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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeApr 13th 2011
   • (edited Apr 13th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have touched [[formal group]] a bit, but don't have time to do anything substantial. I need to adjust some of the terminology that I had been setting up at [[cohesive (infinity,1)-topos]] related to _infinitesimal cohesion_ : the abstract notion currently called "$\infty$-Lie algebroid" there should be called "formal cohesive $\infty$-groupoid". The actual [[L-infinity algebroid]]s are (just) the first order formal smooth $\infty$-groupoids. While on the train I started expanding some other entries on this point, but I need to quit now and continue after a little interruption.

   I have touched formal group a bit, but don’t have time to do anything substantial.

   I need to adjust some of the terminology that I had been setting up at cohesive (infinity,1)-topos related to infinitesimal cohesion : the abstract notion currently called “$\infty$-Lie algebroid” there should be called “formal cohesive $\infty$-groupoid”. The actual L-infinity algebroids are (just) the first order formal smooth $\infty$-groupoids.

   While on the train I started expanding some other entries on this point, but I need to quit now and continue after a little interruption.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeApr 13th 2011
   • (edited Apr 13th 2011)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexIn the usual context there is both an algebroid **and** a formal groupoid, te latter can be first order, second order, whatever. In any case they are different beasts there. I learned a version of the latter from [[Tomasz Maszczyk]]. For example, for infinite neighborhood of a diagonal $\Delta$, there is a sequence of embeddings $\Delta\hookrightarrow\Delta^{(\infty)}\hookrightarrow X\times X$. Now consider a groupoid $G$ with $X = G_0$, with target and source put together into one map $G\to X\times X$. If one pulls it back one gets $G^{\infty}$ over $\Delta^{(\infty)}$ equipped with a composition induced from the composition on $G$, satisfying certain associativity etc. relations. This is sort of infinitesimal groupoid. And it is not an algebroid, just an infinitesimal version of the groupoid whose structure is as close as possible, even in structure, to the original.

   In the usual context there is both an algebroid and a formal groupoid, te latter can be first order, second order, whatever. In any case they are different beasts there. I learned a version of the latter from Tomasz Maszczyk.

   For example, for infinite neighborhood of a diagonal $\Delta$, there is a sequence of embeddings $\Delta\hookrightarrow\Delta^{(\infty)}\hookrightarrow X\times X$. Now consider a groupoid $G$ with $X = G_0$, with target and source put together into one map $G\to X\times X$. If one pulls it back one gets $G^{\infty}$ over $\Delta^{(\infty)}$ equipped with a composition induced from the composition on $G$, satisfying certain associativity etc. relations. This is sort of infinitesimal groupoid. And it is not an algebroid, just an infinitesimal version of the groupoid whose structure is as close as possible, even in structure, to the original.

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeApr 13th 2011
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI mean $G^\infty$ which I mentioned above has the appeal that it is **supported** at $\Delta^\infty$, just as one wants: one has composition of things which are close one to another, that is belong to a neighborhood of the diagonal.

   I mean $G^\infty$ which I mentioned above has the appeal that it is supported at $\Delta^\infty$, just as one wants: one has composition of things which are close one to another, that is belong to a neighborhood of the diagonal.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeApr 13th 2011
   • (edited Apr 13th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes, that's what I said. A Lie algebroid is a first order instance of a formal groupoid. Do we have a special term for "finite order formal"?

   Yes, that’s what I said. A Lie algebroid is a first order instance of a formal groupoid.

   Do we have a special term for “finite order formal”?

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeApr 13th 2011
   • (edited Apr 13th 2011)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexUrs, maybe you said, but just for the clearing-up record, I said that there exists a notion of infinitesimal groupoid of **any order**. Even the first order formal tangent groupoid is a different notion from the first order tangent algebroid. The matter is not in the order but in the packing of data. For a formal groupoid one still has analogues of the source and target map, but they get factored through the infinitesimal neighborhood and do not live on the whole of $X\times X$ like the original groupoid, nor live on the strict diagonal as the tangent algebroid. Finite order formal, well, for me it is just $n$-th order truncation or the formal groupoid supported at $n$-th neighborhood.

   Urs, maybe you said, but just for the clearing-up record, I said that there exists a notion of infinitesimal groupoid of any order. Even the first order formal tangent groupoid is a different notion from the first order tangent algebroid. The matter is not in the order but in the packing of data. For a formal groupoid one still has analogues of the source and target map, but they get factored through the infinitesimal neighborhood and do not live on the whole of $X\times X$ like the original groupoid, nor live on the strict diagonal as the tangent algebroid.

   Finite order formal, well, for me it is just $n$-th order truncation or the formal groupoid supported at $n$-th neighborhood.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeApr 13th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexOkay, so probably this is just a matter of repackaging data. At [[infinity-Lie algebroid]] is a fairly detailed discussion of how Lie algebroids are first order formal groupoids.

   Okay, so probably this is just a matter of repackaging data. At infinity-Lie algebroid is a fairly detailed discussion of how Lie algebroids are first order formal groupoids.

7. • CommentRowNumber7.
   • CommentAuthorTodd_Trimble
   • CommentTimeFeb 10th 2018
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI added to [[formal group]] the fundamental class of examples of a 1-dimensional formal group law, given by $\mu(x, y) = f^{-1}(f(x) + f(y))$ where $f(x)$ is any power series of the form $x + a_2 x^2 + a_3 x^3 +\ldots$ in $R[ [x] ]$. (That the universal formal group law arises in this way should be in some sense the content of [[Lazard ring|Lazard's theorem]], right? So the same observation should probably go there.)

   I added to formal group the fundamental class of examples of a 1-dimensional formal group law, given by $\mu(x, y) = f^{-1}(f(x) + f(y))$ where $f(x)$ is any power series of the form $x + a_2 x^2 + a_3 x^3 +\ldots$ in $R[ [x] ]$. (That the universal formal group law arises in this way should be in some sense the content of Lazard’s theorem, right? So the same observation should probably go there.)

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeFeb 10th 2018
   • (edited Feb 10th 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks, Todd!! You seem to be looking into formal power series things now. I wonder what motivates you? But it's great. > should be in some sense the content of [[Lazard ring|Lazard's theorem]], right? Yes, looks like it should. I haven't thought about it this way. Or if I have, I forgot.

   Thanks, Todd!!

   You seem to be looking into formal power series things now. I wonder what motivates you? But it’s great.

   should be in some sense the content of Lazard’s theorem, right?

   Yes, looks like it should. I haven’t thought about it this way. Or if I have, I forgot.

9. • CommentRowNumber9.
   • CommentAuthorTodd_Trimble
   • CommentTimeFeb 10th 2018
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexThe immediate source of this is that Jim Dolan is back in the States and he and I are talking about a bunch of things. So I'm digging underneath and establishing some infrastructure to help boost my understanding in these discussions, and there's a good chance I'll continue touching quite a few entries in the process.

   The immediate source of this is that Jim Dolan is back in the States and he and I are talking about a bunch of things. So I’m digging underneath and establishing some infrastructure to help boost my understanding in these discussions, and there’s a good chance I’ll continue touching quite a few entries in the process.

10. • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 10th 2018
    • (edited Feb 10th 2018)
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItex>back in the States A visit? Or moved back?

    back in the States

    A visit? Or moved back?

11. • CommentRowNumber11.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 10th 2018
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexWell, I think moved back, but I don't know the details.

    Well, I think moved back, but I don’t know the details.

12. • CommentRowNumber12.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 11th 2018
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexThanks, Todd. Pass on my regards.

    Thanks, Todd. Pass on my regards.

13. • CommentRowNumber13.
    • CommentAuthorMaximUrschumzew
    • CommentTimeMay 11th 2019
    • PermaLink
    Author: MaximUrschumzew
    Format: MarkdownItexHello, I just came across [Example 1.5](https://ncatlab.org/nlab/show/formal+group#Commutative1DimFormalGroupLaw). It says that the unitality condition corresponds to requiring $$a_{i,0} = \left\{ \array{ 1 & if i = 0 \\ 0 & otherwise } \right. $$ I think it should say $i = 1$ instead, as it does further down the page. (Is posting here the right way to go about this? Since I am not familiar with this topic, I thought that I shouldn't be editing the page.)

    Hello, I just came across Example 1.5. It says that the unitality condition corresponds to requiring

    $$a_{i,0} = \left\{ \array{ 1 & if i = 0 \\ 0 & otherwise } \right.$$

    I think it should say $i = 1$ instead, as it does further down the page.

    (Is posting here the right way to go about this? Since I am not familiar with this topic, I thought that I shouldn’t be editing the page.)

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeMay 11th 2019
    • (edited May 11th 2019)
    • PermaLink
    Author: Urs
    Format: MarkdownItexThanks for the alert. Yes that's the right way to go about this. I have fixed it now in the entry

    Thanks for the alert. Yes that’s the right way to go about this.

    I have fixed it now in the entry

15. • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJan 22nd 2021
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to: * Antonia W. Bluher, _Formal groups, elliptic curves, and some theorems of Couveignes_, in: J.P. Buhler (eds.) _Algorithmic Number Theory_ ANTS 1998. Lecture Notes in Computer Science, vol 1423. Springer 1998 ([arXiv:math/9708215](https://arxiv.org/abs/math/9708215), [doi:10.1007/BFb0054887]( https://doi.org/10.1007/BFb0054887)) <a href="https://ncatlab.org/nlab/revision/diff/formal+group/43">diff</a>, <a href="https://ncatlab.org/nlab/revision/formal+group/43">v43</a>, <a href="https://ncatlab.org/nlab/show/formal+group">current</a>

    added pointer to:

    • Antonia W. Bluher, Formal groups, elliptic curves, and some theorems of Couveignes, in: J.P. Buhler (eds.) Algorithmic Number Theory ANTS 1998. Lecture Notes in Computer Science, vol 1423. Springer 1998 (arXiv:math/9708215, doi:10.1007/BFb0054887)

    diff, v43, current