Typo correction: “…condition that $u$ is *flat* with respect to $u$” should read “…condition that $u$ is *flat* with respect to $A$”

Keirn

]]>Thanks! Fixed now.

]]>Typo in the link: Maurer-Cartan form.

]]>At *Maurer-Cartan form* I have added a brief further paragraph stating the general cohesive definition more explicitly (previously it was stated in the context of smooth cohesion, but of course the general definition is the same).

Also added a similar cohesive paragraph to *de Rham complex*.

OK, I will do that. Maybe, I may wait a little bit (I would like to think a bit more on synthetic geometry in next couple of days, anyway –there are few points which I would like to sort out beforehand).

One of the points there I would like to have your opinion on. You see, in algebraic geometry when one has a smooth reduced scheme, it is given without nilpotents, and the infinitesimal thickening is a product of a construction, using external infinitesimal object. Kock’s books insist on putting $D$ as a part of the real line, but then he needs another version of $D$ for vector space etc. and axioms for each version and axioms for each version. This becomes cumbersome, to introduces several $D$-s in the game. Then Kock remarks in a footnote at the beginning that, unlike the approach in the book, Lawvere was insisting that one should not start with the extended line, what is about what I sense as better as well. How do you prefer to organize ?

On the other hand, indeed, at the level of categories of qcoh sheaves, one has the thickenings as categories from the very beginning. Somehow the category of sheaves has more interesting subobjects than one can see at the ring level. Indeed, if one takes spectra then infinitesimal neighborhood does not contribute with new points, just with new formal functions. In a special case of a neighborhood of a point, this is to say that the formal functions have zero radius of convergence, hence their common locus is just a point.

]]>Sure, that would be good.

]]>I did not propose removing. I proposed writing that $\theta$ is the same as $Id + \epsilon$ and that $\epsilon$ is in the tangent space, literaly, by the definition in Kock. $\theta$, by the definition in Kock is an element of a monad around identity. The two are in a nontrivial bijection described in Kock. This would be perfect.

]]>No, I am just saying that what it is the entry is correct. When you edit it, please rather expand on what is there, than remove what is there.

]]>in the entry is correct in its sense, and should not be replaced

Are you saying that my proposal (Id + epsilon or exp(epsilon)) is any less correct ? And it agrees with the language of formal groups what is the same thing. Apealing to a “sense” is not necessary if the entities have their unique mathematical identity, but can be used in practice freely, with identity kept in the definition.

]]>I’d be happy to see more discussion of the additive picture under the exp/log isomorphism added. All I meant to say is that the present statement in the entry is correct in its sense, and should not be replaced. But it certainly deserves to be expanded on.

]]>I understand that there is a bijection. This is not new with the synthetic geometry. In algebraic geometry one has the isomorphism between the underlying spaces of the formal group and of the Lie algebra. But it is not the same, it is just isomorphic (at the level of underlying spaces), and the precise formulas do involve the identification maps which are identification for just part of the structure. For example, the Hopf structure for the enveloping algebra and for the formal group are not isomorphic; there is a difference between the group likes and primitive elements. For nilpotent Lie algebras, where the exponentials are finite, both group likes and primitive elements live in the enveloping algebra, and they are different parts of the enveloping algebra, so we are talking about identifying different subsetws of the same set, what one needs really to be careful with. We have the same in the paper with Durov, where the 4 different tangent functors are in the interplay to take care of the various kinds of neighborhoods.

]]>Sometimes

If you let me know explicitly where you find what ambiguous, I’ll be happy to do something about it.

But if you just say it in this generality I can’t do more than say “Sure.”

]]>Trivial example: a matrix is not a linear transformation.

Significant example: connection vs conection form vs parallel transport versus... ]]>

Group element $\theta$ is not a Lie algebra element

Well, the point of the synthetic reasoning is that the difference disappears. The Lie algebra of a group $G$ is the infinitesimal neighbourhood of the neutral element in $G$.

]]>Well, I know there is a bijection, but writing $id + \epsilon$ or writing $exp(\epsilon)$ is the same thing for such infinitesimals. In precise notation one needs to take care of all identifications. So if you want to say $log(x^{-1}\cdot y)$, I agree. Group element $\theta$ is not a Lie algebra element (like what the entry says), but its exponential.Thus the entry needs either writing $log$ or writing $id+\epsilon$, it is the same. I agree, and saw at the beginning that I can write $y = x \theta(x,y)$ just **I disagree in 13 with the entry that $\theta$ is in the Lie algebra**. Instead, $\theta(x,y) = id+\epsilon(x,y)$ where $\epsilon(x,y)$ is in the Lie algebra. That is what I am saying in 13 and still think so. What I see in Kock, agrees with my attitude about this formula (e.g. proof of 4.3.1). Now $\omega_{MC}(y-x) := \epsilon(x,y)$ where $y-x$ is the tangent vector from $x$ to $y$. This is well defined and $\mathfrak{g}$-valued. The entry misleads by stating that $\omega_{MC}(y-x) = \theta(x,y)$ which is *not* true in my opinion (nor makes sense as this would be a $G$-valued form).

Hi Zoran,

thanks for alerting me of #13. I had not seen that! I guess we overlapped again when I typed #14.

The formula is correct, because this is in terms of the multiplicative description. Instead of the “id plus epsilon” that you are expecting to see we here have “id times epsilon = epsilon”.

But it is easiest to just trust the synthetic logic and let it guide you. That means, we just work as if with finite group elements, and let the formalism take care of making sense of infinitesimals.

So we have two group elements $x$ and $y$ and want to consider their difference. That’s the group element $x^{-1} \cdot y$. If $x$ and $y$ are infinitesimally close, we give that a name: $\theta(x,y)$. That’s it.

To see the relation to the expression $id + \epsilon$ that you are expecting to see, check out in Anders Kock’s book (the second one) the section that is called something like “the exp/log relation” or the like. He explains it in much detail.

]]>Urs 14: this answers my 12, but not more important 13.

]]>We had settled on “smooth infinity-group” a good while back. I have made the corresponding terminology changes to *Maurer-Cartan form*.

But, generally, there should not be any confusion, either way. (In 50 years from now we’ll be dropping all the “$\infty$-“s anyway ;-)

]]>Caveat emptor! see comments in recent Notices AMS on the benefits of good notation ]]>

Feel free to change it.

There is a host of $\mathbb{Z}_2$-choices here. Left or right $G$-actions, $\theta$ left or right invariant, etc.

]]>Wait a second, I am aa bit confused with the synthetic definition. It says that for $x$ infinitesimally close to $y$ in $G$, $y = x \tau(x,y)$ gives element $\tau(x,y)\in G$ which is actually in the Lie algebra. In my view, it should be $id$ plus an element in a Lie algebra.

]]>Section “gauge transformation” in Maurer-Cartan form, near the bottom, has $Ad_g A$ instead of $g^{-1}A g$ in one of the formulas. Is this an (unusual) convention or a typo ? If the latter, I will change it to $Ad_{g^{-1}} A$.

]]>Dropping in early, I would say that floating TOCs are one of those things that no two people are going to agree on completely, and so it is something where user customisability comes to the fore. Firefox, and probably other modern browsers that aren’t completely broken, has the ability to impose a style on the top of the given one. In firefox, the *Stylish* extension is one that does this. Using that, one could make the floating TOCs appear wherever one liked, make them a fixed height, make them “drop-down” like menus, … the possibilities, whilst not quite endless, are nonetheless quite large.

The *default* view should be one that is most useful to someone happening by the nLab for the first time. Whilst the current style is not pretty, nothing is hidden from sight and that’s a good thing. But it wouldn’t be hard to build up a stock of little style tweaks that people who think it worth the (small) effort can install and so customise the look as they like.

Urs wrote:

Which reminds me: haven’t heard of Andrew Stacey in a long while…

Andrew has been on vacation for a while. He’s coming back on July 19th. Then he’s gonna help me set up a high-tech blog and forum for Azimuth, based in part on the stuff he’s done here!

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