Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthorzskoda
   • CommentTimeMay 1st 2011
   • PermaLink
   Author: zskoda
   Format: MarkdownItexNew entry [[characteristic class of a structure]] to complement [[characteristic class]] and [[historical note on characteristic classes]]. I did not link to it from outside so far.

   New entry characteristic class of a structure to complement characteristic class and historical note on characteristic classes. I did not link to it from outside so far.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 1st 2011
   • (edited May 1st 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexHi Zoran, thanks for the reference. Notice that, once again, under the [[Grothendieck construction]] this comes down to the same story as at [[characteristic class]]: what you write $\mathcal{H}_H$ is just the Grothendieck construction of the presheaf $H$ (its [[category of elements]]) (notice you need $H$ to be contravariant for the formula you give to make sense) and $\mathcal{S}$ should also be assumed to be a fibered category, I assume, corresponging under the reverse Grothendieck construction to a sheaf/stack $F_{\mathcal{S}}$ on $\mathcal{T}$. So under the Grothendieck construction a characteristic class in the sense of the article by Fuks that you mention is the same as a morphism $$ F_{\mathcal{S}} \to H $$ in the topos over $\mathcal{T}$. So it's exactly as defined at [[characteristic class]]. I'll add the reference with that interpretation now to the latter entry, if you allow.

   Hi Zoran,

   thanks for the reference.

   Notice that, once again, under the Grothendieck construction this comes down to the same story as at characteristic class:

   what you write $\mathcal{H}_H$ is just the Grothendieck construction of the presheaf $H$ (its category of elements) (notice you need $H$ to be contravariant for the formula you give to make sense) and $\mathcal{S}$ should also be assumed to be a fibered category, I assume, corresponging under the reverse Grothendieck construction to a sheaf/stack $F_{\mathcal{S}}$ on $\mathcal{T}$.

   So under the Grothendieck construction a characteristic class in the sense of the article by Fuks that you mention is the same as a morphism

   $$F_{\mathcal{S}} \to H$$

   in the topos over $\mathcal{T}$. So it’s exactly as defined at characteristic class.

   I’ll add the reference with that interpretation now to the latter entry, if you allow.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 1st 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexokay, I have added the discussion to [[characteristic class]]. I was going to add also a criticism about how Fuks's definition is not local/excisive as long as it restricts to cohomology classes instead of cocycles, but then I figured I shouldn't do that with having seen just a second-hand summary of one part of the paper.

   okay, I have added the discussion to characteristic class.

   I was going to add also a criticism about how Fuks’s definition is not local/excisive as long as it restricts to cohomology classes instead of cocycles, but then I figured I shouldn’t do that with having seen just a second-hand summary of one part of the paper.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeMay 1st 2011
   • (edited May 1st 2011)
   • PermaLink
   Author: zskoda
   Format: MarkdownItex> I'll add the reference with that interpretation now to the latter entry, if you allow. Of course, this is why I did not write in the personal part of the $n$Lab. It is good for students to have access to 1-categorical approach. I saw immediately that it is about fibered categories (does this reformulation make it possible without "concrete" assumption ?; originally Fuks works just with abelian groups in the target of cohomology), and your explanation in 2 is useful to me as well.

   I’ll add the reference with that interpretation now to the latter entry, if you allow.

   Of course, this is why I did not write in the personal part of the $n$Lab. It is good for students to have access to 1-categorical approach. I saw immediately that it is about fibered categories (does this reformulation make it possible without “concrete” assumption ?; originally Fuks works just with abelian groups in the target of cohomology), and your explanation in 2 is useful to me as well.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMay 1st 2011
   • (edited May 1st 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> (does this reformulation make it possible without "concrete" assumption ? The concreteness is what allows us to interpret the situation in terms of sheaves/stacks with values in sets/categories (or groupoids). If we drop the concreteness assumption, we might still be able to proceed, but would step into the far more general and far less explored territory of (higher) categories of (higher) sheaves with coefficients not in the standard coefficient object. I'd hesitate to go in that direction without a strong motivating example that makes it necessary. Is there any chance to see an electronic version of an English (or French or German) translation of Fuks' article? Or else, can you recount further what he discusses in his article? What's his main theorem with his definition?

   (does this reformulation make it possible without “concrete” assumption ?

   The concreteness is what allows us to interpret the situation in terms of sheaves/stacks with values in sets/categories (or groupoids). If we drop the concreteness assumption, we might still be able to proceed, but would step into the far more general and far less explored territory of (higher) categories of (higher) sheaves with coefficients not in the standard coefficient object. I’d hesitate to go in that direction without a strong motivating example that makes it necessary.

   Is there any chance to see an electronic version of an English (or French or German) translation of Fuks’ article?

   Or else, can you recount further what he discusses in his article? What’s his main theorem with his definition?