nForum - Discussion Feed (symbol for infinitesimal shape) 2022-05-23T08:07:54-04:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher Urs comments on "symbol for infinitesimal shape" (63559) https://nforum.ncatlab.org/discussion/6443/?Focus=63559#Comment_63559 2017-06-22T10:54:20-04:00 2022-05-23T08:07:53-04:00 Urs https://nforum.ncatlab.org/account/4/ Thanks for the alert. That’s really interesting.

Thanks for the alert. That’s really interesting.

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David_Corfield comments on "symbol for infinitesimal shape" (63558) https://nforum.ncatlab.org/discussion/6443/?Focus=63558#Comment_63558 2017-06-22T10:39:22-04:00 2022-05-23T08:07:53-04:00 David_Corfield https://nforum.ncatlab.org/account/20/ Re #79 and the superalgebra approach to the de Rham complex, I see that Buium uses this approach in arithmetic differential geometry. If you can see page 33 of his book, he points out that there is ...

Re #79 and the superalgebra approach to the de Rham complex, I see that Buium uses this approach in arithmetic differential geometry. If you can see page 33 of his book, he points out that there is no de Rham calculus available, but he can then use the Lie superalgebra formalism.

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David_Corfield comments on "symbol for infinitesimal shape" (63242) https://nforum.ncatlab.org/discussion/6443/?Focus=63242#Comment_63242 2017-06-09T04:10:52-04:00 2022-05-23T08:07:53-04:00 David_Corfield https://nforum.ncatlab.org/account/20/ Ok, have put something in as Definition 0.17.

Ok, have put something in as Definition 0.17.

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Urs comments on "symbol for infinitesimal shape" (63231) https://nforum.ncatlab.org/discussion/6443/?Focus=63231#Comment_63231 2017-06-08T06:50:03-04:00 2022-05-23T08:07:53-04:00 Urs https://nforum.ncatlab.org/account/4/ Maybe we could say “co-quotient” for the fiber.

Maybe we could say “co-quotient” for the fiber.

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David_Corfield comments on "symbol for infinitesimal shape" (63226) https://nforum.ncatlab.org/discussion/6443/?Focus=63226#Comment_63226 2017-06-08T05:13:14-04:00 2022-05-23T08:07:53-04:00 David_Corfield https://nforum.ncatlab.org/account/20/ Ok, so what could we have to parallel the comonadic “is what remains after taking away the piece of..” ? Or just drop it and have The negative of a monadic moment &bigcirc;\bigcirc is the ...

Ok, so what could we have to parallel the comonadic “is what remains after taking away the piece of..” ? Or just drop it and have

The negative of a monadic moment $\bigcirc$ is the fiber (nlab) of the unit map

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Urs comments on "symbol for infinitesimal shape" (63220) https://nforum.ncatlab.org/discussion/6443/?Focus=63220#Comment_63220 2017-06-08T04:05:08-04:00 2022-05-23T08:07:53-04:00 Urs https://nforum.ncatlab.org/account/4/ Yes. Maybe not “quotienting out” for the fiber. On the other hand if we do this on stable types, then the fiber of a morphism is of course also the cofiber of a shifted morphism.

Yes. Maybe not “quotienting out” for the fiber. On the other hand if we do this on stable types, then the fiber of a morphism is of course also the cofiber of a shifted morphism.

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David_Corfield comments on "symbol for infinitesimal shape" (63219) https://nforum.ncatlab.org/discussion/6443/?Focus=63219#Comment_63219 2017-06-08T03:40:39-04:00 2022-05-23T08:07:53-04:00 David_Corfield https://nforum.ncatlab.org/account/20/ Since we don’t have an official definition for monads, can we just imitate def 0.13: The negative of a comonadic moment &square;\Box is what remains after taking away the piece of pure ...

Since we don’t have an official definition for monads, can we just imitate def 0.13:

The negative of a comonadic moment $\Box$ is what remains after taking away the piece of pure $\Box$-quality, hence is the cofiber (nlab) of the counit map:

$\overline{\Box}(X) \coloneqq cofib(\Box X \to X) \,.$

The negative of a monadic moment $\bigcirc$ is what results from quotienting out the pure $\bigcirc$-quality, hence is the fiber (nlab) of the unit map:

$\overline{\bigcirc}(X) \coloneqq fib(X \to \bigcirc X) \,.$
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Urs comments on "symbol for infinitesimal shape" (63218) https://nforum.ncatlab.org/discussion/6443/?Focus=63218#Comment_63218 2017-06-08T03:19:58-04:00 2022-05-23T08:07:53-04:00 Urs https://nforum.ncatlab.org/account/4/ what do &flat;¯E\overline{\flat}E and ʃ¯E\overline{&amp;#643;} E mean here? Sorry, my bad. Yes, as David says, I am using this sometimes to denote the homotopy (co-)fiber of the ...

what do $\overline{\flat}E$ and $\overline{ʃ} E$ mean here?

Sorry, my bad. Yes, as David says, I am using this sometimes to denote the homotopy (co-)fiber of the (co-)unit of the given (co-)monad. (Def. 2.7 in arXiv:1402.7041 or def. 0.13 here).

And the statement which I was referring to, that for a stable cohesive type $E$ the types $\overline{\flat}E$ and $\overline{ʃ} E$ have the interpretation of the (co-)closed $E$-valued differential forms is discussed in some detail at differential cohomology diagram. Unfortunately, there I write instead ${\flat}_{dR} E$ and ${ʃ}_{dR} E$ for these (co-)fibers.

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Urs comments on "symbol for infinitesimal shape" (63217) https://nforum.ncatlab.org/discussion/6443/?Focus=63217#Comment_63217 2017-06-08T03:10:57-04:00 2022-05-23T08:07:53-04:00 Urs https://nforum.ncatlab.org/account/4/ is it provable synthetically that they are? I haven’t tried. That would certainly be a worthwhile question to look into. The key step in the proof would presumeably not be so much concerned with ...

is it provable synthetically that they are?

I haven’t tried. That would certainly be a worthwhile question to look into.

The key step in the proof would presumeably not be so much concerned with the nature of the differential forms as such, but with the interalization of the Dold-Kan corespondence: One will need way to turn a chain complex of stable types into a stable type and show that this construction behaves properly.

Once this is established the remainder should be immediate.

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David_Corfield comments on "symbol for infinitesimal shape" (63214) https://nforum.ncatlab.org/discussion/6443/?Focus=63214#Comment_63214 2017-06-08T01:41:52-04:00 2022-05-23T08:07:53-04:00 David_Corfield https://nforum.ncatlab.org/account/20/ For a comonadic modality &square;¯X&colone;cofib(&square;X&rightarrow;X). \overline{\Box} X \coloneqq cofib(\Box X \to X). (See Definition 2.2.14 of dcct.) For a monadic one, ...

$\overline{\Box} X \coloneqq cofib(\Box X \to X).$

(See Definition 2.2.14 of dcct.)

For a monadic one, it’s the fiber of the unit, though I’m not sure this is made explicit anywhere. It appears in the differential cohomology diagram in Proposition 2.2.17 at the same point in dcct.

Notation is certainly not consistent across nLab.

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maxsnew comments on "symbol for infinitesimal shape" (63209) https://nforum.ncatlab.org/discussion/6443/?Focus=63209#Comment_63209 2017-06-07T20:13:56-04:00 2022-05-23T08:07:53-04:00 maxsnew https://nforum.ncatlab.org/account/1534/ Sorry, but what do &flat;¯E\overline{\flat}E and ʃ¯E\overline{&amp;#643;} E mean here?

Sorry, but what do $\overline{\flat}E$ and $\overline{ʃ} E$ mean here?

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Mike Shulman comments on "symbol for infinitesimal shape" (63207) https://nforum.ncatlab.org/discussion/6443/?Focus=63207#Comment_63207 2017-06-07T19:02:48-04:00 2022-05-23T08:07:53-04:00 Mike Shulman https://nforum.ncatlab.org/account/3/ Are either (1) or (2) an instance of &flat;¯E\overline{\flat}E or ʃ¯E\overline{&amp;#643;} E (when restricted to closed or co-closed forms)? I mean, I presume so in the model, but is it ...

Are either (1) or (2) an instance of $\overline{\flat}E$ or $\overline{ʃ} E$ (when restricted to closed or co-closed forms)? I mean, I presume so in the model, but is it provable synthetically that they are?

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Urs comments on "symbol for infinitesimal shape" (63196) https://nforum.ncatlab.org/discussion/6443/?Focus=63196#Comment_63196 2017-06-07T11:12:56-04:00 2022-05-23T08:07:53-04:00 Urs https://nforum.ncatlab.org/account/4/ That pullback is just the fiberwise &sharp;\sharp Ah, right. So that’s some improvement already! I bet you’ll see many more improvements. I wish I had time to further think about this stuff ...

That pullback is just the fiberwise $\sharp$

Ah, right. So that’s some improvement already! I bet you’ll see many more improvements. I wish I had time to further think about this stuff with you right now.

I don’t think we’ll try solid cohesion yet.

I understand, this was in reaction to David’s suggestion.

The punchline is just that if you do want to see explicit differential forms in a synthetic treatment (as opposed to just knowing that for every stable type $E$ the types $\overline{\flat}E$ and $\overline{ʃ} E$ behave like those of closed and co-closed $E$-valued differential forms, respectively ) then there are these two options:

1. axiomatize $D^1$ via a Kock-Lawvere axiom scheme and proceed in the established fashion – this works but is not elegant,

2. pass to solid cohesion, require that there is an object $\mathbb{R}^{0 \vert 1}$ such that the “rheonomy modality” $Rh$ is $\mathbb{R}^{0\vert 1}$-localization and then obtain the full de Rham complex in one go as

$\Omega^\bullet(-) \;\coloneqq\; \mathbb{R}^{ \left( (-)^{\mathbb{R}^{0\vert 1}} \right)}$

I am not saying that you should do (either of) this. On the contrary, I have been advocating all along the perspective that it is not urgent to consider a synthetic axiomatization of exactly the standard differential forms if we already have the “general differential forms” $\overline{\flat} E$ and $\overline{ʃ} E$.

This is maybe analogous to the story of Euclid’s synthetic axioms: There it is not too urgent to impose the parallel postulate, even if classical experience seems to suggest otherwise. On the contrary, it is useful to first prove things in more generality and only consider the axiom later.

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Mike Shulman comments on "symbol for infinitesimal shape" (63195) https://nforum.ncatlab.org/discussion/6443/?Focus=63195#Comment_63195 2017-06-07T10:00:50-04:00 2022-05-23T08:07:54-04:00 Mike Shulman https://nforum.ncatlab.org/account/3/ Re #75: okay, great, thanks. That pullback is just the fiberwise &sharp;\sharp, so it has a nice type-theoretic interpretation if we regard the filtration as a tower of dependent types. Re #77: ...

Re #75: okay, great, thanks. That pullback is just the fiberwise $\sharp$, so it has a nice type-theoretic interpretation if we regard the filtration as a tower of dependent types.

Re #77: that’s neat, but it’s turning out to be hard enough to write down a type theory for differential cohesion, so I don’t think we’ll try solid cohesion yet.

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Urs comments on "symbol for infinitesimal shape" (63192) https://nforum.ncatlab.org/discussion/6443/?Focus=63192#Comment_63192 2017-06-07T05:36:11-04:00 2022-05-23T08:07:54-04:00 Urs https://nforum.ncatlab.org/account/4/ Right, let me recall: If we do have the “first oder infinitesimal interval” D 1={x&Element;&Ropf;&vert;x 2=0}&subset;&Ropf;D^1 = \{x \in \mathbb{R}\vert x^2 = 0\} \subset ...

Right, let me recall:

If we do have the “first oder infinitesimal interval” $D^1 = \{x \in \mathbb{R}\vert x^2 = 0\} \subset \mathbb{R}$ in hand, then there is the usual SDG Yoga: define microlinear spaces $X$, then say what it means for a map $T X \coloneqq X^{D^1} \longrightarrow \mathbb{R}$ to be fiberwise linear. These fiberwise linear maps $T X \to \mathbb{R}$ then are the differential 1-forms on $X$. Then one defines the wedge products and with this finally higher degree forms.

This is not very elegant, though, first of all due to that requirement to restrict by hand to fiberwise linearity. (Which is also why the “amazing right adjoint$(-)_{D^1}$ to $(-)^{D^1}$ is not all that useful: $\mathbb{R}_{D^1}$ does not modulate differential 1-forms, only a sub-object $\Omega^1 \subset \mathbb{R}_{D^1}$ of it does, which takes work to carve out.)

If however we pass to the super-geometric version of $D^1$, the superpoint $\mathbb{R}^{0\vert 1}$, then this problem goes away: For every $X$ then the full type of functions $X^{\mathbb{R}^{0\vert 1}} \longrightarrow \mathbb{R}$ is, externally, the full de Rham complex of differential forms on $X$ (regarded as a super-space by its canonical $\mathbb{Z}/2$-grading of even/odd-degree forms). It even automatically knows the de Rham differential: that is given by the canonical action of the odd component of $Aut(\mathbb{R}^{0\vert 1})$ on $X^{\mathbb{R}^{0\vert 1}} \to \mathbb{R}$.

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David_Corfield comments on "symbol for infinitesimal shape" (63191) https://nforum.ncatlab.org/discussion/6443/?Focus=63191#Comment_63191 2017-06-07T04:00:49-04:00 2022-05-23T08:07:54-04:00 David_Corfield https://nforum.ncatlab.org/account/20/ It sounds like this is heading close to some of the issues appearing in that inconclusive discussion we had on FTC. Perhaps there is something there for eager young differentially cohesive HoTT-ers ...

It sounds like this is heading close to some of the issues appearing in that inconclusive discussion we had on FTC. Perhaps there is something there for eager young differentially cohesive HoTT-ers in Snowbird to develop, such as that intriguing idea that the solid cohesion of the super-world might be used to characterise constructions at the differential (elastic) stage. (But perhaps the solid modalities are for a later date.)

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Urs comments on "symbol for infinitesimal shape" (63189) https://nforum.ncatlab.org/discussion/6443/?Focus=63189#Comment_63189 2017-06-07T03:17:32-04:00 2022-05-23T08:07:54-04:00 Urs https://nforum.ncatlab.org/account/4/ Sorry, Mike, if I am being unclear. I see that the note in the Sandbox was a bit terse. The general formula needs as input a type AA equipped with a filtration A=A n&rightarrow;A ...

Sorry, Mike, if I am being unclear. I see that the note in the Sandbox was a bit terse.

The general formula needs as input a type $A$ equipped with a filtration $A = A_n \to A_{n-1} \to \cdots \to A_0$, thought of as degreewise forgetting connection data (a “Hodge filtration”). Given this, then the concretification of $A$ on a geometrically contractible type $\Sigma$ is defined starting with

$A \mathbf{Conn}_0(\Sigma) \coloneqq [\Sigma, A_0]$ and

and then inductively

$A \mathbf{Conn}_{k+1}(\Sigma) \coloneqq im_{n-k}\left( [\Sigma, A_{k+1}] \to \sharp [\Sigma, A_{k+1}] \underset{\sharp [\Sigma, A_k]}{\times} A \mathbf{Conn}_k(\Sigma)\right)$.

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Mike Shulman comments on "symbol for infinitesimal shape" (63185) https://nforum.ncatlab.org/discussion/6443/?Focus=63185#Comment_63185 2017-06-06T15:44:50-04:00 2022-05-23T08:07:54-04:00 Mike Shulman https://nforum.ncatlab.org/account/3/ How does that formula apply generally? It uses the Deligne complex.

How does that formula apply generally? It uses the Deligne complex.

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Urs comments on "symbol for infinitesimal shape" (63180) https://nforum.ncatlab.org/discussion/6443/?Focus=63180#Comment_63180 2017-06-06T11:53:52-04:00 2022-05-23T08:07:54-04:00 Urs https://nforum.ncatlab.org/account/4/ That formula which I had pointed you to above (now here) gives the right answer for ordinary differential cocycles. That’s the only case for which we know a priori what the answer should be. So ...

That formula which I had pointed you to above (now here) gives the right answer for ordinary differential cocycles. That’s the only case for which we know a priori what the answer should be. So then from there one would be inclined to turn this around and take that formula, which now applies generally, to be the definition of differential concretification.

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Mike Shulman comments on "symbol for infinitesimal shape" (63170) https://nforum.ncatlab.org/discussion/6443/?Focus=63170#Comment_63170 2017-06-06T09:37:33-04:00 2022-05-23T08:07:54-04:00 Mike Shulman https://nforum.ncatlab.org/account/3/ The theorem involves differential concretification, and you said in #64 that your formula for differential concretification was wrong. Is there a correct formula for differential concretification ...

The theorem involves differential concretification, and you said in #64 that your formula for differential concretification was wrong. Is there a correct formula for differential concretification that works for arbitrary cocycles?

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Urs comments on "symbol for infinitesimal shape" (63152) https://nforum.ncatlab.org/discussion/6443/?Focus=63152#Comment_63152 2017-06-06T04:30:52-04:00 2022-05-23T08:07:54-04:00 Urs https://nforum.ncatlab.org/account/4/ The result I was pointing to expresses the obstruction to a definite globalization of any (differential) cocycle, it is not specific to cocycles in Deligne cohomology.

The result I was pointing to expresses the obstruction to a definite globalization of any (differential) cocycle, it is not specific to cocycles in Deligne cohomology.

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Mike Shulman comments on "symbol for infinitesimal shape" (63110) https://nforum.ncatlab.org/discussion/6443/?Focus=63110#Comment_63110 2017-06-03T18:28:49-04:00 2022-05-23T08:07:54-04:00 Mike Shulman https://nforum.ncatlab.org/account/3/ Can you say more, then, about what you were suggesting in #61 as a goal for formalization? If one of the main inputs to the result can’t be defined synthetically, what exactly would you like to ...

Can you say more, then, about what you were suggesting in #61 as a goal for formalization? If one of the main inputs to the result can’t be defined synthetically, what exactly would you like to see done synthetically?

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Urs comments on "symbol for infinitesimal shape" (63065) https://nforum.ncatlab.org/discussion/6443/?Focus=63065#Comment_63065 2017-06-03T03:49:05-04:00 2022-05-23T08:07:54-04:00 Urs https://nforum.ncatlab.org/account/4/ Right, so the boldface GConn(X)G\mathbf{Conn}(X) is used for the moduli stacks of connections, while the connconn-subscript on BG conn\mathbf{B}G_{conn} denotes the classifying stacks. So GConn(X)G ...

Right, so the boldface $G\mathbf{Conn}(X)$ is used for the moduli stacks of connections, while the $conn$-subscript on $\mathbf{B}G_{conn}$ denotes the classifying stacks. So $G \mathbf{Conn}(X)$ is to denote the concretification of $[X, \mathbf{B}G_{conn}]$.

I don’t know that these Deligne chain complexes may be defined synthetically. The argument in the Sandbox is less ambitious: It just means to show that there is a construction formulated abstractly using just cohesion (that def 3.1) which in the standard model applied to these chain complexes yields the expected result.

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Mike Shulman comments on "symbol for infinitesimal shape" (63062) https://nforum.ncatlab.org/discussion/6443/?Focus=63062#Comment_63062 2017-06-02T14:28:16-04:00 2022-05-23T08:07:54-04:00 Mike Shulman https://nforum.ncatlab.org/account/3/ Oh, no, wait, I see, it is something else, some chain complex. Can that be defined synthetically?

Oh, no, wait, I see, it is something else, some chain complex. Can that be defined synthetically?

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