to \circ\}, C)$

with $\{\bullet \to \circ\} := 2$ .

Oh, I am being stupid. You wrote $C^2$ and literally did mean that: $C^2 = \mathrm{Hom}(2,C)$ I guess.

Okay, all the better.

Posted by: Urs Schreiber on September 3, 2007 8:41 AM | Permalink | Reply to this

Re: More on Tangent Categories

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I wrote:

Ah, is that the very definition of $[1+n]$ ?

Sorry, I was confused (my fault). I think now I understand what you meant (repeating just for my own benefit):

You gave an abstract argument for why a sequence of $n$ -composable morphisms in $T C$ is the same as a sequence of $n+1$ composable morphisms in $C$ .

For instance, for $n=2$ we have that a sequence of two composable morphisms in $T C$ looks like $\array{ a &\to& b &\to& c \\ & \nwarrow &\uparrow& \nearrow \\ && x } \,,$ which (since all these triangles commute) is the same as a sequence of three morphisms $\array{ a &\to& b &\to& c \\ & \nwarrow && \\ && x }$ in $C$ , clearly. This is the

essentially trivial calculation

which you set out to do

reasonably carefully and functorially

All right. Sorry for being slow. This idea is then encoded in the pushout $\array{ [n] &\stackrel{i_0}{\to}& 2 \times [n] \\ \downarrow && \downarrow \\ [0] &\to& [n+1] }$ which you consider after diagram (3): here the functor $2 \times [n] \to [n+1]$ sends $\array{ a &\to& b &\to& c \\ \uparrow && \uparrow && \uparrow \\ x_1 &\to& x_2 &\to& x_3 }$ to $\array{ a &\to& b &\to& c \\ & \nwarrow &\uparrow& \nearrow \\ && x } \,,$ by sending all the lower horizontal morphisms to identities, and then reads the result as $\array{ a &\to& b &\to& c \\ & \nwarrow && \\ && x } \,.$

All right, I get it. Very nice.

Posted by: Urs Schreiber on September 3, 2007 1:46 PM | Permalink | Reply to this

Re: More on Tangent Categories

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Urs, it looks like we’re pretty much on the same page here; in particular you’ve filled in some gaps I’d left in there, so that it may now be easier to follow.

On this $[1+n]$ thing. (I see you’ve just pipped me! but let me continue anyway.) Literally, it’s just the finite ordinal obtained by the ordinal $[n] = (0 \leq 1 \leq ... \leq n)$ by throwing in a new element at the beginning, or tensoring with the ordinal $[0]$ on the left (in $\Delta$ as a monoidal category). [NB: $[n]$ has $n+1$ elements; it’s supposed to denote the dimension, cf. the fact that the $n$ -dimensional affine simplex has $n+1$ vertices.]

But the tensor product in $\Delta$ (“ordinal sum”), which string-diagrammatically is just juxtaposition, geometrically corresponds to simplicial join. Recall that the simplicial join of two spaces $X$ and $Y$ is the space of all line segments between them – more formally, the pushout of the following diagram:

$X \stackrel{\pi_1}{\leftarrow} X \times Y \stackrel{i_0}{\to} X \times I \times Y \stackrel{i_1}{\leftarrow} X \times Y \stackrel{\pi_2}{\to} Y.$

What I’m claiming is that geometric realization $R: Set^{\Delta^{op}} \to Top$ takes the Day convolution product, induced by ordinal sum on $\Delta$ , to the monoidal product on $Top$ given by simplicial join. (I should probably be careful here: simplicial join $X \star Y$ isn’t cocontinuous in $X$ and $Y$ . But I think the statement’s okay if I restrict attention to connected simplicial sets.)

So, to finish this shaggy dog story: when I said that $[1+n]$ is the cone on $[n]$ , it was really in reference to the idea that the cone of a space $X$ is the simplicial join of $X$ with a point $[0]$ . (But you, Urs, already understood all this!)

Let me reiterate that the tangent category construction, as I now see it, is something familiar from closed model category theory: basically, we’re turning the inclusion

$i: C_0 \to C$

into a weak equivalence followed by a fibration,

$C_0 \to T C \to C,$

in a standard way. I would expect something similar could be said in the $n$ -category situation, if we live to see $n$ -Cat made into a closed model category.

Posted by: Todd Trimble on September 3, 2007 2:15 PM | Permalink | Reply to this

Re: More on Tangent Categories

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we’re turning the inclusion $i : C_0 \to C$ into a weak equivalence followed by a fibration, $C_0 \to T C \to C$ in a standard way.

Ah, thanks for offering this point of view!

I would expect something similar could be said in the $n$ -category situation

I think it should be true for any $n$ and any flavor of $n$ -category that the canonical $C_0 \to T C$ is an equivalence, i.e. that the “categorical $n$ -tangent bundle” $p : T C \to C_0$ is a “deformation retract” of $C_0$ in that we have an equivalence $T C \simeq C_0$ (which, by the way, is smooth when we are in the context of $n$ -categories internal to smooth spaces).

Thanks for sharing this point of view concerning factoring $C_0 \to C$ through an equivalence (which harmonizes nicely with my intuition about “arrow-theoretic tangency” in terms of categorically “puffed up” points).

I’ll try to return the favor by sharing some possibly useful remarks about categorical cotangents, which I am in the process of writing up.

Posted by: Urs Schreiber on September 3, 2007 2:46 PM | Permalink | Reply to this

Re: More on Tangent Categories

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What I should have said was that $T$ is a comonad on the category of strict 2-categories, not on the category of simplicial objects.

Posted by: David Roberts on September 3, 2007 5:29 AM | Permalink | Reply to this

Re: More on Tangent Categories

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Hey, Urs — you may find this talk interesting:

Kathryn Hess, Bundle theory for categories.

On page 28 you’ll see she says “bundles of categories can be thought of as functors endowed with a flat connection”. She goes up to pseudofunctors to get rid of the flatness.

Posted by: John Baez on August 29, 2007 2:55 PM | Permalink | Reply to this

Re: More on Tangent Categories

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Thanks for the link.

I rember Kathryn Hess talked about similar things in Toronto.

I think onbe needs to be really careful with terminology here, paying attention to what degree the notion of “bundle”, “connection” and “flatness” correspond to the standard usage.

Consider this example:

an ordinary principal $G$ -bundle $P$ over base $X$ .

The corresponding Atiyah-groupoid is

$\mathrm{At} = P \times_G P \,.$ This comes equipped with the obvious projection $\mathrm{At} \to X \times X \,.$ Judging from slides 13 and 14 of Hess’s talk, this does qualify as a “bundle of categories” in the terminology used there. (The uniqueness required on slide 14 holds simply because $\mathrm{At}$ is a groupoid.)

If I understand correctly the notation, then the lift on p. 13 is regarded not just as a property, but as extra structure: i.e. we choose once and for all these lifts.

In my example, this amounts to choosing functors $\beta : X \times X \to \mathrm{At} \,.$

And indeed, (if everything is smooth), these correspond precisely to the flat connections on $P$ .

We have two options to get around this:

either we pass from the functor $X \times X \to \mathrm{At}$ to the pseudo functor $X \times X \to \mathrm{AUT}(\mathrm{Ad}_P) \,.$ That’s my notation for what happens on slide 4. I am emphasizing here the point of view that we are really splitting the integrated Atiyah sequence $\mathrm{Ad}P \to \mathrm{At} \to X \times X$ and then making use of Schreier theory. I talked about that in Curvature, the Atiyah Sequence and Inner Automorphisms.

Or we do something else: we pull back the Atiyah groupoid to the full path groupoid along $\array{ && \mathrm{At} \\ && \downarrow \\ P_1(X) &\to& X \times X } \,,$ where now $P_1(X)$ denotes (as always here on the $n$ -Café) the groupoid whose morphisms are just thin-homotopy classes of paths.

Writing $\tilde \mathrm{At}$ for the corresponding pullback, we get now the “bundle of categories” $\tilde \mathrm{At} \to P_1(X) \,.$ And choices oflifts in this case $\mathrm{tra} : P_1(X) \to \tilde\mathrm{At}$ are precisely in bijection with unrestricted connections on $P$ .

(That follows for instance from Parallel Transport and Functors).

I am saying this because it shows that one might need to be a little more careful with the statement on slide 27, which asserts that strict functoriality of parallel transport means that the parallel transport is flat.

This is true in the ordinary sense only if the domain of the transport is just homotopy classes of paths – as one would expect.

Also, the last statement of slide 28, which says that “bundles of categories can be thought of as functors endowed with flat connections” I wouldn’t put this way. First, it seems that one should make explicit that – as far as I understand, see above – we seem to be talking about “bundles of categories equipped with chosen lifts”, and becuase “flat” here doens’t seem to be quite the suggestive term, in general.

By the way, the kind of classification result on slide 23 for “bundles of categories” does in fact have an analogue for smooth principal $n$ -bundles (i.e. for the cases where one has more structure around then the fibers just being categories without smooth structure and/or group action): the map $n\mathrm{Cat}_* \to \mathrm{Cat}$ which Hess considers is then replaced by $\mathrm{INN}(G_{(n)}) \to \Sigma G_{(n)} \,,$ for $G_{(n)}$ the structure $n$ -group, and $\mathrm{INN}(G_{(n)})$ its $(n+1)$ -group of inner automorphisms. This is discussed in the last section of this.

Posted by: Urs Schreiber on August 29, 2007 3:54 PM | Permalink | Reply to this

Re: More on Tangent Categories

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Jim asks:

How about the COtangent version? since I’m more familiar with that in its deformation theory avatar.

I really don’t know. I quickly glanced back at some of the posts on cotangent categories… is there a consensus yet on what these should be? Or at least a best guess?

Posted by: Todd Trimble on September 2, 2007 5:35 PM | Permalink | Reply to this

Read the post Arrow-Theoretic Differential Theory IV: Cotangents
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Tracked: September 3, 2007 4:18 PM

Re: More on Tangent Categories

Until we went to cats, the only tangents
were either on a smooth manifold or purely algebraic: tangent complexes. For manifolds that are only topological, Milnor came up withthe idea of a microbundle, for which the inspiring example was a germ of the diagonal
M \subset M x M.

Inspire any categorical thoughts?

Posted by: jim stasheff on September 7, 2007 2:06 AM | Permalink | Reply to this

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It looks like the tangent microbundle of a topological manifold $M$ is related to the tangent category of $codisc(M)$ . Of course, in this case we know that $M$ has a `presentation’ by a disjoint union of $\mathbf{R}^n$ ’s for some $n$ , and this is what is used to make the tangent microbundle have the local triviality condition $V \stackrel{\sim}{\to} U \times \mathbf{R}^n$ (mentioned here, for example).

Here’s a naive proposal: If $C$ is a category of things with presentations by things with trivial homotopy/cohomology/usw, the tangent microbundle of an object $X$ of $C$ should have some local triviality condition like the one above for topological manifolds, with $\mathbf{R}^n$ replaced by the thing with trivial homotopy/etc. Then we imitate the tangent category construction for the internal category which is the codiscrete one on objects $X$ , and we should get some sort of analogous relationship.

Posted by: David Roberts on September 7, 2007 3:23 AM | Permalink | Reply to this

Re: More on Tangent Categories

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David Roberts wrote:

It looks like the tangent microbundle of a topological manifold $M$ is related to the tangent category of $\mathrm{codisc}(M)$ .

Invaluable, David. Thanks.

For the record, let me first reproduce here the definition Marc Nardman gives at the post you linked to:

Definition (microbundle) A microbundle of rank $n$ is a topological bundle $p : E \to B$ with section $i : B \to E$ which is locally trivial with fibers locally looking like $\mathbb{R}^n$ in that $B$ can be covered with open sets $U$ such that for each $U \subset B$ of base space there is a subset $V \subset E$ of the total space above it, which looks like $U \times \mathbb{R}^n$ :

$\array{ E &&& V &&\stackrel{\sim}{\to}&& U \times \mathbb{R}^n \\ \downarrow^p &&&& {}_i\nwarrow && \nearrow \\ B &&&&& U }$ and $\array{ E &&& V &&\stackrel{\sim}{\to}&& U \times \mathbb{R}^n \\ \downarrow^p &&&& {}_p\searrow && \swarrow \\ B &&&&& U } \,.$

Every ordinary vector bundle is hence a microbundle. But the motivating example is apparently $\array{ E &&&& X \times X \\ \downarrow^p &&:=&& \;\downarrow^{p_1} \\ B &&&& X }$ for $X$ any manifold, where the section is $i : x \mapsto (x,x)$ .

And indeed, as David Roberts indicated, this example is the object part of the tangent category of the pair groupoid over $X$ :

$\array{ X \times X \\ \downarrow \\ X } \;\;\;\; = \mathrm{Obj} \left( \array{ T(\mathrm{codisc}(X)) \\ \downarrow \\ \mathrm{codisc}(X)_0 } \right) \,.$

Notice that also the notion of the canonical section coincides.

One difference is this:

a microbundle is not in general a deformation retract of its base: as opposed to ordinary vector bundles, for a general microbundle we don’t in general have $E \simeq B ,,$ as far as I can see. But that’s always true for a tangent ( $n$ -)groupoid. In the above example one sees what’s going on:

the space of objects of each fiber itself may not be contractible as a topological space (it may look like any manifold $$X$), but when we do take into account the morphisms present in the tangent category, then the result is contractible as a topological category.

That’s good I think. This is a property we want to associate with something whose fibers are supposed to be “infinitesimal deviations”.

Posted by: Urs Schreiber on September 7, 2007 11:10 AM | Permalink | Reply to this

Re: More on Tangent Categories

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I suppose in general we could take the germ of the identity map of an internal groupoid, much as a microbundle of a topological manifold is the germ of the diagonal - this is then a better analogy with existing microbundle theory

Posted by: David Roberts on September 10, 2007 2:30 AM | Permalink | Reply to this

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I suppose in general we could take the germ of the identity map of an internal groupoid, much as a microbundle of a topological manifold is the germ of the diagonal - this is then a better analogy with existing microbundle theory

I am not entirely sure what you have in mind here. Could you expand on that?

But let me try to make a guess:

As I keep emphasizing, the “tangent category” $T C$ does become “infinitesimal” in a possibly generalized sense once we look at suitable families in $\Gamma(T C) \,.$ When $C$ is a smooth groupoid, then smooth group homomorphisms $\mathbb{R} \to \Gamma(T C)$ do pick out the “true tangents” of $C$ : the sections of the corresponding Lie algebroid.

But as we exchange $\mathbb{R}$ for something else, we get different notions of infinitesimals.

Here we want a topological notion.

So for $C$ topological, we could look at continuous maps from the interval into $\Gamma(T C)$ $I \to \Gamma(T C)$ restricted to map the origin to the identity section. Then germs of such maps, at the origin, correspond to what I imagine you are imagining as “germs of the identity map” on $C$ . Is that right?

I am wondering: does it technically make sense to regard $\mathbb{R}$ as a topological group and consider group (or maybe just monoid) homomorphisms $\mathbb{R} \to \Gamma(T C)$ internal to Top? I mean: does this have nontrivial examples for cases when $C$ is a nontrivial topological category? What would the space of such homomorphisms be like?

Posted by: Urs Schreiber on September 10, 2007 8:03 PM | Permalink | Reply to this

Re: More on Tangent Categories

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It seems though that definitions will start to become tautological: The tangent bundle of a smooth manifold $X$ is the Lie algebroid of $X\times X \rightrightarrows X$ , but if it is only a topological manifold, we take the microbundle $X \times X \to X$ , where the projection is the target map. We could then say that the microalgebroid of a groupoid $G$ internal to topological manifolds is just the microbundle $G_1 \stackrel{t}{\to} G_0$ . This would have a map to the tangent microbundle of $G_0$ , and what would be interesting is any other algebraic structure we could pull out. Then, and only then, let us relate this to the tangent groupoid of $G$ .

Also interesting would be investigating microbundles with other fibres, which is certainly considered in the literature.

One could then say that a internal groupoid admitting microstructure is one such that the target map is a microbundle in the sense desired (not with just fibre $\mathbf{R}^n$ , say).

Complementing the definition above, I’ve started to think about microbundles in terms of descent data, not specifying the fibre to be $\mathbf{R}^n$ , and which brings out the notion of `germ of the diagonal’ a bit more.

Posted by: David Roberts on September 11, 2007 4:55 AM | Permalink | Reply to this

Re: More on Tangent Categories

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One thing which looks like it would be necessary is the existence of fine enough covers: I certainly wouldn’t try to do what I said above for varieties with Zariski covers!

Posted by: David Roberts on September 7, 2007 3:52 AM | Permalink | Reply to this

Read the post Lie n-Algebra Cohomology
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Tracked: September 13, 2007 6:52 PM

Re: More on Tangent Categories

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The algebraic geometry and complex spaces definition of cotangent cone of a space is defined (roughly speaking) by modding out all but the infinitesimal neighborhoods of the diagonal in the product. From this point of view, the germs of the diagonal micro bundle “tangent space” of a topological manifold should perhaps better be considered as the topological version of the _cotangent space_. For example like for the cotangent cone in algebraic geometry we are not restricted to locally uniform spaces like topological manifolds. Germs of neighborhoods of the diagonal make perfect sense for more singular topological spaces, they just don’t necessarily make up nice bundles.

Posted by: Rogier Brussee on September 18, 2007 11:39 PM | Permalink | Reply to this

Read the post Obstructions, tangent categories and Lie n-tegration
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Excerpt: Thoughts on n-bundle theory in terms of Lie n-algebras.
Tracked: September 24, 2007 10:00 PM

Read the post Detecting Higher Order Necklaces
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Excerpt: Nils Baas on higher order structures,Enrico Vitale on weak cokernels and a speculation on weak Lie n-algebras triggered by discussion with Pavol Severa.
Tracked: September 28, 2007 12:44 AM

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Tracked: January 30, 2008 9:10 PM

Read the post Doctrinal and Tannakian Reconstruction
Weblog: The n-Category Café
Excerpt: Seeking what is common between Gabriel-Ulmer style dualities and Tannaka duality
Tracked: July 11, 2011 11:36 AM

The n-Category Café

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August 21, 2007