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Atrium > Mathematics, Physics & Philosophy: Classifying spaces and cohomology

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- CommentRowNumber1.
- CommentAuthordomenico_fiorenza
- CommentTimeDec 24th 2009
- (edited Dec 28th 2009)
- PermaLink
Author: domenico_fiorenza
Format: HtmlIn the [[Cech cohomology]] entry one finds that <latex>H^1(X;G)</latex> is naturally identified with (the colimit over all Cech covers of) <latex>SSh(C(U),\mathbf{B}G)/homotopy</latex>, where <latex>C(U)</latex> is a Cech cover of <latex>X</latex>. On the other hand it is well known that, for a nice topological space <latex>X</latex>, <latex>H^1(X;G)</latex> is identified with <latex>Top(X,|N(\mathbf{B}G)|)/homotopy</latex>. The topological realization of the nerve of <latex>\mathbf{B}G</latex> is, if I'm not wrong here, the homotopy colimit of the groupoid <latex>\mathbf{B}G</latex> as a topological groupoid, whereas <latex>X</latex> is the colimit of <latex>C(U)</latex>. With this in mind one ends up with an isomorphism <latex>Ssh(C(U),\mathbf{B}G)/homotopy \leftrightarrow Top(colim C(U), hocolim \mathbf{B}G)/homotopy</latex> which suggests an equivalence <latex>Ssh(C(U),\mathbf{B}G) \leftrightarrow Top(colim(C(U), hocolim \mathbf{B}G)</latex> This should follow directly from some abstract argument, and reading all of the above backwords this would provide a construction of the classifying space. But I'm unable to see which abstract argument one should invoke here :(
In the Cech cohomology entry one finds that is naturally identified with (the colimit over all Cech covers of) , where is a Cech cover of . On the other hand it is well known that, for a nice topological space , is identified with . The topological realization of the nerve of is, if I'm not wrong here, the homotopy colimit of the groupoid as a topological groupoid, whereas is the colimit of .

With this in mind one ends up with an isomorphism

which suggests an equivalence

This should follow directly from some abstract argument, and reading all of the above backwords this would provide a construction of the classifying space. But I'm unable to see which abstract argument one should invoke here :(
- CommentRowNumber2.
- CommentAuthordomenico_fiorenza
- CommentTimeDec 26th 2009
- (edited Dec 28th 2009)
- PermaLink
Author: domenico_fiorenza
Format: HtmlIn the [[principal bundle]] entry one finds "Recall that for every group there is the the one-object groupoid <latex>\mathbf{B}G</latex>. Under the Yoneda embedding this represents a prestack. Write <latex>\overline{\mathbf{B}G}</latex> for the corresponding stack obtained by stackification. This is our <latex>GBund(-)</latex>". I totally agree that <latex>GBund(-)</latex> is obtained by stackification of the "trivial $G$-bundle" functor (which is a prestack), but I'm unable to see this as an issue of Yoneda embedding. This basically because I am only able to see the presheaf represented by <latex>\mathbf{B}G</latex> as a presheaf on the same (infinity-)category <latex>\mathbf{B}G</latex> is an object of. This seems to agree with what I've written a few lines later, where seems that both <latex>X</latex> and <latex>\mathbf{B}G</latex> should be objects in the same <latex>(\infty,1)</latex>-topos <latex>H</latex>. Yet, <latex>H(X,\mathbf{B}G)</latex> now seems to be the simplicial presheaf represented by <latex>\mathbf{B}G</latex>, and sayng that this is already a simplical sheaf then seems to be a statement on <latex>\mathbf{B}G</latex>. In any case, making both <latex>X</latex> and <latex>\mathbf{B}G</latex> objects of the same topos seems to require an explicitation of some missing (to me) ingredient. For instance, working with topological spaces, one could trade <latex>X</latex> for the nerve of a Cech cover <latex>C(U)</latex>, in order to have two groupoids and consider <latex>Hom(C(U),\mathbf{B}G)</latex>, or see <latex>\mathbf{B}G</latex> as the topological realization of its nerve, and look at <latex>Hom(X,|N(\mathbf{B}G)|)</latex>. Both pictures seems to be valid -and actually to be equivalent-, but I'm unable to see them as <latex>H(X,\mathbf{B}G)</latex> without saying explicitly how <latex>X</latex> and <latex>\mathbf{B}G</latex> are seen in the same topos. I apologize for such a confused post, with something clearer in my mind I could have tried editing the entry in the nLab instead.. :)
In the principal bundle entry one finds "Recall that for every group there is the the one-object groupoid . Under the Yoneda embedding this represents a prestack. Write for the corresponding stack obtained by stackification. This is our ". I totally agree that is obtained by stackification of the "trivial $G$-bundle" functor (which is a prestack), but I'm unable to see this as an issue of Yoneda embedding. This basically because I am only able to see the presheaf represented by as a presheaf on the same (infinity-)category is an object of. This seems to agree with what I've written a few lines later, where seems that both and should be objects in the same -topos . Yet, now seems to be the simplicial presheaf represented by , and sayng that this is already a simplical sheaf then seems to be a statement on . In any case, making both and objects of the same topos seems to require an explicitation of some missing (to me) ingredient. For instance, working with topological spaces, one could trade for the nerve of a Cech cover , in order to have two groupoids and consider , or see as the topological realization of its nerve, and look at . Both pictures seems to be valid -and actually to be equivalent-, but I'm unable to see them as without saying explicitly how and are seen in the same topos.

I apologize for such a confused post, with something clearer in my mind I could have tried editing the entry in the nLab instead.. :)
- CommentRowNumber3.
- CommentAuthordomenico_fiorenza
- CommentTimeDec 26th 2009
- PermaLink
Author: domenico_fiorenza
Format: TextI've now (finally..) read the [[cohomology]] entry, and everything became clear.. :)
I've now (finally..) read the cohomology entry, and everything became clear.. :)
- CommentRowNumber4.
- CommentAuthordomenico_fiorenza
- CommentTimeDec 27th 2009
- (edited Dec 28th 2009)
- PermaLink
Author: domenico_fiorenza
Format: HtmlNow that I've read [[cohomology]] the nonsense of the above posts has finally come into focus, so I'll try to restate it from the beginning, in a particular example to try to make it the clearer as possible. Let <latex>X</latex> be a (as nice as needed) topological space. I want to compute <latex>H^n(X;\mathbb{Z})</latex>. The nPOV on cohomology tells me what I have to do is to look at both <latex>X</latex> and <latex>\mathbb{Z}</latex> as objects in a suitable <latex>(\infty,1)</latex>-topos <latex>H</latex>, and then consider the <latex>\infty</latex>-groupoid <latex>H(X;\mathbb{Z})</latex>. To make this concrete I have two possibilities here. First, I can choose the prototypical <latex>(\infty,1)</latex>-topos <latex>Top</latex>. Now <latex>X</latex> is naturally an object in <latex>Top</latex>, and one can look at "<latex>\mathbb{Z}</latex> in degree <latex>n</latex>" as the strict <latex>\infty</latex>-groupoid <latex>B^n\mathbb{Z}</latex>, and so as the topological space <latex>K(\mathbb{Z};n)</latex>. That is I can recover <latex>H^n(X;\mathbb{Z})</latex> from <latex>Top(X;K(\mathbb{Z};n))</latex>. But I also have a second possibiliy. namely, I can look at <latex>\mathbb{Z}</latex> as to an abelian sheaf, and choose the topos of <latex>\infty</latex>-stacks on some site (topological spaces with open immersions? the open subsets of <latex>X</latex>?). This time <latex>\mathbb{Z}</latex> and is shifted version <latex>B^n\mathbb{Z}</latex> are naturally simplicial sheaves, and one can look at <latex>X</latex> as a simplicial sheaf via Yoneda embedding, and what one ends up with is that <latex>H^n(X;\mathbb{Z})</latex> is recovered from <latex>SSh(X;B^n\mathbb{Z})</latex> or, if one wants to stress the Yoneda embedding, from <latex>SSh(Y(X);B^n\mathbb{Z})</latex>. The above tells in particular that the two <latex>\infty</latex>-groupoids <latex>Top(X;K(\mathbb{Z};n))</latex> and <latex>SSh(Y(X);B^n\mathbb{Z})</latex> have the same <latex>\pi_0</latex>. So the natural question is: is there a homotopy equivalence <latex>Top(X;K(\mathbb{Z};n)) \leftrightarrow SSh(Y(X);B^n\mathbb{Z})</latex> ?
Now that I've read cohomology the nonsense of the above posts has finally come into focus, so I'll try to restate it from the beginning, in a particular example to try to make it the clearer as possible.

Let be a (as nice as needed) topological space. I want to compute . The nPOV on cohomology tells me what I have to do is to look at both and as objects in a suitable -topos , and then consider the -groupoid .

To make this concrete I have two possibilities here. First, I can choose the prototypical -topos . Now is naturally an object in , and one can look at " in degree " as the strict -groupoid , and so as the topological space . That is I can recover from .

But I also have a second possibiliy. namely, I can look at as to an abelian sheaf, and choose the topos of -stacks on some site (topological spaces with open immersions? the open subsets of ?). This time and is shifted version are naturally simplicial sheaves, and one can look at as a simplicial sheaf via Yoneda embedding, and what one ends up with is that is recovered from or, if one wants to stress the Yoneda embedding, from .

The above tells in particular that the two -groupoids and have the same . So the natural question is: is there a homotopy equivalence ?
- CommentRowNumber5.
- CommentAuthorMike Shulman
- CommentTimeDec 27th 2009
- PermaLink
Author: Mike Shulman
Format: MarkdownWell, <latex>\pi_1</latex> of each of them can be identified with <latex>Top(X,K(Z,n+1))</latex> and <latex>SSh(X,B^{n+1}Z)</latex>, respectively, and thereby with <latex>H^{n+1}(X;Z)</latex>, and similarly for all higher homotopy groups. So all we need is a map inducing those isomorphisms. Perhaps there is an adjunction.

Well, $\pi_1$ of each of them can be identified with $Top(X,K(Z,n+1))$ and $SSh(X,B^{n+1}Z)$ , respectively, and thereby with $H^{n+1}(X;Z)$ , and similarly for all higher homotopy groups. So all we need is a map inducing those isomorphisms. Perhaps there is an adjunction.
- CommentRowNumber6.
- CommentAuthordomenico_fiorenza
- CommentTimeDec 28th 2009
- (edited Dec 28th 2009)
- PermaLink
Author: domenico_fiorenza
Format: HtmlI'm a bit confused here. It seems to me that the chain of isomorphisms <latex>\pi_1Top(X,Y)=\pi_0\Omega Top(X,Y)=\pi_0Top(\Sigma X, Y)=\pi_0Top(X,\Omega Y)</latex> (<latex>Top</latex> stands for pointed topologicla spaces here) would rather imply that <latex>\pi_1 Top(X, K(\mathbb{Z},n))</latex> is isomorphic to <latex>H^{n-1}(X;\mathbb{Z})</latex>. The reasoning above however is not affected by a +1 in place of -1, so I agree it says that not only <latex>\pi_0</latex> but all the <latex>\pi_i</latex>'s of the two infinity-groupoids involved are isomorphic. As far as concerns finding a map inducing these isomorphisms, a strong enought version of Yoneda lemma (I've seen there's some discussion on this going on at [[Yoneda lemma for (infinity,1)-categories]] would imply that <latex>Top(X;K(\mathbb{Z},n))</latex> is equivalent to <latex>SSh(X,K(\mathbb{Z},n))</latex> (I mean, consider both <latex>X</latex> and <latex>K(\mathbb{Z},n)</latex> as presheaves on <latex>Top</latex>). So (Yoneda again) one is reduced to looking for an equivalence of simplicial preheaves between <latex>B^n\mathbb{Z}</latex> and <latex>K(\mathbb{Z},n)</latex>. Since the latter is the presheaf associated to the topological realization of the first, all this hints to a topos version of the [[geometric realization]]/[[fundamental infinity-groupoid]] adjunction.
I'm a bit confused here. It seems to me that the chain of isomorphisms ( stands for pointed topologicla spaces here) would rather imply that is isomorphic to . The reasoning above however is not affected by a +1 in place of -1, so I agree it says that not only but all the 's of the two infinity-groupoids involved are isomorphic.

As far as concerns finding a map inducing these isomorphisms, a strong enought version of Yoneda lemma (I've seen there's some discussion on this going on at Yoneda lemma for (infinity,1)-categories would imply that is equivalent to (I mean, consider both and as presheaves on ). So (Yoneda again) one is reduced to looking for an equivalence of simplicial preheaves between and . Since the latter is the presheaf associated to the topological realization of the first, all this hints to a topos version of the geometric realization/fundamental infinity-groupoid adjunction.
- CommentRowNumber7.
- CommentAuthorUrs
- CommentTimeDec 28th 2009
- PermaLink
Author: Urs
Format: MarkdownI have come out of some Christmas vacation and have now worked on all my accumulated email and all the accumulated discussion here. I should reply to this here, too, but now I am running out of time. Domenico, I understand that in the course of the above messages some of your questions found their answers already. Could you maybe summarize in a sentence or two what the remaining questions are? (And include your LaTeX formulas in double dollar signs (inline), then they display pretty-printed here and are easier to read).

I have come out of some Christmas vacation and have now worked on all my accumulated email and all the accumulated discussion here. I should reply to this here, too, but now I am running out of time.

Domenico, I understand that in the course of the above messages some of your questions found their answers already. Could you maybe summarize in a sentence or two what the remaining questions are?

(And include your LaTeX formulas in double dollar signs (inline), then they display pretty-printed here and are easier to read).
- CommentRowNumber8.
- CommentAuthordomenico_fiorenza
- CommentTimeDec 28th 2009
- (edited Dec 28th 2009)
- PermaLink
Author: domenico_fiorenza
Format: HtmlAll of the above can probably be summarized in some suitably corrected version of following question: let <latex>A</latex> be an abelian group and denote by the same symbol the corresponding constant simplicial presheaf over topological spaces (with open covers as covers); let <latex>|N\mathbf{B}A|</latex> be the topological realization of the nerve of <latex>\mathbf{B}A</latex>. Look at <latex>|N\mathbf{B}A|</latex> as a presheaf on <latex>Top</latex> via Yoneda embedding. Are the two presheaves <latex>A</latex> and <latex>|N\mathbf{B}A|</latex> equivalent? (I edited the above posts with double dollars)
All of the above can probably be summarized in some suitably corrected version of following question: let be an abelian group and denote by the same symbol the corresponding constant simplicial presheaf over topological spaces (with open covers as covers); let be the topological realization of the nerve of . Look at as a presheaf on via Yoneda embedding. Are the two presheaves and equivalent?

(I edited the above posts with double dollars)
- CommentRowNumber9.
- CommentAuthorMike Shulman
- CommentTimeDec 28th 2009
- PermaLink
Author: Mike Shulman
Format: MarkdownOh, yeah, I added instead of subtracted, sorry. In too much of a hurry.

Oh, yeah, I added instead of subtracted, sorry. In too much of a hurry.
- CommentRowNumber10.
- CommentAuthorMike Shulman
- CommentTimeDec 28th 2009
- PermaLink
Author: Mike Shulman
Format: MarkdownDomenico, I think maybe you want to compare <latex>|N\mathbf{B}A|</latex> not to the constant simplicial presheaf <latex>A</latex> but to its delooping relative to the homotopy theory of simplicial (pre)sheaves? I feel like something of this sort must be somewhere in HTT, since the introduction talks about identifying these two kinds of cohomology.

Domenico, I think maybe you want to compare $|N\mathbf{B}A|$ not to the constant simplicial presheaf $A$ but to its delooping relative to the homotopy theory of simplicial (pre)sheaves?

I feel like something of this sort must be somewhere in HTT, since the introduction talks about identifying these two kinds of cohomology.
- CommentRowNumber11.
- CommentAuthordomenico_fiorenza
- CommentTimeDec 29th 2009
- (edited Dec 29th 2009)
- PermaLink
Author: domenico_fiorenza
Format: HtmlActually I do not exactly know what I want to compare, that's my main problem.. I'll try to look into HTT. Basically these last days speculations on all came out from the idea of cohomology as it is stated at [[cohomology]] and, as a beatiful slogan, at [[nPOV]]. My point is that if we really want the nPOV on cohomology to be *the* point of view on cohomology (and I guess we want), we do not only need to identify each single cohomology theory with the <latex>\pi_0</latex> of a space of infinity-morphisms in some (infinity,1)-topos, but also (and I find this to be the crucial point) exhibit all natural transformations of cohomology theories as morphisms of topoi. I have two basic examples in mind. One is the subject of this forum entry: there are several possible natural topoi computations of <latex>H^n(X;\mathbb{Z})</latex>; how are they related? The other, I'm trying to write something about in my area, but I get more confuded each time I think to it: what are K-theory and the Chern character in the topos description of cohomology?
Actually I do not exactly know what I want to compare, that's my main problem.. I'll try to look into HTT.

Basically these last days speculations on all came out from the idea of cohomology as it is stated at cohomology and, as a beatiful slogan, at nPOV. My point is that if we really want the nPOV on cohomology to be *the* point of view on cohomology (and I guess we want), we do not only need to identify each single cohomology theory with the of a space of infinity-morphisms in some (infinity,1)-topos, but also (and I find this to be the crucial point) exhibit all natural transformations of cohomology theories as morphisms of topoi.

I have two basic examples in mind. One is the subject of this forum entry: there are several possible natural topoi computations of ; how are they related? The other, I'm trying to write something about in my area, but I get more confuded each time I think to it: what are K-theory and the Chern character in the topos description of cohomology?
- CommentRowNumber12.
- CommentAuthorUrs
- CommentTimeDec 29th 2009
- (edited Dec 29th 2009)
- PermaLink
Author: Urs
Format: Markdown<blockquote> Basically these last days speculations on all came out from the idea of [[cohomology]] as it is stated at cohomology and, as a beatiful slogan, at [[nPOV]]. </blockquote> Thanks for that feedback. This made me include this slogan now in a standout box at [[cohomology]]. <blockquote> also (and I find this to be the crucial point) exhibit all natural transformations of cohomology theories as morphisms of topoi. </blockquote> Thanks, now I see what you are getting at. This deserves to be discussed, yes. As a first quick remark: what is ordinarily thought of as a morphism of cohomology theories happens inside a single (oo,1)-category, say between two functors represneted by two different spectra. Another thing I want to eventually discuss, which is relevant in this context, is this: for defining an (oo,1)-categorical hom-space of course all we need is an (oo,1)-category. So what is the point of requiring this to be an (oo,1)-topos? I think the answer is: if it is a topos, then we have the expected relation between cocycles and the objects they classify: every cocycles with coefficints in a pointed object classifies, by definition, its [[homotopy fiber]]. If the ambient (oo,1)-category is an (oo,1)-topos, then we have Giraud's axioms about effective groupoid objects, and using this we can conclude that this homotopy fiber behaves like a principal bundle. Parts of this is discussed at [[principal infinity-bundle]]. But I think it will make sense to also regard hom-spaces in non-toposes as cohomologies, of sorts, in particular in stable (oo,1)-categories. (Though the point of view keeps being expressed that every stable (oo,1)-category is somehow as good as an (oo,1)-topos.) <blockquote> The other, I'm trying to write something about in my area, but I get more confuded each time I think to it: what are K-theory and the Chern character in the topos description of cohomology? </blockquote> That's a very intersting point. The best abstract-nonsense description of what really happens with K-theory that I have seen is the one staated in the definition section at [[K-theory]]. This is taken from Lurie's "Stable (oo,1)-categories", equippedd with some additional remarks. Concerning Chern-character: this is something I am very much thinking about currently in the context of my work on [[schreiber:theory of differential nonabelian cohomology]]. I thinkl there is a good abstract-nonsense formulation using the <a href="http://ncatlab.org/schreiber/published/deRham+theorem+for+%E2%88%9E-Lie+groupoids">deRham theorem for oo-Lie groupoids</a>. I am currently busy with writing up more details on this. As soon as I am done replying to all sorts of forum discussions, I might actually have time to type this, then I get back to you!
This comment is invalid XHTML+MathML+SVG; displaying source. <div> <blockquote> Basically these last days speculations on all came out from the idea of <a href="https://ncatlab.org/nlab/show/cohomology">cohomology</a> as it is stated at cohomology and, as a beatiful slogan, at <a href="https://ncatlab.org/nlab/show/nPOV">nPOV</a>. </blockquote> <p>Thanks for that feedback. This made me include this slogan now in a standout box at <a href="https://ncatlab.org/nlab/show/cohomology">cohomology</a>.</p> <blockquote> also (and I find this to be the crucial point) exhibit all natural transformations of cohomology theories as morphisms of topoi. </blockquote> <p>Thanks, now I see what you are getting at.</p> <p>This deserves to be discussed, yes. As a first quick remark: what is ordinarily thought of as a morphism of cohomology theories happens inside a single (oo,1)-category, say between two functors represneted by two different spectra.</p> <p>Another thing I want to eventually discuss, which is relevant in this context, is this:</p> <p>for defining an (oo,1)-categorical hom-space of course all we need is an (oo,1)-category. So what is the point of requiring this to be an (oo,1)-topos?</p> <p>I think the answer is: if it is a topos, then we have the expected relation between cocycles and the objects they classify: every cocycles with coefficints in a pointed object classifies, by definition, its <a href="https://ncatlab.org/nlab/show/homotopy+fiber">homotopy fiber</a>. If the ambient (oo,1)-category is an (oo,1)-topos, then we have Giraud's axioms about effective groupoid objects, and using this we can conclude that this homotopy fiber behaves like a principal bundle. Parts of this is discussed at <a href="https://ncatlab.org/nlab/show/principal+infinity-bundle">principal infinity-bundle</a>.</p> <p>But I think it will make sense to also regard hom-spaces in non-toposes as cohomologies, of sorts, in particular in stable (oo,1)-categories. (Though the point of view keeps being expressed that every stable (oo,1)-category is somehow as good as an (oo,1)-topos.)</p> <blockquote> The other, I'm trying to write something about in my area, but I get more confuded each time I think to it: what are K-theory and the Chern character in the topos description of cohomology? </blockquote> <p>That's a very intersting point.</p> <p>The best abstract-nonsense description of what really happens with K-theory that I have seen is the one staated in the definition section at <a href="https://ncatlab.org/nlab/show/K-theory">K-theory</a>. This is taken from Lurie's "Stable (oo,1)-categories", equippedd with some additional remarks.</p> <p>Concerning Chern-character: this is something I am very much thinking about currently in the context of my work on <a href="https://ncatlab.org/schreiber/show/theory+of+differential+nonabelian+cohomology">theory of differential nonabelian cohomology (schreiber)</a>. I thinkl there is a good abstract-nonsense formulation using the <a href="http://ncatlab.org/schreiber/published/deRham+theorem+for+%E2%88%9E-Lie+groupoids">deRham theorem for oo-Lie groupoids</a>. I am currently busy with writing up more details on this. As soon as I am done replying to all sorts of forum discussions, I might actually have time to type this, then I get back to you!</p> </div>
- CommentRowNumber13.
- CommentAuthordomenico_fiorenza
- CommentTimeDec 29th 2009
- PermaLink
Author: domenico_fiorenza
Format: HtmlI'll look at your work on the Chern character, then :) by the way, on second thought, the [[nPOV]] point of view on cohomology, 'thousands of definitions of notions of cohomology and its variants. From the nPOV, just a single concept: an infinity-categorical hom-space' immediately leads to the following rephrase: 'cohomology is the theory of representable functors in infinity-topoi', and then Yoneda allows a further rephrasement: 'cohomology is infinity-topos theory'. but I prefer the slogan the way you formulated it.
I'll look at your work on the Chern character, then :)

by the way, on second thought, the nPOV point of view on cohomology, 'thousands of definitions of notions of cohomology and its variants. From the nPOV, just a single concept: an infinity-categorical hom-space' immediately leads to the following rephrase: 'cohomology is the theory of representable functors in infinity-topoi', and then Yoneda allows a further rephrasement: 'cohomology is infinity-topos theory'. but I prefer the slogan the way you formulated it.
- CommentRowNumber14.
- CommentAuthorUrs
- CommentTimeDec 29th 2009
- (edited Dec 29th 2009)
- PermaLink
Author: Urs
Format: Markdown<blockquote> I'll look at your work on the Chern character, then :) </blockquote> Okay. Maybe you can help me with one gap in my development that I still have: currently I am taking a detour through some oo-Lie theory for the full story: given an object <latex> A </latex> in a smooth (oo,1)-topos, I pass to its oo-Lie algebroid <latex> \mathfrak{a} </latex>. For that I know how to nicely produce another oo-Lie algebroid <latex> \mathfrak{a}_{ab} </latex>, which is something like the free abelianization of <latex> \mathfrak{a} </latex>. For instance for the case that <latex> A = \mathbf{B}G </latex> is a delooped Lie group, <latex> \mathfrak{a}_{ab} </latex> is the thing such that its function algebra is the algebra of invariant polynomials on the Lie algebra of <latex> G </latex>. Then I claim that there is a canonical morphism <latex> H(X,A) \to H_{dR}(X, \mathfrak{a}_{ab})</latex> from cohomology with coefficients in <latex> A </latex> to deRham cohomology with coefficients in <latex> \mathfrak{a}_{ab} </latex> (which is a collection of differential forms on X, one for each invariant polynomial) and that further postcomposed with the oo-deRham theorem that sends the deRham cohomology to real cohomology, this map is the general Chern character map for cohomology with coefficients in <latex> A </latex>. What I am lacking currently is a good way to make this statement without the detour throuh oo-Lie algebroids and deRham cohomology. Somehow I should be able to associate with <latex> A </latex> directly an oo-Lie groupoid that is something like the direct product of lots of Eilenber-MacLane spaces, one for each homotopy group of A, and then shifted up by one. I am not sure yet if I have found the best way to say this.
This comment is invalid XHTML+MathML+SVG; displaying source. <div> <blockquote> I'll look at your work on the Chern character, then :) </blockquote> <p>Okay. Maybe you can help me with one gap in my development that I still have:</p> <p>currently I am taking a detour through some oo-Lie theory for the full story:</p> <p>given an object <img src="/extensions/vLaTeX/cache/latex_e9fcd5f411ac341411f03f46e55c0ad7.png" title=" A " style="vertical-align: -20%;" class="tex" alt=" A "/> in a smooth (oo,1)-topos, I pass to its oo-Lie algebroid <img src="/extensions/vLaTeX/cache/latex_1b0319bca12e05b3240248dc1b06065b.png" title=" \mathfrak{a} " style="vertical-align: -20%;" class="tex" alt=" \mathfrak{a} "/>. For that I know how to nicely produce another oo-Lie algebroid <img src="/extensions/vLaTeX/cache/latex_335744e9417addee97dab8a30751e23c.png" title=" \mathfrak{a}_{ab} " style="vertical-align: -20%;" class="tex" alt=" \mathfrak{a}_{ab} "/>, which is something like the free abelianization of <img src="/extensions/vLaTeX/cache/latex_1b0319bca12e05b3240248dc1b06065b.png" title=" \mathfrak{a} " style="vertical-align: -20%;" class="tex" alt=" \mathfrak{a} "/>. For instance for the case that <img src="/extensions/vLaTeX/cache/latex_7907e1320a1ebed1f468d87d10815ead.png" title=" A = \mathbf{B}G " style="vertical-align: -20%;" class="tex" alt=" A = \mathbf{B}G "/> is a delooped Lie group, <img src="/extensions/vLaTeX/cache/latex_335744e9417addee97dab8a30751e23c.png" title=" \mathfrak{a}_{ab} " style="vertical-align: -20%;" class="tex" alt=" \mathfrak{a}_{ab} "/> is the thing such that its function algebra is the algebra of invariant polynomials on the Lie algebra of <img src="/extensions/vLaTeX/cache/latex_e9eec6c9fb23c40613a213b34d8c085f.png" title=" G " style="vertical-align: -20%;" class="tex" alt=" G "/>.</p> <p>Then I claim that there is a canonical morphism</p> <img src="/extensions/vLaTeX/cache/latex_2e4f0dd2228ab139e239aeae48d5ac9e.png" title=" H(X,A) \to H_{dR}(X, \mathfrak{a}_{ab})" style="vertical-align: -20%;" class="tex" alt=" H(X,A) \to H_{dR}(X, \mathfrak{a}_{ab})"/> <p>from cohomology with coefficients in <img src="/extensions/vLaTeX/cache/latex_e9fcd5f411ac341411f03f46e55c0ad7.png" title=" A " style="vertical-align: -20%;" class="tex" alt=" A "/> to deRham cohomology with coefficients in <img src="/extensions/vLaTeX/cache/latex_335744e9417addee97dab8a30751e23c.png" title=" \mathfrak{a}_{ab} " style="vertical-align: -20%;" class="tex" alt=" \mathfrak{a}_{ab} "/> (which is a collection of differential forms on X, one for each invariant polynomial) and that further postcomposed with the oo-deRham theorem that sends the deRham cohomology to real cohomology, this map is the general Chern character map for cohomology with coefficients in <img src="/extensions/vLaTeX/cache/latex_e9fcd5f411ac341411f03f46e55c0ad7.png" title=" A " style="vertical-align: -20%;" class="tex" alt=" A "/>.</p> <p>What I am lacking currently is a good way to make this statement without the detour throuh oo-Lie algebroids and deRham cohomology. Somehow I should be able to associate with <img src="/extensions/vLaTeX/cache/latex_e9fcd5f411ac341411f03f46e55c0ad7.png" title=" A " style="vertical-align: -20%;" class="tex" alt=" A "/> directly an oo-Lie groupoid that is something like the direct product of lots of Eilenber-MacLane spaces, one for each homotopy group of A, and then shifted up by one. I am not sure yet if I have found the best way to say this.</p> </div>
- CommentRowNumber15.
- CommentAuthordomenico_fiorenza
- CommentTimeDec 29th 2009
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Author: domenico_fiorenza
Format: HtmlI see your point, but I'm not fully confident a product of Eleinberg-Mac Lane spaces will work. Rather, I'd think to the Postnikov system for <latex>A</latex>. I'll think to this. The general picture, however seems quite clear: by Yoneda, a canonical morphism <latex>H(X,A)\to H(X,{\mathbb R})</latex> must be induced, by composition, by an object in <latex>H(A,{\mathbb R})</latex>. so what we are interested in is the cohomology of <latex>A</latex>.
I see your point, but I'm not fully confident a product of Eleinberg-Mac Lane spaces will work. Rather, I'd think to the Postnikov system for . I'll think to this.

The general picture, however seems quite clear: by Yoneda, a canonical morphism

must be induced, by composition, by an object in . so what we are interested in is the cohomology of .
- CommentRowNumber16.
- CommentAuthorUrs
- CommentTimeDec 29th 2009
- (edited Dec 29th 2009)
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Author: Urs
Format: MarkdownRight, exactly, we want an object consisting of lots of copies of <latex> \mathbb{R} </latex> in various degrees, something like <latex> \prod_i \mathbf{B}^{n_i} \mathbb{R} </latex> such that the entire real cohomology content of <latex> A </latex> is encoded in a single morphism <latex> A \to \prod_i \mathbf{B}^{n_i} \mathbb{R} </latex> and then the Chern character map should be postcomposition with this morphism. I'd think. Possibly this is looking for the rationalization of <latex> A </latex>...

Right, exactly, we want an object consisting of lots of copies of $\mathbb{R}$ in various degrees, something like $\prod_i \mathbf{B}^{n_i} \mathbb{R}$ such that the entire real cohomology content of $A$ is encoded in a single morphism $A \to \prod_i \mathbf{B}^{n_i} \mathbb{R}$ and then the Chern character map should be postcomposition with this morphism. I'd think.

Possibly this is looking for the rationalization of $A$ ...
- CommentRowNumber17.
- CommentAuthordomenico_fiorenza
- CommentTimeDec 29th 2009
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Author: domenico_fiorenza
Format: HtmlI wouldn't see things this way. Namely, I wouldn't look for a single morphism from <latex>A</latex> to (some version of) <latex>\mathbb{R}</latex> encoding the whole cohomology <latex>H(A,\mathbb{R})</latex>. Rather, I would look at special (in some sense) objects of <latex>H(A,\mathbb{R})</latex>. For instance to go from K-theory of a (nice) topological space <latex>X</latex> to its singular cohomology, one can use any polynomial (or even formal series) in the Chern classes, i.e. any element in the singular cohomology of <latex>\mathbf{B}U</latex> to define a pullback morphism <latex>K(X)\to H(X;\mathbb{Z})</latex>. But the Chern character is special since <latex>Ch:K(X)\to H(X;\mathbb{Z})</latex> is a ring homomorphism. So I'd start with a (commutative?) ring object <latex>A</latex> in <latex>H</latex> and look for objects in <latex>H(A,\mathbb{R})</latex> inducing (<latex>E_\infty</latex>?) ring homomorphisms by composition. I see in your consruction one begins with an arbitrary object <latex>A</latex>, but maybe there's a groupoidification (and so maybe a ringification) hidden there in going from <latex>A</latex> to <latex>\mathfrak{a}</latex> (and then maybe integrating <latex>\mathfrak{a}</latex> and taking some sort of ring of functions on the <latex>\infty</latex>-groupoid one obtains.
I wouldn't see things this way. Namely, I wouldn't look for a single morphism from to (some version of) encoding the whole cohomology . Rather, I would look at special (in some sense) objects of . For instance to go from K-theory of a (nice) topological space to its singular cohomology, one can use any polynomial (or even formal series) in the Chern classes, i.e. any element in the singular cohomology of to define a pullback morphism . But the Chern character is special since is a ring homomorphism. So I'd start with a (commutative?) ring object in and look for objects in inducing (?) ring homomorphisms by composition. I see in your consruction one begins with an arbitrary object , but maybe there's a groupoidification (and so maybe a ringification) hidden there in going from to (and then maybe integrating and taking some sort of ring of functions on the -groupoid one obtains.
- CommentRowNumber18.
- CommentAuthorUrs
- CommentTimeDec 29th 2009
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Author: Urs
Format: MarkdownI see your point. I am hoping for a description that does not need to assume that <latex> A </latex> is a ring object, but maybe that'll be necessary. I'll have to go offline very soon now. But I'll think about it.

I see your point. I am hoping for a description that does not need to assume that $A$ is a ring object, but maybe that'll be necessary.

I'll have to go offline very soon now. But I'll think about it.
- CommentRowNumber19.
- CommentAuthorTobyBartels
- CommentTimeDec 30th 2009
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Author: TobyBartels
Format: MarkdownReading your new slogan at [[cohomology]], Urs, I'm reminded of Taylor & Wheeler, _Spacetime Physics_.

Reading your new slogan at cohomology, Urs, I'm reminded of Taylor & Wheeler, Spacetime Physics.
- CommentRowNumber20.
- CommentAuthordomenico_fiorenza
- CommentTimeDec 30th 2009
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Author: domenico_fiorenza
Format: HtmlI finally managed to write something at [[domenicofiorenza:Chern character|Chern character]] in my nLab area.
I finally managed to write something at Chern character in my nLab area.
- CommentRowNumber21.
- CommentAuthorUrs
- CommentTimeDec 30th 2009
- (edited Jan 4th 2010)
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Author: Urs
Format: MarkdownThanks. I'll try to reply in a little while, after I have something else out of the way.

Thanks. I'll try to reply in a little while, after I have something else out of the way.
- CommentRowNumber22.
- CommentAuthorUrs
- CommentTimeJan 4th 2010
- (edited Jan 4th 2010)
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Author: Urs
Format: Markdown...

...
- CommentRowNumber23.
- CommentAuthorUrs
- CommentTimeJan 4th 2010
- (edited Jan 4th 2010)
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Author: Urs
Format: MarkdownOkay, now I'll come back to this discussion here. I am beginning to think that the morphism that we are looking for above, that from some object <latex>A</latex> in an (oo,1)-topos to some other object <latex> \tilde A </latex>, such that postcomposition with this map maps <latex>A</latex>-valued cohomology to something like the corresponding "Chern character" is essentially the morphism <latex>A \to \Omega^\infty \Sigma^\infty \Sigma A</latex> from <latex>A</latex> to the free [[stabilization]] of its suspension. Or more precisely, as I indicated above, the construction is this: let <latex>\mathbf{H}^I</latex> by the arrow category of the smooth ambient (oo,1)-topos. For <latex> [X \hookrightarrow \Pi(X)] </latex> the object in <latex>\mathbf{H}^I</latex> given by the inclusion of <latex>X</latex> into its path oo-groupoid, and <latex> [A \to *] </latex> the map to the terminal object, we have that ordinary <latex>A</latex>-cohomology of <latex>X</latex> with coefficients in <latex> A </latex> may be expressed in <latex> \mathbf{H}^I </latex> as <latex> \mathbf{H}(X,A) \simeq \mathbf{H}^I([X \to \Pi(X)], [A \to *])</latex>. Now let <latex> A_{ch} := \Omega^\infty \Sigma^\infty \Sigma A </latex> be the [[stabilization]] of the [[suspension object]] of <latex> A</latex>. The morphism <latex> A\to \Sigma A \to A_{ch}</latex> in <latex> \mathbf{H} </latex> induces a morphism <latex> [A \to *] \to [* \to A_{ch}]</latex> in <latex> \mathbf{H}^I</latex>. So we get a morphism in cohomology <latex>\mathbf{H}(X,A) \to \mathbf{H}^I([X \to \Pi(X)], [*\to A_{ch}])</latex>. A cocycle in the thing on the right is a flat differential (i.e. on <latex> \Pi(X) </latex>) <latex>A_{ch}</latex> cocycle, whose underlying ordinary cocycle (i.e. on <latex>X</latex>) is trivial. This makes this a cocycle in the "real-ization" of <latex>A</latex>: to see what I mean consider the archetypical case for instance that <latex> A = \mathbf{B}^n U(1)</latex>. Then an object in <latex> \mathbf{H}^I([X \to \Pi(X)], [*\to A_{ch}]) </latex> is effectively a cocycle in <latex>U(1)</latex>-[[Deligne cohomology]] whose 0-form part trivializes. But that's then the same as an <latex> \mathbb{R}</latex>-Deligne cocycle (since the differential forms don't see the nontrivial topology of the coefficient object, only the 0-form part does, if we kill that, we are left with the real version). So there is an object <latex>A_{ch}^{\mathbb{R}} </latex> such that <latex>\mathbf{H}^I([X \to \Pi(X)], [* \to A_{ch}])\simeq\mathbf{H}(\Pi(X), A_{ch}^{\mathbb{R}})</latex> I think. On the right we have a souped-up version of deRham cohomology. I think we may apply the deRham theorem for oo-groupoids to it to pass to a souped-up version of equivalent real cohomology. But since for the present purpose this just amounts to more notation, let me not do it. Then the punchline is that I am saying that the full generalization of the "Chern cheracter" that in ordinary [[differential cohomology]] would be suposed to map an Eilenberg-Steenrod cohomology theory <latex> \Gamma </latex> to ordinary real cohomology with coefficients in <latex> \Gamma(pt) \otimes_{\mathbb{Z}} \mathbb{R}</latex> is the composite of the above maps <latex> H(X,A) \simeq H^I([X\to \Pi(X)], [A\to *]) \to H^I([X \to \Pi(X)], [* \to A_{ch}]) \simeq H(\Pi(X),A_{ch}^\mathbb{R}) \,. </latex> Here is something to check this statement. I am beginning to think the following is true, but I don't have a full proof yet: for <latex>\mathfrak{a}</latex> the oo-Lie groupoid of <latex>A</latex> let <latex> St(\Sigma \mathfrak{a})</latex> be the oo-Lie groupoid of the stabilization of <latex> \Sigma A</latex>. Then I think the [[schreiber:Chevalley-Eilenberg algebra]] <latex> CE(St(\Sigma\mathfrak{a})) </latex> is the algebra of [[invariant polynomial]]s on <latex> \mathfrak{a}</latex>. I was looking for such a completely intrinsic oo-categorical definition of invariant polynomials for a while, and now I think this is it. But I still need to think about it. Notice that, if right, this means for instance that for <latex> A = \mathbf{B} U \times \mathbb{Z}</latex> (the coefficient object for degree 0 K-theory) we have that <latex> A_{ch}^{\mathbb{R}} </latex> is the oo-Lie groupoid integrating the oo-Lie algebroid whose CE-algebra is that of invariant polynomials on <latex> \mathfrak{u} </latex>. But that's <latex> \prod_n \mathbf{B}^n \mathbb{R} </latex>, hence the above generalized Chern-character for <latex> A = \mathbf{B}U</latex> would take values in even graded real cohomology, which is the right answer. Hm, Ill better stop here and dicsuss this in more detail on my web. I'll also have a look at the entry on your web now.

Okay, now I'll come back to this discussion here.

I am beginning to think that the morphism that we are looking for above, that from some object $A$ in an (oo,1)-topos to some other object $\tilde A$ , such that postcomposition with this map maps $A$ -valued cohomology to something like the corresponding "Chern character" is essentially the morphism $A \to \Omega^\infty \Sigma^\infty \Sigma A$

from $A$ to the free stabilization of its suspension.

Or more precisely, as I indicated above, the construction is this:

let $\mathbf{H}^I$ by the arrow category of the smooth ambient (oo,1)-topos. For $[X \hookrightarrow \Pi(X)]$ the object in $\mathbf{H}^I$ given by the inclusion of $X$ into its path oo-groupoid, and $[A \to *]$ the map to the terminal object, we have that ordinary $A$ -cohomology of $X$ with coefficients in $A$ may be expressed in $\mathbf{H}^I$ as $\mathbf{H}(X,A) \simeq \mathbf{H}^I([X \to \Pi(X)], [A \to *])$ .

Now let $A_{ch} := \Omega^\infty \Sigma^\infty \Sigma A$ be the stabilization of the suspension object of $A$ . The morphism $A\to \Sigma A \to A_{ch}$ in $\mathbf{H}$ induces a morphism $[A \to *] \to [* \to A_{ch}]$ in $\mathbf{H}^I$ . So we get a morphism in cohomology

$\mathbf{H}(X,A) \to \mathbf{H}^I([X \to \Pi(X)], [*\to A_{ch}])$ .

A cocycle in the thing on the right is a flat differential (i.e. on $\Pi(X)$ ) $A_{ch}$ cocycle, whose underlying ordinary cocycle (i.e. on $X$ ) is trivial. This makes this a cocycle in the "real-ization" of $A$ : to see what I mean consider the archetypical case for instance that $A = \mathbf{B}^n U(1)$ . Then an object in $\mathbf{H}^I([X \to \Pi(X)], [*\to A_{ch}])$ is effectively a cocycle in $U(1)$ -Deligne cohomology whose 0-form part trivializes. But that's then the same as an $\mathbb{R}$ -Deligne cocycle (since the differential forms don't see the nontrivial topology of the coefficient object, only the 0-form part does, if we kill that, we are left with the real version).

So there is an object $A_{ch}^{\mathbb{R}}$ such that
$\mathbf{H}^I([X \to \Pi(X)], [* \to A_{ch}])\simeq\mathbf{H}(\Pi(X), A_{ch}^{\mathbb{R}})$
I think. On the right we have a souped-up version of deRham cohomology. I think we may apply the deRham theorem for oo-groupoids to it to pass to a souped-up version of equivalent real cohomology. But since for the present purpose this just amounts to more notation, let me not do it. Then the punchline is that I am saying that the full generalization of the "Chern cheracter" that in ordinary differential cohomology would be suposed to map an Eilenberg-Steenrod cohomology theory $\Gamma$ to ordinary real cohomology with coefficients in $\Gamma(pt) \otimes_{\mathbb{Z}} \mathbb{R}$ is the composite of the above maps $H(X,A) \simeq H^I([X\to \Pi(X)], [A\to *]) \to H^I([X \to \Pi(X)], [* \to A_{ch}]) \simeq H(\Pi(X),A_{ch}^\mathbb{R}) \,.$

Here is something to check this statement. I am beginning to think the following is true, but I don't have a full proof yet:

for $\mathfrak{a}$ the oo-Lie groupoid of $A$ let $St(\Sigma \mathfrak{a})$ be the oo-Lie groupoid of the stabilization of $\Sigma A$ . Then I think the Chevalley-Eilenberg algebra (schreiber) $CE(St(\Sigma\mathfrak{a}))$ is the algebra of invariant polynomials on $\mathfrak{a}$ .

I was looking for such a completely intrinsic oo-categorical definition of invariant polynomials for a while, and now I think this is it. But I still need to think about it.

Notice that, if right, this means for instance that for $A = \mathbf{B} U \times \mathbb{Z}$ (the coefficient object for degree 0 K-theory) we have that $A_{ch}^{\mathbb{R}}$ is the oo-Lie groupoid integrating the oo-Lie algebroid whose CE-algebra is that of invariant polynomials on $\mathfrak{u}$ . But that's $\prod_n \mathbf{B}^n \mathbb{R}$ , hence the above generalized Chern-character for $A = \mathbf{B}U$ would take values in even graded real cohomology, which is the right answer.

Hm, Ill better stop here and dicsuss this in more detail on my web. I'll also have a look at the entry on your web now.

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