Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 12th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am working on [[Lie infinity-algebroid]]. So far I have completely reworked the old Idea- and Definition-section to one new Idea-section. More to come.

   I am working on Lie infinity-algebroid.

   So far I have completely reworked the old Idea- and Definition-section to one new Idea-section. More to come.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 12th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded Examples-sections * <a href="http://ncatlab.org/nlab/show/Lie+infinity-algebroid#TangentLieAlgebroid">On tangent Lie algebroids</a>; * <a href="http://ncatlab.org/nlab/show/Lie+infinity-algebroid#LieAlgebra">On Lie algebras</a> * <a href="http://ncatlab.org/nlab/show/Lie+infinity-algebroid#TangentOfLieAlgebra">On tangent Lie 2-algebroids of Lie algebras </a> (inclomplete and stubby, though)

   added Examples-sections

   • On tangent Lie algebroids;

   • On Lie algebras

   • On tangent Lie 2-algebroids of Lie algebras (inclomplete and stubby, though)

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeAug 25th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexfew months back finally somebody else noticed that the obvious term to coin is "$\infty$-Lie algebroid". I added the reference to [[infinity-Lie algebroid]]. Found out about this by coming by chance across the new blog <a href="http://blogs.scienceforums.net/ajb/">Mathematical Ramblings</a>.

   few months back finally somebody else noticed that the obvious term to coin is “-Lie algebroid”. I added the reference to infinity-Lie algebroid.

   Found out about this by coming by chance across the new blog Mathematical Ramblings.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeFeb 24th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am about to bring [[infinity-Lie algebroid]] into shape. Currently I am reworking it and right now it is even more of a mess than it was earlier. But it should get better soon. I think I am now ready to give a decent discussion using monoidal Dold-Kan. Notice that the DK functor $\Xi$ from cochain dg-algebras $A$ in non-negative degree to cosimplicial algebras sends $$ \Xi A : [n] \mapsto \bigoplus_{i = 0}^n A_i \otimes \wedge^i \mathbb{R}^n $$ with the product given by $$ (a, x) \cdot (b, y) = (a b , x \wedge y) \,. $$ If $A = CE(\mathfrak{g}^*)$ is semi-free and graded commutative, then this gives the commutative algebra with $$ (a, x) \cdot (b, y) = (a \wedge b , x \wedge y) \,. $$ To see what this means geometrically, it is useful to realize that this reproduces Anders Kock's smooth loci of infinitesimal simplices (currently discussed at [[infinitesimal object]]). Kock defines loci $\tilde D(k,n)$ as formal duals to the real commutative algebras on generators $\{\epsilon^{i}_j\}$ with $0 \leq i \leq k$ and $0 \leq j \leq n$, subject to the relation $$ \epsilon^{i}_j \epsilon^{i'}_{j'} + \epsilon^{i}_{j'} \epsilon^{i'}_{j} = 0 $$ for all index combinations. He observes that this characterizes precisely the function algebras on simplices in $\mathbb{R}^n$ all whose vertices are infinitesimal neighbours of the origin and of each other. But now let $\mathfrak{a}$ be an ordinary Lie algebroid and $A = CE(\mathfrak{a})$ its Chevalley-Eilenberg algebra, generated locally from a basis $\{t^i\}$ of generators in degree 1. Then with $\{v_j\}$ a basis of $\mathbb{R}^n$ consider the commutative algebra $$ \Xi CE(\mathfrak{g})_n \,. $$ Under the identification $$ \epsilon^i_j := t^i \otimes v_j $$ this is just Kock's algebra for functions on $\tilde D(k,n)$ (relative over a base) and in fact this construction gives a bijection between Kock's algebras and those degreewise in the image of $\Xi$ for $A = CE(\mathfrak{a})$. So the upshot is that we have in full generality (not ust for the examples of Lie algebras and tangent Lie algebroids that I had discussed so far on the entry) that Lie algebroids regarded as infinitesimal Lie $\infty$-groupoids are locally simplicial smooth loci of the form $$ [n] \mapsto U \times \tilde D(k,n) $$ Sorry for the incoherent rambling. I think what I just want to say is: 1. there is a nice story that wraps up and generalizes comprehensively what used to be discussed at [[infinity-Lie algebroid]]; 1. I'll get to it tomorrow, after having slept a bit. ;-)

   I am about to bring infinity-Lie algebroid into shape.

   Currently I am reworking it and right now it is even more of a mess than it was earlier. But it should get better soon.

   I think I am now ready to give a decent discussion using monoidal Dold-Kan. Notice that the DK functor from cochain dg-algebras in non-negative degree to cosimplicial algebras sends

   with the product given by

   If is semi-free and graded commutative, then this gives the commutative algebra with

   To see what this means geometrically, it is useful to realize that this reproduces Anders Kock’s smooth loci of infinitesimal simplices (currently discussed at infinitesimal object).

   Kock defines loci as formal duals to the real commutative algebras on generators with and , subject to the relation

   for all index combinations. He observes that this characterizes precisely the function algebras on simplices in all whose vertices are infinitesimal neighbours of the origin and of each other.

   But now let be an ordinary Lie algebroid and its Chevalley-Eilenberg algebra, generated locally from a basis of generators in degree 1.

   Then with a basis of consider the commutative algebra

   Under the identification

   this is just Kock’s algebra for functions on (relative over a base) and in fact this construction gives a bijection between Kock’s algebras and those degreewise in the image of for .

   So the upshot is that we have in full generality (not ust for the examples of Lie algebras and tangent Lie algebroids that I had discussed so far on the entry) that Lie algebroids regarded as infinitesimal Lie -groupoids are locally simplicial smooth loci of the form

   Sorry for the incoherent rambling. I think what I just want to say is:

   1. there is a nice story that wraps up and generalizes comprehensively what used to be discussed at infinity-Lie algebroid;

   2. I’ll get to it tomorrow, after having slept a bit. ;-)

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeFeb 24th 2011
   • (edited Feb 24th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexokay, I have been working on the entry a bit. It's taking shape now, I think. I have * streamlined the Definition section: first recalling the general abstract definition in any cohesive $\infty$-topos, then giving a detailed account of the presentation by homological algebra means; * worked on the first two subsection of the Properties section: * In <a href="http://ncatlab.org/nlab/show/Lie+infinity-algebroid#ModelsForTheAbstractAxioms">As models for the abstract axioms</a> is the statement and proof that the concrete model by dg-algebras is indeed a model for the abstractly defined notion (actually, currently this is disucssed only for $L_\infty$-algebras, not for general $L_\infty$-algebroids, but the generalization is clear, i think, but a bit more tedious) * In <a href="http://ncatlab.org/nlab/show/Lie+infinity-algebroid#Cohomology">Cohomology of oo-Lie algebroids</a> is statement and proof of the fact that the intrinsic real cohomology of an $\infty$-Lie algebroid in the cohesive $\infty$-topos coincides with the ordinary cochain cohomology of its Chevalley-Eilenberg algebra (again, this is currently written out in fact only for $L_\infty$-algebras, out of laziness). From the next subsection on the entry is still pieced together from loosely related pieces that I still need to go over. Which I'll do next, maybe after some lunch break or so.

   okay, I have been working on the entry a bit. It’s taking shape now, I think. I have

   • streamlined the Definition section: first recalling the general abstract definition in any cohesive -topos, then giving a detailed account of the presentation by homological algebra means;

   • worked on the first two subsection of the Properties section:

     • In As models for the abstract axioms is the statement and proof that the concrete model by dg-algebras is indeed a model for the abstractly defined notion (actually, currently this is disucssed only for -algebras, not for general -algebroids, but the generalization is clear, i think, but a bit more tedious)

     • In Cohomology of oo-Lie algebroids is statement and proof of the fact that the intrinsic real cohomology of an -Lie algebroid in the cohesive -topos coincides with the ordinary cochain cohomology of its Chevalley-Eilenberg algebra (again, this is currently written out in fact only for -algebras, out of laziness).

   From the next subsection on the entry is still pieced together from loosely related pieces that I still need to go over. Which I’ll do next, maybe after some lunch break or so.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeFeb 24th 2011
   • (edited Feb 24th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexhave been further working my way through the entry. I'd dare say it is beginning to look good -- but let me know if you disagree! All edits since the previous message affect the Examples-subsection <a href="LieAlgebroidsAsInfinLieAlgebroids">Lie algebroids as oo-Lie algebroids</a> * I have tried to streamline the discussion of Anders Kock's smooth loci of infinitesimal simplices and the observation and proof that these are precisely what the monoidal Dold-Kan correspondence produces degreewise on CE-algebras of Lie algebroids: in <a href="SmoothLociOfInfinitesimalSimplices">Smooth loci of infinitesimal simplices</a>. (This uses material that I had previously typed into [[infinitesimal object]]. Over there are still a few more details that I haven't moved over yet.) * Using that I have then polished the Examples-section <a href="http://ncatlab.org/nlab/show/Lie+infinity-algebroid#TangentLieAlgebroid">The tangent Lie algebroid</a>. * The Examples-section <a href="http://ncatlab.org/nlab/show/Lie+infinity-algebroid#LieAlgebra">on Lie algebras</a> as the infinitesimal neighbourhood of the point in $\mathbf{B}G$ I have left pretty much untouched, though I am thinking if I had more energy left, I would want to polish that, too.

   have been further working my way through the entry. I’d dare say it is beginning to look good – but let me know if you disagree!

   All edits since the previous message affect the Examples-subsection Lie algebroids as oo-Lie algebroids

   • I have tried to streamline the discussion of Anders Kock’s smooth loci of infinitesimal simplices and the observation and proof that these are precisely what the monoidal Dold-Kan correspondence produces degreewise on CE-algebras of Lie algebroids: in Smooth loci of infinitesimal simplices.

     (This uses material that I had previously typed into infinitesimal object. Over there are still a few more details that I haven’t moved over yet.)

   • Using that I have then polished the Examples-section The tangent Lie algebroid.

   • The Examples-section on Lie algebras as the infinitesimal neighbourhood of the point in I have left pretty much untouched, though I am thinking if I had more energy left, I would want to polish that, too.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeApr 14th 2011
   • (edited Apr 14th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have incorporated into the entry [[L-infinity algebroid]] the change of the terminology in the general abstract case to [[formal cohesive infinity-groupoid]]. Then I have added in the section <a href="http://nlab.mathforge.org/nlab/show/Lie+infinity-algebroid#ModelsForTheAbstractAxioms">Models for the abstract axioms</a> detailed proofs that $L_\infty$-algebroids in the traditional sense are indeed presentations of formal cohesive $\infty$-groupoids in $SynthDiff\infty Grpd$ (and similarly in lots of similar cohesive $\infty$-toposes, but I am concentrating on this one). Previously there had only been a similar argument for delooped $L_\infty$-algebras. One would hope that $L_\infty$-algebroids in the traditional sense are precisely the first order formal synthetic-differential $\infty$-groupoids. But I don't know (yet) how to give a general abstract $\infty$-topos theoretic axiomatization of a formal cohesive $\infty$-groupoid being "first order" .

   I have incorporated into the entry L-infinity algebroid the change of the terminology in the general abstract case to formal cohesive infinity-groupoid. Then I have added in the section Models for the abstract axioms detailed proofs that -algebroids in the traditional sense are indeed presentations of formal cohesive -groupoids in (and similarly in lots of similar cohesive -toposes, but I am concentrating on this one). Previously there had only been a similar argument for delooped -algebras.

   One would hope that -algebroids in the traditional sense are precisely the first order formal synthetic-differential -groupoids. But I don’t know (yet) how to give a general abstract -topos theoretic axiomatization of a formal cohesive -groupoid being “first order” .

8. • CommentRowNumber8.
   • CommentAuthorjim stasheff
   • CommentTimeAug 4th 2017
   • PermaLink
   Author: jim stasheff
   Format: TextBeware a confusion of names L-infinity algebroid versus L∞ -algebroid versus infinity Lie algebroid versus... There are two important inequivalent definitions - with and without higher anchors. Not sure which should have which name but we need consensus.

   Beware a confusion of names L-infinity algebroid versus L∞ -algebroid versus infinity Lie algebroid versus...
   There are two important inequivalent definitions - with and without higher anchors. Not sure which should have which name
   but we need consensus.
9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeAug 4th 2017
   • PermaLink
   Author: Urs
   Format: MarkdownItexMy impression is that I had introduced the definition for _$L_\infty$-algebroid_ as definition A.1 in [Commun. Math. Phys. 315 (2012), 169-213](https://arxiv.org/abs/0910.4001v2) at a time when no comparable definition had been in use.

   My impression is that I had introduced the definition for -algebroid as definition A.1 in Commun. Math. Phys. 315 (2012), 169-213 at a time when no comparable definition had been in use.