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1. • CommentRowNumber1.
   • CommentAuthordomenico_fiorenza
   • CommentTimeOct 15th 2010
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   Author: domenico_fiorenza
   Format: MarkdownItexIn the discussion on [[Lie integration]] for circle Lie $n$-groups, more precisely <a href="http://ncatlab.org/nlab/show/Lie+infinity-groupoid#LieIntDiffCoeffsForLineG roups">here</a> we describe a relation between a simplicial preshef arising from Dold-Kan applied to a differential complex and another one obtained by Lie integration. The relation between the two should actually be more evident than how we describe it: any differential complex is an $L_\infty$-algebra (with trivial higher brackets), and the exp construction for these particular $L_\infty$-algebras is precisely the Dold-Kan map. (if I'm not wrong).

   In the discussion on Lie integration for circle Lie $n$-groups, more precisely here we describe a relation between a simplicial preshef arising from Dold-Kan applied to a differential complex and another one obtained by Lie integration. The relation between the two should actually be more evident than how we describe it: any differential complex is an $L_\infty$-algebra (with trivial higher brackets), and the exp construction for these particular $L_\infty$-algebras is precisely the Dold-Kan map. (if I’m not wrong).

2. • CommentRowNumber2.
   • CommentAuthordomenico_fiorenza
   • CommentTimeOct 15th 2010
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   Author: domenico_fiorenza
   Format: MarkdownItexmore precisely, fo a fixed Cartesian space $U$, we can first consider the $L_\infty$-algebra $\Omega^1(U)\to\cdots\to \Omega^n(U)_{closed}$ given by the truncated de Rham complex of $U$ shifted so to be in degrees $[-n,0]$ and then exponentiate it to find that a $k$-simplex is a closed degree 1 element in $\Omega^{[1,n]}(U)[n-1]\otimes\Omega^\bullet(\Delta^k)$, i.e. a degree $n$ closed form in $\Omega^\bullet(U\times\Delta^k)$ with at least "a leg along $U$". or we can directly consider the $L_\infty$-algebra $b^{n-1}\mathbb{R}$ and consider in $exp(b^{n-1}\mathbb{R})(U,[k])$ the subpresheaf consisting of elements with at least a leg along $U$. the two are manifestly the same.

   more precisely, fo a fixed Cartesian space $U$, we can first consider the $L_\infty$-algebra $\Omega^1(U)\to\cdots\to \Omega^n(U)_{closed}$ given by the truncated de Rham complex of $U$ shifted so to be in degrees $[-n,0]$ and then exponentiate it to find that a $k$-simplex is a closed degree 1 element in $\Omega^{[1,n]}(U)[n-1]\otimes\Omega^\bullet(\Delta^k)$, i.e. a degree $n$ closed form in $\Omega^\bullet(U\times\Delta^k)$ with at least “a leg along $U$”.

   or we can directly consider the $L_\infty$-algebra $b^{n-1}\mathbb{R}$ and consider in $exp(b^{n-1}\mathbb{R})(U,[k])$ the subpresheaf consisting of elements with at least a leg along $U$. the two are manifestly the same.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeOct 15th 2010
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   Author: Urs
   Format: MarkdownItex> and the exp construction for these particular $L_\infty$-algebras is precisely the Dold-Kan map. (if I'm not wrong). Let#s see: the Dold-Kan map $Ch_\bullet \to sSet$ goes $$ V_\bullet \mapsto ([k] \mapsto Hom_{Ch_\bullet} ( N_\bullet(\Delta^k) , V_\bullet )) \,, $$ where $N_\bullet(\Delta^k)$ is the normalized chains complex of the simplicial homology $\Delta^k \times \mathbb{Z}$. I guess what you are saying is that if we dualize (supposing we are of finite type) we get $$ ([k] \mapsto Hom_{Ch^\bullet\bullet} ( V^\bullet , N^\bullet(\Delta^k) )) \,, $$ where now $N^\bullet(\Delta^k )$ is the simplicial cochain complex. Now, that't very similar to the Lie integration map, which has $$ ([k] \mapsto Hom_{Ch^\bullet\bullet} ( V^\bullet , \Omega^\bullet(\Delta^k) )) \,. $$ While similar, this latter is a much bigger model. One still has to discuss a little that both are actually weakly equivalent. But let me know if I am misunderstanding what you have in mind.

   and the exp construction for these particular $L_\infty$-algebras is precisely the Dold-Kan map. (if I’m not wrong).

   Let#s see: the Dold-Kan map $Ch_\bullet \to sSet$ goes

   $$V_\bullet \mapsto ([k] \mapsto Hom_{Ch_\bullet} ( N_\bullet(\Delta^k) , V_\bullet )) \,,$$

   where

   $N_\bullet(\Delta^k)$ is the normalized chains complex of the simplicial homology $\Delta^k \times \mathbb{Z}$.

   I guess what you are saying is that if we dualize (supposing we are of finite type) we get

   $$([k] \mapsto Hom_{Ch^\bullet\bullet} ( V^\bullet , N^\bullet(\Delta^k) )) \,,$$

   where now $N^\bullet(\Delta^k )$ is the simplicial cochain complex.

   Now, that’t very similar to the Lie integration map, which has

   $$([k] \mapsto Hom_{Ch^\bullet\bullet} ( V^\bullet , \Omega^\bullet(\Delta^k) )) \,.$$

   While similar, this latter is a much bigger model. One still has to discuss a little that both are actually weakly equivalent.

   But let me know if I am misunderstanding what you have in mind.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeOct 15th 2010
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   Author: zskoda
   Format: TextIs this exp remark related to something behind the talk of Joyal discussed recently about the relation between Newton interpolation formula and Dold-Kan ?

   Is this exp remark related to something behind the talk of Joyal discussed recently about the relation between Newton interpolation formula and Dold-Kan ?
5. • CommentRowNumber5.
   • CommentAuthordomenico_fiorenza
   • CommentTimeOct 15th 2010
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   Author: domenico_fiorenza
   Format: MarkdownItex> While similar, this latter is a much bigger model. One still has to discuss a little that both are actually weakly equivalent. Right. The fact that they are equivalent is the starting point in Getzler's <a href="http://arxiv.org/abs/math/0404003">Lie theory for nilpotent L-infinity algebras</a>. It's me having a distorted way of looking at this: since I first met Dold-Kan in Getzler's paper, I'm used to think of it in terms of exp, but that's not the original definition. By the way, Getzler's proof, as you know, is based on Dupont and on integration on simplices :)

   While similar, this latter is a much bigger model. One still has to discuss a little that both are actually weakly equivalent.

   Right. The fact that they are equivalent is the starting point in Getzler’s Lie theory for nilpotent L-infinity algebras. It’s me having a distorted way of looking at this: since I first met Dold-Kan in Getzler’s paper, I’m used to think of it in terms of exp, but that’s not the original definition.

   By the way, Getzler’s proof, as you know, is based on Dupont and on integration on simplices :)

6. • CommentRowNumber6.
   • CommentAuthordomenico_fiorenza
   • CommentTimeOct 16th 2010
   • (edited Oct 17th 2010)
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   Author: domenico_fiorenza
   Format: MarkdownItexMore precisely, in section 3 of Getzler's paper you find Whitney's formula for a simplicial morphism $P_k:\Omega^\bullet(\Delta^k) \to N^\bullet(\Delta^k)$ in terms of integration of differential forms along simplices. This induces a simplicial morphism $Hom_{Ch}(V^\bullet,\Omega^\bullet(\Delta^k))\to Hom_{Ch}(V^\bullet,N^\bullet(\Delta^k))$ which is the one we are interested in. The fact that it is a weak equivalence follows by the fact $P_\bullet$ is a weak equivalence, and here one has an explicit way of seeing this by Dupont formulas. Note that here one sees the relevance of $\Omega^\bullet(\Delta^k)$ being acyclic (we have discussed this somewhere). Since in nLab we're preferring differential forms with sitting instants on the simplex to polyinomial differential forms, we should check Poincare' lemma holds for forms with sitting instants. For instance, realizing the $n$-simplex as $\{x^i\geq 0, \sum x^i=1 \}\subseteq \mathbb{R}^{n+1}$, and choosing as homotopy $(x,t)\mapsto (1/n+t(x^0-1/n),\dots, 1/n+t(x^n-1/n))$ one could try to check that, if $\omega=\omega_{i_0\dots i_k}(x)dx^{i^0}\wedge\cdots\wedge dx^{i_k}$ has sitting instants, then also $\sum_j(-1)^{j}(x^{i_j}-1/n)(\int_0^1t^{k-1}\omega_{i_0\dots i_k}(1/n+t(x^0-1/n),\dots, 1/n+t(x^n-1/n))dt)\wedge dx^{i^0}\wedge\cdots\wedge \hat{dx^{i_j}}\wedge\cdots\wedge dx^{i_k}$ has sitting instants. e.g, for $n=1$ this is $ (x-1/2)\int_0^1(\omega_0(1/2+t(x-1/2),1/2-t(x-1/2))-\omega_1(1/2+t(x-1/2),1/2-t(x-1/2))dt=\int_{1/2}^x(\omega_0(s,1-s)-\omega_1(s,1-s))ds $ one has $ \frac{d}{dx}\int_{1/2}^x(\omega_0(s,1-s)-\omega_1(s,1-s))ds=\omega_0(x,1-x)-\omega_1(x,1-x) $ which is zero in a neighborhood of $0$ and of $1$, so that $\int_{1/2}^x(\omega_0(s,1-s)-\omega_1(s,1-s))ds$ is locally constant there, and so has sitting instants.

   More precisely, in section 3 of Getzler’s paper you find Whitney’s formula for a simplicial morphism $P_k:\Omega^\bullet(\Delta^k) \to N^\bullet(\Delta^k)$ in terms of integration of differential forms along simplices. This induces a simplicial morphism $Hom_{Ch}(V^\bullet,\Omega^\bullet(\Delta^k))\to Hom_{Ch}(V^\bullet,N^\bullet(\Delta^k))$ which is the one we are interested in. The fact that it is a weak equivalence follows by the fact $P_\bullet$ is a weak equivalence, and here one has an explicit way of seeing this by Dupont formulas.

   Note that here one sees the relevance of $\Omega^\bullet(\Delta^k)$ being acyclic (we have discussed this somewhere). Since in nLab we’re preferring differential forms with sitting instants on the simplex to polyinomial differential forms, we should check Poincare’ lemma holds for forms with sitting instants. For instance, realizing the $n$-simplex as $\{x^i\geq 0, \sum x^i=1 \}\subseteq \mathbb{R}^{n+1}$, and choosing as homotopy $(x,t)\mapsto (1/n+t(x^0-1/n),\dots, 1/n+t(x^n-1/n))$ one could try to check that, if $\omega=\omega_{i_0\dots i_k}(x)dx^{i^0}\wedge\cdots\wedge dx^{i_k}$ has sitting instants, then also

   $\sum_j(-1)^{j}(x^{i_j}-1/n)(\int_0^1t^{k-1}\omega_{i_0\dots i_k}(1/n+t(x^0-1/n),\dots, 1/n+t(x^n-1/n))dt)\wedge dx^{i^0}\wedge\cdots\wedge \hat{dx^{i_j}}\wedge\cdots\wedge dx^{i_k}$

   has sitting instants. e.g, for $n=1$ this is

   $(x-1/2)\int_0^1(\omega_0(1/2+t(x-1/2),1/2-t(x-1/2))-\omega_1(1/2+t(x-1/2),1/2-t(x-1/2))dt=\int_{1/2}^x(\omega_0(s,1-s)-\omega_1(s,1-s))ds$

   one has

   $\frac{d}{dx}\int_{1/2}^x(\omega_0(s,1-s)-\omega_1(s,1-s))ds=\omega_0(x,1-x)-\omega_1(x,1-x)$

   which is zero in a neighborhood of $0$ and of $1$, so that $\int_{1/2}^x(\omega_0(s,1-s)-\omega_1(s,1-s))ds$ is locally constant there, and so has sitting instants.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeOct 16th 2010
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   Author: Urs
   Format: MarkdownItexOkay, good. My proof of the weak equivalence even when using forms with sitting instants over at [[Lie infinity-groupoid]] is more indirect: there I use that I have shown at [[circle n-bundle with connection]] that both complexes model the same homotopy pullback and hence must have the same homology groups. But, yes, let's think about a proof of the Poinaré-lemma for forms with sitting instants, sure. That would be good to have in any case. Let's create some page for discussing this. How about creating [[cochain on a simplex]]? (Notice we also have [[cochain on a simplicial set]] )

   Okay, good.

   My proof of the weak equivalence even when using forms with sitting instants over at Lie infinity-groupoid is more indirect: there I use that I have shown at circle n-bundle with connection that both complexes model the same homotopy pullback and hence must have the same homology groups.

   But, yes, let’s think about a proof of the Poinaré-lemma for forms with sitting instants, sure. That would be good to have in any case.

   Let’s create some page for discussing this. How about creating cochain on a simplex?

   (Notice we also have cochain on a simplicial set )

8. • CommentRowNumber8.
   • CommentAuthorzskoda
   • CommentTimeOct 16th 2010
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   Author: zskoda
   Format: MarkdownI got interested in the beginning of this thread and than a turn-off appeared...I can not help it but I am so allergic to sitting instants. I never understood it as natural. It makes me a huge psychological obstacle.

   I got interested in the beginning of this thread and than a turn-off appeared...I can not help it but I am so allergic to sitting instants. I never understood it as natural. It makes me a huge psychological obstacle.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeOct 16th 2010
   • (edited Oct 16th 2010)
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   Author: Urs
   Format: MarkdownItexSitting instants is part of the general toolset for working in the smooth context. Piecewise smooth is not smooth.

   Sitting instants is part of the general toolset for working in the smooth context. Piecewise smooth is not smooth.

10. • CommentRowNumber10.
    • CommentAuthorzskoda
    • CommentTimeOct 17th 2010
    • (edited Oct 17th 2010)
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    Author: zskoda
    Format: MarkdownThis is one of the approaches toward connections. The approach via Ehresmann distributions of hyperplanes for example does not need them. I am not convinced after so many hours of reading in vain articles with sitting instants that I can ever feel that technique and that it is necessary. You see for example, people compute homotopy groups. Do you know of any computation whatsoever where thin homotopy groups are computed ? To me it looks just that they appear at certain level of abstraction to make formal certain point of view.

    This is one of the approaches toward connections. The approach via Ehresmann distributions of hyperplanes for example does not need them. I am not convinced after so many hours of reading in vain articles with sitting instants that I can ever feel that technique and that it is necessary. You see for example, people compute homotopy groups. Do you know of any computation whatsoever where thin homotopy groups are computed ? To me it looks just that they appear at certain level of abstraction to make formal certain point of view.

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2010
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    Author: Urs
    Format: MarkdownItex> Do you know of any computation whatsoever where thin homotopy groups are computed ? I know that Kapranov was wanting to understand them at some point. But I am not sure how you got from sitting instants to thin homotopy.

    Do you know of any computation whatsoever where thin homotopy groups are computed ?

    I know that Kapranov was wanting to understand them at some point. But I am not sure how you got from sitting instants to thin homotopy.

12. • CommentRowNumber12.
    • CommentAuthorzskoda
    • CommentTimeOct 17th 2010
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    Author: zskoda
    Format: MarkdownThe definition of thin homotopy which I was tortured with involves sitting instants. That is where i started disliking them. In the business of general manifolds with boundary it is OK, but by complaint is the sitting instances for defining the thin homotopy. That is the point where my intuition broke more than once.

    The definition of thin homotopy which I was tortured with involves sitting instants. That is where i started disliking them. In the business of general manifolds with boundary it is OK, but by complaint is the sitting instances for defining the thin homotopy. That is the point where my intuition broke more than once.

13. • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2010
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    Author: Urs
    Format: MarkdownItexHere is the definition: [[thin homotopy]]. No sitting instants there. I am guessing you are thinking of the definition of [[path groupoid]], where both thin homotopy and sitting instants are invoked. But anyway, thin homotopy is not what this thread here is about.

    Here is the definition: thin homotopy. No sitting instants there.

    I am guessing you are thinking of the definition of path groupoid, where both thin homotopy and sitting instants are invoked.

    But anyway, thin homotopy is not what this thread here is about.

14. • CommentRowNumber14.
    • CommentAuthorzskoda
    • CommentTimeOct 18th 2010
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    Author: zskoda
    Format: MarkdownSorry for disturbing the thread then.

    Sorry for disturbing the thread then.

15. • CommentRowNumber15.
    • CommentAuthorTim_Porter
    • CommentTimeOct 18th 2010
    • (edited Oct 18th 2010)
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    Author: Tim_Porter
    Format: MarkdownItex@Zoran The obvious thing to do is to try to put your block into words in a new thread! :-) I understand thin homotopies but do not really like sitting instants, I understand them but I don't like them and there should be an easier way.

    @Zoran The obvious thing to do is to try to put your block into words in a new thread! :-) I understand thin homotopies but do not really like sitting instants, I understand them but I don’t like them and there should be an easier way.