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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 12th 2010
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   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
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   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
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   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 “$\infty$-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
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   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 $\Xi$ 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;

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

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeFeb 24th 2011
   • (edited Feb 24th 2011)
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   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 $\infty$-topos, then giving a detailed account of the presentation by homological algebra means;

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

     • In 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 $L_\infty$-algebras, not for general $L_\infty$-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 $\infty$-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.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeFeb 24th 2011
   • (edited Feb 24th 2011)
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   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 $\mathbf{B}G$ I have left pretty much untouched, though I am thinking if I had more energy left, I would want to polish that, too.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeApr 14th 2011
   • (edited Apr 14th 2011)
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   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 $L_\infty$-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” .

8. • CommentRowNumber8.
   • CommentAuthorjim stasheff
   • CommentTimeAug 4th 2017
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   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
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   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 $L_\infty$-algebroid as definition A.1 in Commun. Math. Phys. 315 (2012), 169-213 at a time when no comparable definition had been in use.