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March 14, 2007

Differentiating Lie Groupoids to Lie Algebroids

Posted by Urs Schreiber

The concept of a groupoid is a rather natural one. As is that of a Lie groupoid.

Every Lie groupoid may be differentiated to yield a Lie algebroid. However, maybe somewhat surprisingly, the standard definition of a Lie algebroid has an appearence which is nowhere close to the simple elegance of the definition of a Lie groupoid.

While one may tend to accept this as a sad fact of life, it becomes increasingly annoying as one tries to categorify these concepts: passing from (Lie) groupoids to (Lie) 2-groupoids is, again, the most natural thing in the world. But the analogous step on the Lie algebroid side – which surely ought to exist – is, when using the standard definition of a Lie algebroid, quite non-obvious.

In fact, to the best of my knowledge, no direct definition of Lie 2-algebroid has ever appeared.

(What does exists is an indirect definition, using a detour through Baez-Crans Lie-2-algebras, their relation to L L_\infty-algebras, the relation of those to quasi-free differential algebras and finally their known relation to Lie 1-algebroids.)

I would like to try to improve on this situation by re-formulating the definition of the Lie-algebroid Lie(Gr) \mathrm{Lie}(\mathrm{Gr}) associated to any Lie groupoid Gr \mathrm{Gr} using only canonical and natural ingredients.

In order to accomplish this, I invoke the point of view that

every Lie groupoid, Gr\mathrm{Gr}, is canonically a Gr\mathrm{Gr}-equivariant principal Gr\mathrm{Gr}-bundle over its space of objects.

While possibly still sounding a little intricate, this is a very natural point of view, since it is, as I shall make explicit, nothing but the “integrated Yoneda embedding” of the Lie groupoid, which gives rise to the functor tra Gr:GrC \mathrm{tra}_{\mathrm{Gr}} : \mathrm{Gr} \to C^\infty that sends objects to the target fibers over them and morphisms to the postcomposition with these: tra Gr:(xfy)(t 1(x)ft 1(y)). \mathrm{tra}_{\mathrm{Gr}} : (x \stackrel{f}{\to} y) \mapsto ( t^{-1}(x) \stackrel{f \circ \cdot}{\to} t^{-1}(y)) \,.

The following is taken from Differentiating Lie Groupoids, which is slightly more detailed.

Canonical Ingredients

In this section I simply list a couple of standard facts and constructions. These will then be used in the next section to swiftly say how a Lie algebroid arises from a Lie groupoid.

Fact. Every Lie groupoid, when regarded as a span Mor(Gr) t s Obj(Gr) Obj(Gr) \array{ & & \mathrm{Mor}(\mathrm{Gr}) \\ & {}^{t}\swarrow & & \searrow^s \\ \mathrm{Obj}(\mathrm{Gr}) &&&& \mathrm{Obj}(\mathrm{Gr}) } internal to smooth manifolds, canonically becomes a Gr\mathrm{Gr}-principal bundle Mor(Gr) p Obj(Gr) \array{ \mathrm{Mor}(\mathrm{Gr}) \\ {}^p \downarrow \\ \mathrm{Obj}(\mathrm{Gr}) } (also known as a Gr\mathrm{Gr}-torsor) over its own space of objects, with the target map playing the role of the bundle projection and the source map that of the “momentum map’” (or “anchor map”).

This bundle is equivariant with respect to the canonical Gr\mathrm{Gr}-action on its own space of objects.

In the language of parallel transport functors, the same fact has the following, maybe more immediate, formulation (where GrTor\mathrm{GrTor} denotes the category of Gr\mathrm{Gr}-torsors over a point).

Fact. We have a smoothly locally trivializable Gr\mathrm{Gr}-principal parallel transport R:GrGrTor R : \mathrm{Gr} \to \mathrm{GrTor} acting by “right translation” R:(xfy)(t 1(x)ft 1(y)). R : (x \stackrel{f}{\to} y) \mapsto (t^{-1}(x) \stackrel{f \circ \cdot}{\to} t^{-1}(y) ) \,.

(Notice that, while the Gr\mathrm{Gr}-bundle Mor(Gr)Obj(Gr)\mathrm{Mor}(\mathrm{Gr}) \to \mathrm{Obj}(\mathrm{Gr}) does have a global section, it has no equivariant global section.)

This functor encodes the target map and the composition in the groupoid, by way of an “integrated Yoneda embedding”. The source map in Gr\mathrm{Gr} appears, from this point of view, as a natural transformation on this functor:

Fact. Write S:GrC S : \mathrm{Gr} \to C^\infty for the functor that sends everything to Id Obj(Gr)\mathrm{Id}_{\mathrm{Obj}(\mathrm{Gr})}. Then the source map, ss, of Gr\mathrm{Gr} is a natural transformation s:RS. s : R \to S \,.

(Here the application of the faithful forgetful functor GrTorC \mathrm{GrTor} \to C^\infty, which just forgets the groupoid action on a smooth manifold, is to be understood implicitly.)

Fact. We have the following three functors.

1)

The tangent bundle functor T:C VectBun T : C^\infty \to \mathrm{VectBun} sends smooth spaces to their tangent bundle and sends smooth maps to their differential.

That this assignmnet respects composition is nothing but the chain rule of calculus.

2)

The section functor Γ:VectBun(M)Vect \Gamma : \mathrm{VectBun}(M) \to \mathrm{Vect} sends a vector bundle to its space of sections and sends a morphism of vector bundles to the induced map on their sections.

3)

The composition of both, defined on each isomorphism class, ΓT:C | MVect \Gamma \circ T : C^\infty|_{\sim M} \to \mathrm{Vect} in fact factors through the forgetful functor LieAlgVect, \mathrm{LieAlg} \to \mathrm{Vect} \,, since the space of section of a tangent vector bundle TXT X canonically carries the structure of the Lie algebra of vector fields on XX.

In order to combine these facts neatly, consider the following definition.

Write I:P 1(X)Vect I : P_1(X) \to \mathrm{Vect} for the tensor unit in the category [P 1(X),Vect][P_1(X),\mathrm{Vect}] of functors into vector spaces, whose monoidal structure is inherited from the standard monoidal structure on Vect\mathrm{Vect}.

Definition.

Let tra:P 1(X)Vect\mathrm{tra} : P_1(X) \to \mathrm{Vect} be a smoothly locally trivializable vector bundle with connection. A flat section or covariantly constant section of tra\mathrm{tra} is a morphism e:Itra. e : I \to \mathrm{tra} \,. We write Γ fl(tra):=[I,tra] \Gamma_\mathrm{fl}(\mathrm{tra}) := [I,\mathrm{tra}] for the vector space of flat sections of tra\mathrm{tra}.

It follows that to any parallel transport with values in smooth spaces we may canonically associate the Lie algebra of flat sections of the associated vector bundle of vector fields on the fibers.

Central Definition. Given a parallel transport with values in smooth spaces tra:P 1(X)C \mathrm{tra} : P_1(X) \to C^\infty write

(1)Lie(tra):=Γ fl(ΓTtra) \mathrm{Lie}(\mathrm{tra}) := \Gamma_\mathrm{fl}(\Gamma\circ T\circ\mathrm{tra})

for the associated Lie algebra of flat sections of the associated vector bundle of vector fields on the fibers.


Lie Algebroids

We have seen that, essentially by the Yoneda embedding, any Lie groupoid Gr\mathrm{Gr} is encoded in a functor R:GrC , R : \mathrm{Gr} \to C^\infty \,, giving the right action of the groupoid on itself (encoding target and composition maps), together with a transformation s:RS, s : R \to S \,, (encoding the source map).

Applying the definition (1) to this transformation yields a morphism of Lie algebras ρ:=ds:Lie(R)Lie(S). \rho := ds : \mathrm{Lie}(R) \to \mathrm{Lie}(S) \,.

This is the Lie algebroid obtained from differentiating the Lie groupoid Gr\mathrm{Gr}.

To see more clearly how this reproduces the standard way in which the defintion of a Lie algebroid is formulated, notice that Lie(S)Γ(TObj(Gr)) \mathrm{Lie}(S) \simeq \Gamma(T \mathrm{Obj}(\mathrm{Gr})) and Lie(R)Γ( xObj(Gr)T Id xt 1(x)). \mathrm{Lie}(R) \simeq \Gamma( \cup_{x \in \mathrm{Obj}(\mathrm{Gr})} T_{\mathrm{Id}_x}t^{-1}(x) ) \,.

Posted at March 14, 2007 9:16 PM UTC

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Re: Differentiating Lie Groupoids to Lie Algebroids

That made my brain hurt :)

Posted by: Eric on March 14, 2007 10:09 PM | Permalink | Reply to this

Re: Differentiating Lie Groupoids to Lie Algebroids

That made my brain hurt :)

Sorry for that!

(I think yours is the quickest reply to an entry that I ever received. :-)

Let’s see, maybe I can find a dosage of this potion which you’ll find medicative instead of toxic:

The action of a groupoid on itself (like any action) is a functor. I baptized it RR.

Doing nothing on the space of objects of the groupoid is a trivial action, hence also a functor. This I called S.

The fact that the right action of a groupoid on itself leaves the left ends of its morphisms alone leads to a natural transformation s:RS. s : R \to S \,.

Whenever you see a sufficiently well behaved functor, it may be beneficial to think of it as encoding a parallel transport in a bundle.

If the functor takes values in smooth spaces, there is naturally a Lie algebra associated to such a functor.

This assignment is itself functorial. Hence from the morphism of functors s:RS s : R \to S which expresses how a groupoid acts on itself, we obtain a morphism of Lie algebras ρ:Lie(R)Lie(S). \rho : \mathrm{Lie}(R) \to \mathrm{Lie}(S) \,.

This is supposed to make you feel good! It’s smooth and neat and leaves a sweet taste after swallowing it.

What may hurt is the transition from this last formula to the equivalent standard definition of a Lie algebroid.

Your doctor’s advice: don’t attempt to do that unless you feel in good shape for it!

Posted by: urs on March 14, 2007 10:28 PM | Permalink | Reply to this

Re: Differentiating Lie Groupoids to Lie Algebroids

The only problem is that Lie algebroids are the underlying vector bundles, not their Lie algebras of sections.

I suppose one could just work with coherent sheaves of Lie algebras and take the best of both worlds.

On categorifying, one needs to know what object gives rise to a Lie 2-aglebra upon taking sections. I’ve followed the magical mystery tour this far, but I don’t recall seeing such a thing. This is the next installment, I wager.

Posted by: David Roberts on March 15, 2007 3:39 AM | Permalink | Reply to this

Re: Differentiating Lie Groupoids to Lie Algebroids

Hi David R.,

thanks for your comment!

Concerning the problem you see, maybe I was not being explicit enough towards the end:

The only problem is that Lie algebroids are the underlying vector bundles, […]

… which are given by what I called ΓTR\Gamma\circ T \circ R

[…] not their Lie algebras of sections.

… which are extracted by hitting the above with Γ fl\Gamma_{\mathrm{fl}}!

On categorifying, one needs to know what object gives rise to a Lie 2-aglebra upon taking sections.

Yes. We need to understand the categorification of a vector field.

2-Vector fields should automatically come with a bracket operation on them, like ordinary vector fields give you the bracket on them for free.

This involves knowing what the flow exp(v)(t)\mathrm{exp}(v)(t) generated by a vector field vv categorifies to. If we know that, we should be able to form the bracket ddtexp(v)(t)exp(w)(t)exp(v)(t) 1exp(w)(t) 1. \frac{d}{d t} \exp(v)(t)\circ \exp(w)(t)\circ \exp(v)(t)^{-1} \circ \exp(w)(t)^{-1} \,.

I believe the right way to think of nn-flows of nn-vector fields is the one described here.

If we have this, the procedure should be clear:

A Lie 2-groupoid Gr\mathrm{Gr} is canonically a Gr\mathrm{Gr}-equivariant Gr\mathrm{Gr}-2-bundle over its space of objects.

More manifestly, its (right) action on itself is a functor R:GrCat C R : \mathrm{Gr} \to \mathrm{Cat}_{C^\infty} with values in categories internal to smooth spaces.

Given any smooth category, we can form its tangent 2-bundle: T:Cat C 2VectBun. T : \mathrm{Cat}_{C^\infty} \to 2\mathrm{VectBun} \,.

Given any 2-vector bundle, we can form its space of 2-sections. Γ2VectBun2Vect. \Gamma 2\mathrm{VectBun} \to 2\mathrm{Vect} \,.

The composition of these two 2-functors ΓT \Gamma \circ T should naturally factor through the forgetful 2-functor 2LieAlg2Vect, 2\mathrm{LieAlg} \to 2\mathrm{Vect} \,, by way of the bracket of 2-vector fields on themselves, discussed above.

Combining this, we turn the 2-groupoid action on itself into an equivariant 2-vector bundle ΓTR \Gamma \circ T \circ R whose fibers are actually not just 2-vector spaces but in fact Lie 2-algebras.

This induces a Lie 2-algebra structure on the space of sections of this bundle. But we shall only be interested in the sub Lie 2-algebra of flat sections (i.e. of morphisms e:IΓTRe : I \to \Gamma\circ T \circ R) and hence form Γ fl(ΓTR). \Gamma_{\mathrm{fl}}(\Gamma \circ T \circ R) \,.

That would be the natural route for the Magical Mystery Tour to proceed.

Posted by: urs on March 15, 2007 11:53 AM | Permalink | Reply to this

Re: Differentiating Lie Groupoids to Lie Algebroids

Ah, I see - I was confused in thinking you’d ended up with one end product, but just as in the usual definition of algebroid, we need two: the vector bundle and how the anchor map behaves on sections.

Am I right in supposing the tangent 2-bundle of a smooth category is the one mentioned here? Then I assume you’ve gone to its transport description - wait, no you haven’t. tsk tsk - where’s that all pervasive philosphy of anafunctors? ;-)

Also, I suppose these are BC 2-vector spaces - a good place to start, but where are the bimodules? ;-)

(only joking)

Posted by: David Roberts on March 16, 2007 6:28 AM | Permalink | Reply to this

Re: Differentiating Lie Groupoids to Lie Algebroids

we need two: the vector bundle and how the anchor map behaves on sections.

Yes. Let me maybe emphasize: the origin of what I wrote above was the desire to equivalently rewrite in more component-independent language the description in Mackenzie’s book for how to differentiate a Lie groupoid to obtain a Lie algebroid .

I am claiming that the description in terms of flat section of equivariant vector bundles canonically associated to a groupoid regarded as a groupoid bundle over itself, which I gave, reproduces, when you spell it out in components, step-by-step the prescription Mackenzie gives. That at least was the intention.

Am I right in supposing the tangent 2-bundle of a smooth category is the one mentioned here?

Yes, I think so. Actually I am still thinking about the 1-groupoid case at the moment. But, yes, I think as we move to 2-groupoids, that is what will replace the tangent bundle operation TT.

Then I assume you’ve gone to its transport description

Sorry: what is “it” now?

- wait, no you haven’t.

Oh, you think there is anything I don’t conceive in terms of transport? ;-)

tsk tsk - where’s that all pervasive philosphy of anafunctors? ;-)

All hidden in the qualification “locally smooth” which I sprinkle in here and there.

Also, I suppose these are BC 2-vector spaces - a good place to start, but where are the bimodules? ;-)

(only joking)

Right, you are rightly teasing me, who never tires of ranting on how we should not ignore more general 2-vector spaces than Baez-Crans 2-vector spaces.

But don’t get this wrong: if something comes to us as a BC 2-vector space, then that’s how it is. I am just saying that when we set up a theory of this or that, we should beware of not throwing lots of useful 2-vector spaces out of the window before even starting.

As an analogous example: if something comes to you as a complex vector space, then so be it. But if you never allow yourself to think of real vector spaces, you are bound to miss a couple of fun examples for useful vector spaces.

When we do nn-Lie theory, everything naturally comes to us in terms of Baez-Crans nn-vector spaces. It’s not at us to make a choice here.

Posted by: urs on March 16, 2007 5:24 PM | Permalink | Reply to this

Re: Differentiating Lie Groupoids to Lie Algebroids

David mentions the two fold aspect of vector bundle and anchor; this is essentially the
two-fold aspect of Lie Rinehart pairs (A,L)

jim

Posted by: jim stasheff on March 16, 2007 6:29 PM | Permalink | Reply to this

Re: Differentiating Lie Groupoids to Lie Algebroids

this is essentially the two-fold aspect of Lie Rinehart pairs (A,L)(A,L)

This I don’t know. I guess I could try to make a search for Rinehart…

Posted by: urs on March 16, 2007 6:57 PM | Permalink | Reply to this

Re: Differentiating Lie Groupoids to Lie Algebroids

There is now an extensive literature on or using Lie-Rinehart algebras
but the original is
G. Rinehart: Differential forms for general commutative algebras, Trans AMS 108 pp 195-222 (1963) !!~ the same year as me and Murray!!

There’s really no differentiablity involved,
jsut forms dual to derivations.

Posted by: jim stasheff on March 17, 2007 2:09 AM | Permalink | Reply to this

Re: Differentiating Lie Groupoids to Lie Algebroids

Since I’m still somewhat challenged by the problem of copying
parts of the comments to which I’m responding,I’ll try to use the convention of extra indentation for the quotes.


DLGLA

The title suggests you are concerned with the passage from Lie Groupoids to Lie Algebroids, but thee is implict some problems just with Lie algebroids per se. Btw, the smooth structure may be obscuring the algebra. Lie-Rinhart algebras consisting of a pair of a commutative
algebra A and Lie algebra L each of which is a module over the other
with suitable compatability codify the algebra from A = smooth functions and L = vector fields on some manifold.

the source map that of the moment map or anchor map

the source map is to Ob(Gr) but the anchor map is supposed to go to the tangent bundle??

by urs: March 14 To see more clearly…Lie(S) =
and Lie(R) = …

as through a cloud darkly - can you rub my nose in how this matches the formal defn of Lie Algebroid??

by David Roberts: what object gives rise to a Lie 2-algebra on taking sections

I take that to mean: what is a tangent 2-bundle? something of order 2?
cf. 2-jets or am I being too niave?

In the Lie 2-algebra context, we have 2-morphisms which are derivation homotopies betwween 1-morphisms - does that help?

Since not all Lie algebroids integrate to Lie groupoids, I worry
about invoking flows - but maybe all `physically reasonable’
Lie algebroids do integrate?

by urs: We shall only be interested in the sub Lie 2-alg of flat sections
why? isn’t that like never going off shell?

I see there have already been further comments while I was preparing this.
I’ll look at them next.

and what is that photo - too time for me to make out

Posted by: jim stasheff on March 16, 2007 6:21 PM | Permalink | Reply to this

Re: Differentiating Lie Groupoids to Lie Algebroids

I’ll answer the easiest question first:

and what is that photo - too time for me to make out

It’s just a joke. David Roberts found himself taken on a Magical Mystery Tour by my discussion of differentiation of Lie groupoids, so I included a photo of four beatles pretending to be walrusses.

(I am just following John’s policy of including pictures whenever possible. Since I don’t have a photo of a Lie algebroid, I am resorting to circumstantial motives…)

Posted by: urs on March 16, 2007 6:54 PM | Permalink | Reply to this

Re: Differentiating Lie Groupoids to Lie Algebroids

can you rub my nose in how this matches the formal defn of Lie Algebroid??

My description was really supposed to be directly paralleling the presciption for how to differentiate a groupoid as given in Mackenzie’s book. I stared at that for a while and then tried to reformulate it in component-free language, i.e. not mentioning elements but only mentioning morphisms.

So, I am claiming, the “Lie algebra of left-invariant sections” which Mackenzie describes is nothing but what I called Lie(R):=Gamma fl(ΓTR), \mathrm{Lie}(R) := \mathrm{Gamma}_{\mathrm{fl}} (\Gamma \circ T \circ R) \,, namely the Lie algebra of equivariant sections of the equivariant vector bundle which we canonically obtain when we regard the groupoid Gr\mathrm{Gr} as a Gr\mathrm{Gr}-equivariant principal Gr\mathrm{Gr}-bundle over itself.

I will write this out in more detail, if desired. But not right now, I will have to catch my train into the week end…

Posted by: urs on March 16, 2007 7:02 PM | Permalink | Reply to this

Re: Differentiating Lie Groupoids to Lie Algebroids

Since not all Lie algebroids integrate to Lie groupoids, I worry about invoking flows

No need to worry: its the very fact that vector fields have flows which make the concept of a Lie algebroid associated to a Lie groupoid work in the first place.

A Lie algebroid arises from a Lie groupoid by looking at certain flat sections of a certain vector bundle.

These sections carry a canonical bracket, because these vectors really arise as “vector fields” on the target-fibers of the groupoid.

The bracket on vector fields comes from differentiating the group-commutator of the flows they induce.

This is a stupid trivial statement for ordinary vector fields, since every child knows how the bracket on vector fields is defined.

But I mentioned it here because this should be the good way to lift all this to the categorified case: as soon as I know what the flow of a 2-vector field is, I have a chance of deriving a notion of bracket functor on 2-vector fields.

Posted by: urs on March 16, 2007 7:18 PM | Permalink | Reply to this

Re: Differentiating Lie Groupoids to Lie Algebroids

But not all Lie algebroids can be obtained that way, unless you mean formally?

I realize you really have the groupiod in mind initially, but categorically aren’t we somewhat better off with algebra?

Posted by: jim stasheff on March 17, 2007 2:11 AM | Permalink | Reply to this

Re: Differentiating Lie Groupoids to Lie Algebroids

Jim,

Yes, I did mean tangent 2-bundles (but really vector 2-bundles), but I never felt I got a satisfactory answer to the point raised here (last paragraph in urs’ september 13 post) about local triviality.

Somehow (I don’t have time now) natural transformations between functors X TX X_\bullet \to TX_\bullet (“2-sections”) should give rise to derivation homotopies.

I like to think of an algebroid as the infinitesimal version of (s,t):G 1G 0×G 0(s,t):G_1 \to G_0 \times G_0 for some Lie groupoid G G_\bullet, we think of these as groupoids over object space G 0G_0, and spaces over G 0G_0 via the source map and projection on 1st factor. (We are just getting more coskeletal here, measuring this with the anchor map)

A stab at a 2-algebroid in this line of thinking is the infinitesimal version of G 2G 1×G 1G 0×G 0,G_2 \to G_1 \times G_1 \to G_0 \times G_0, for G G_\bullet a (globular) Lie 2-groupoid. Think of this as a diagram of spaces over G 0G_0 via source and projections. Also as a diagram of 2-groupoids with object space G 0G_0. I haven’t checked how much sense G 1×G 1G_1 \times G_1 makes as a somewhat coskeletal 2-groupoid. Then we have a pair of vector bundles over G 0G_0 (besides TG 0TG_0): V 2,V 1V_2,V_1 and “anchor maps” V 2V 1V_2 \to V_1 and V 1TG 0V_1 \to TG_0 satisfying all sorts of conditions. Sections here should coalesce into a 3-term complex or something like it.

Just a comment on itegrability - I heard that all algebroids integrate to smooth stacks, if not Lie groupoids. Can anyone confirm or deny this?

This is nothing like your view urs, but I’m starting from further back.

Posted by: David Roberts on March 19, 2007 6:56 AM | Permalink | Reply to this

Re: Differentiating Lie Groupoids to Lie Algebroids

The integration to smooth formal
stacks was the content of my second comment below, I don’t believe there’ll
be any stronger than formal though.
In fact it should be true and essentially is known I think that any Linfty algebroid integrates to a formal derived higher stack - I guess it’s known for positively or negatively graded ones, I don’t know how exactly one deals with arbitrary ones.
If I understand the terminology correctly
the 2-algebra case should correspond to the theory of Courant algebroids, which are the tangent complexes of gerbes, so very nice stacks.


The construction is essentially Koszul duality, or passing to solutions of the Maurer-Cartan equation (aka Deligne groupoid). Hinich discusses this (without the “oid” part) for Linfty algebras in nonnegative degrees – the idea is that the version of Lie’s theorem gives an equivalence between derived formal groupoids (over a point) and Linfty algebras, and it shouldn’t be hard to add parameters for the general oid statement.
Getzler’s recent Annals paper
on Linfty goes into negative gradings, and Toen’s survey on higher and derived stacks explains beautifully the general picture – they were essentially invented to have a place to integrate such things, with arbitrary stackiness for the negative degrees and dg structure for the positive degrees.. in other words you can think of any Linfty algebroid as the tangent complex to something, formally..

Posted by: David Ben-Zvi on March 19, 2007 11:28 AM | Permalink | Reply to this

Re: Differentiating Lie Groupoids to Lie Algebroids

Yes, I did mean tangent 2-bundles (but really vector 2-bundles), but I never felt I got a satisfactory answer to the point raised here (last paragraph in urs’ september 13 post) about local triviality.

One issue is that the tangent 2-bundle of a 2-space (a category internal to smooth spaces) has a proper base 2-space, instead of a base space with only identity morphisms.

This is different from those 2-bundles that are related to gerbes. Local trivializability for 2-bundles over proper 2-spaces requires a little more thought.

Well, let’s see, call base 2-space BB and its tangent 2-bundle TBBT B \to B. We want to find an epimorphism π:UB, \pi : U \to B \,, such that π *TBU×somethign\pi^*T B \simeq U \times \mathrm{somethign}.

An epimorphism in Cat\mathrm{Cat} is a functor whose image generates the codomain.

So the morphisms of UU should be disjoint unions of smooth subspaces of the morphisms of BB such that they generate all of BB while at the same time being topologically trivial as much as possible.

Let’s maybe look at a simple example. Take B=X×XB = X \times X to be the pair groupoid of some nn-dimensional manifold X. Let uXu \to X be any good cover of XX and let UU be the sub-groupoid of the pair groupoid of uu which contains only those pairs both of whose elements have the same projection to XX.

Then UU generates X×XX \times X and π:UB\pi : U \to B is hence epi.

Is π *TBU×something\pi^* T B \simeq U \times \mathrm{something}?

Er, hm, eh, not sure… Can’t seem to find a suitable isomorphism. Maybe requires a proper equivalence.

The other question would be: do we need any notion of local trivializablility for the applications of tangent 2-bundles that we have in mind?

Posted by: urs on March 19, 2007 12:49 PM | Permalink | Reply to this

Re: Differentiating Lie Groupoids to Lie Algebroids

David Roberts wrote:

I like to think of an algebroid as the infinitesimal version of (s,t):G 1G 0×G 0(s,t) : G_1 \to G_0 \times G_0 for some Lie groupoid G G_\bullet

[…]

This is nothing like your view urs

Hm, not sure why you say that this is nothing like the view I have talked about. I thought the issue is mainly how to spell out the details nicely.

Maybe all I am really doing is fighting the fact that I am not really familiar with the point of view that David Ben-Zvi and others are emphasizing. When I read David Ben-Zvi’s comment’s I just feel a little dumb and uneducated.

On the other hand, my current practical needs call for something that works without mentioning either of “formal” or “stack” and uses nothing but some arrow-theory and ordinary differential geometry.

To be concrete, I would like to write a nice discussion of connections on principal groupoid bundles in terms of algebroid morphisms, accessible to the ordinary mortal who knows what a groupoid is and what a differentiable manifold is.

I found the available literature to be wanting, in this respect. But possibly that’s just me.

Anyway, what I want is this: for me a principal groupoid bundle with connection is a locally smooth functor tra:P 1(X)GrTor \mathrm{tra} : P_1(X) \to \mathrm{GrTor} from paths in base space to torsors (over a point) for the given groupoid.

Locally this looks like a functor tra U:P 1(U)Gr \mathrm{tra}_U : P_1(U) \to \mathrm{Gr} from paths to the groupoid itself.

This is a smooth functor of Lie groupoids. I want a “nice” way to say what we get when differentiating this to dtra U:Lie(P 1(U))Lie(Gr). d \mathrm{tra}_U : \mathrm{Lie}(P_1(U)) \to \mathrm{Lie}(\mathrm{Gr}) \,.

Using the point of view that I was advocating in the above entry, I do get something reasonably nice. But I am not convinced yet that I have found the optimal way to do this.

I am thinking of “nice” here in some absolute sense, but of course the risk is that what I consider “nice” is just that which does not strain my own ignorance too much.

I’ll need to keep thinking about this. Am grateful for all the comments here.

Posted by: urs on March 19, 2007 1:59 PM | Permalink | Reply to this

Re: Differentiating Lie Groupoids to Lie Algebroids

urs schreibt

Hm, not sure why you say that this is nothing like the view I have talked about. I thought the issue is mainly how to spell out the details nicely.

I mean: it looks little like the p.o.v. of a groupoid as a trivial bundle over its own object space, and more like a slight restatement of the Mackenzie construction.

Posted by: David Roberts on March 20, 2007 4:37 AM | Permalink | Reply to this

Re: Differentiating Lie Groupoids to Lie Algebroids

I’m sorry if my comments were unpleasantly/obnoxiously terse, didn’t intend them to be so.

What I was trying to express is the following: if you have a Lie algebroid, you can take its universal enveloping algebroid – this is defined in direct analogy with the universal enveloping algebra, or like the definition of differential operators starting from vector fields: you make a unital algebra over functions generated by the algebroid and subject to the relations given by the bracket, and commutation with functions given by the anchor map.

Next the dual to the enveloping algebra on a group is the completion at the identity of functions on the group. You can do the same for an algebroid - the dual to the enveloping algebroid are naturally functions on a groupoid over the base - well, really just the germ of a groupoid near the identity section,
aka formal groupoid.

If the anchor is zero, this just produces
a family of (formal) groups over the manifold.

The claims are 1. this can be thought of as a stack, though that doesn’t really gain anything on this picture.
2. you can do the same game starting from more exotic objects, like Lie 2-algebras or Linfty algebras. what you get when you integrate these objects will be more complicated, but basically be a Z-graded
version of this story.

A nice language for all of this is Koszul duality, which is a fancy way to explain passing from a Lie algebra to the ring of functions on the corresponding group near the identity – ie from Lie things to commutative things, and vice versa.
This is a super powerful and general POV
which has been around I guess since Quillen and I only learned about very recently hence my haranguing about it. Sorry if the tone was not helpful..
David

Posted by: David Ben-Zvi on March 20, 2007 6:35 AM | Permalink | Reply to this

Re: Differentiating Lie Groupoids to Lie Algebroids

I’m sorry […]

Oh, no need! I very much appreciate your comments!

I am roughly aware of the general idea Koszul duality, at least as far as John Baez taught us about it, and I am pretty fond of the duality between L L_\infty algebras and their dual quasi-free differential algebras, which is the example of Koszul duality you seem to be referring to.

Posted by: urs on March 20, 2007 1:26 PM | Permalink | Reply to this

Re: Differentiating Lie Groupoids to Lie Algebroids

I wrote:

A stab at a 2-algebroid in this line of thinking is the infinitesimal version of G 2G 1×G 1G 0×G 0G_2 \to G_1 \times G_1 \to G_0\times G_0, for G G_\bullet a (globular) Lie 2-groupoid. Think of this as a diagram of spaces over G 0G_0 via source and projections. Also as a diagram of 2-groupoids with object space G 0G_0. I haven’t checked how much sense G 1×G 1G_1 \times G_1 makes as a somewhat coskeletal 2-groupoid.

G 1×G 1G_1 \times G_1 here should be G 1× G 0×G 0G 1G_1 \times_{G_0 \times G_0} G_1 - pairs of morphisms with the same source and target. This is the 2-codiscrete 2-groupoid associated to the underlying 1-groupoid of G G_\bullet, with exactly one 2-morphism between any two 1-morphisms. G 0×G 0G_0 \times G_0 is the codiscrete 1-groupoid thought of as a discrete 2-groupoid.

It would help if I knew what the action 2-groupoid of a 2-groupoid acting on something (==itself) looked like. I checked leaked bits of Igor’s work and I could only find the action 1-groupoid of a 2-groupoid - no data of the action of 2-morphisms.

Posted by: David Roberts on March 20, 2007 5:33 AM | Permalink | Reply to this

Re: Differentiating Lie Groupoids to Lie Algebroids

I apologize if this is stupid (or just restating what you said) but
isn’t a Lie algebroid just a sheaf
(vector bundle) with Lie bracket
given by a first order bidifferential operator?

More generally we have the category of vector bundles with
morphisms all differential operators, and we want
a Lie algebra object in here I think,
plus again the requirement
that we want a first order bracket.

Another way to say this would be to look in the
monoidal category of bimodules over functions
(i.e. sheaves on the square) and look for algebra objects in there.
Among these there are LOCAL algebras -
those supported on the formal
neighborhood of the diagonal - and
universal enveloping algebroids of Lie
algebroids are the typical example
(eg the algebra of differential operators,
which comes from the tangent sheaf).

Then if we have a Lie groupoid (group
object in spaces over X x X, or something
like that) taking the tangent space
to the unit section gives a Lie algebroid.
(I’m clearly missing something nice to say
since sheaves on X x X are not symmetric monoidal, so probably one has
to say something in order to make sense of Lie algebra objects, but near
the diagonal I guess that works out
or something… anyone?)

Posted by: David Ben-Zvi on March 15, 2007 4:52 AM | Permalink | Reply to this

Re: Differentiating Lie Groupoids to Lie Algebroids

Okay that comment was probably stupid,
but let’s try a different one. In characteristic zero (where I live) Lie algebras are the same as formal groups, using the Baker-Campbell-Hausdorff theorem. Likewise by formally exponentiating, Lie algebroids are the same as formal groupoids – i.e. groupoids which are just small thickenings of the identity section
(the caveat being they can also grow “vertically” over the diagonal – for example if we take a bundle of Lie algebras,
so the Lie bracket is function-linear,
then it exponentiates to a family of formal groups over X).

So if you like Lie groupoids, and you like
formal neighborhoods, and you like characteristic zero, then you like Lie algebroids.

(likewise if you like formal derived group stacks then you like Linfty algebroids and conversely, though that statement might have somewhat limited appeal..but if not here, where?)

Posted by: David Ben-Zvi on March 15, 2007 5:32 AM | Permalink | Reply to this

Re: Differentiating Lie Groupoids to Lie Algebroids

Hi David Ben-Zvi,

thanks a lot for these comments! Certainly neither stupid (if anyone is stupid here, it’s me) nor restating what I said.

I guess what you describe is a very useful way to look at algebroids. Maybe what I am describing is just a strange way to get to the same point.

Posted by: urs on March 15, 2007 2:14 PM | Permalink | Reply to this

Re: Differentiating Lie Groupoids to Lie Algebroids

I was alerted by private email to the fact that my original statement about the section functor was at best imprecise.

For morphisms of vector bundles where the action on the base space is not iso, taking sections is not really functorial.

Of course in the above application the bases of those vector bundles (assuming without restriction of generality that the groupoid we are talking about is connected) are all diffeomorphic.

This allows to really have a functor Γ:VectBun(M)Vect. \Gamma : \mathrm{VectBun}(M) \to \mathrm{Vect} \,.

Where Mt 1(x)M \simeq t^{-1}(x) for all xx.

So that’s (one way to state) the reason why we can speak of the Lie algebroid of a Lie groupoid, but don’t have a directly analogous construction for the differentiation of arbitrary smooth categories.

Posted by: urs on March 15, 2007 5:53 PM | Permalink | Reply to this

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