Start a new discussion

Not signed in

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

Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorDavid_Corfield
• CommentTimeSep 21st 2010

In differential cohomology in an (∞,1)-topos – survey, I can’t guess what ’nothing’ should be here:

The curvature characteristic forms / Chern characters in the traditional formulation of differential cohomology take values in abelian $\infty$-Lie algebras and are therefore effectively nothing differential forms with values in a complex of vector spaces

• CommentRowNumber2.
• CommentAuthorDavidRoberts
• CommentTimeSep 21st 2010

’nothing but’?

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeSep 21st 2010
• (edited Sep 21st 2010)

That must be it. I’ll add the ’but’.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeSep 21st 2010

Thanks. Yes, I must have meant “nothing but”. Maybe this is a phrase that would better be removed? It doesn’t really add information.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJan 1st 2011
• (edited Jan 1st 2011)

I am working on finalizing some write-ups. Now I have gone through the section Introduction – General abstract theory that means to leisurely survey the main general abstract ideas (whereas the following section Introduction – Concrete implementation in ooGrpd surveys the concrete specific constructions).

The “Introduction – General abstract theory”-section starts out by introducing the notions of toposes and $\infty$-toposes as such and then looks at the definition of cohomology and homotopy inside these, and – combining the two – of differential cohomology. The idea is that a reader with knowledge of basic notions in category theory (I don’t explain adjoint functors) and homtopy theory (I don’t explain homotopies and homotopy groups) can read this and get a useful idea of what the technical discussion in the main section General abstract theory is supposed to accomplish.

(While I am polishing these wiki-pages I am gradually turning this into a classical pdf, file.)

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJan 6th 2011
• (edited Jan 6th 2011)

There is now a “pdf-exceprt” of the writeup. So far it contains the “Introduction” (a kind of survey of the whole thing) and a skeleton of the remainder.

I won’t give the link here, since it will be updated and the link name will change. The current version is the top link at differential cohomology in a cohesive topos (schreiber)

(you see that I keep changing my mind about the working title).

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeJan 6th 2011
• (edited Jan 6th 2011)

Here is the new pdf version for today, now in sections 2.1 - 2.3 with the Yoga of connected/cohesive $\infty$-toposes and $\infty$-connected/$\infty$-cohesive sites.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeJan 11th 2011
• (edited Jan 11th 2011)

I have uploaded a new pdf-version at differential cohomology in a cohesive topos (schreiber): now including

• section 2.3.7: Paths and geometric Postnikov towers

• section 2.3.9: Flat $\infty$-connections and local systems

• section 2.3.10: de Rham cohomology

• section 2.3.11: $\infty$-Lie algebras

• section 2.3.12: Maurer-Cartan forms and curvature characteristic forms

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeJan 17th 2011
• (edited Jan 17th 2011)

I have uploaded a new pdf-version at differential cohomology in a cohesive topos (schreiber): now including

• section 2.3.8 Universal coverings and geometric Whitehead towers

• section 2.3.13 Differential cohomology

• section 2.3.14 Chern-Weil homomorphism

• section 2.3.15 Holonomy and $\infty$-Chern-Simons functional

• section 3.1: Discrete $\infty$-groupoids

• section 3.2: Euclidean-topological $\infty$-groupoids

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeJan 25th 2011

I have uploaded a new pdf-version at differential cohomology in a cohesive topos (schreiber): now including

• in sections 3.3.2 - 3.3.7 the complete derivation of the intrinsic differential cohomology in $Smooth \infty Grpd$ and the proof that it coincides with ordinary differential cohomology
• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeFeb 1st 2011

I have uploaded a new pdf-version at differential cohomology in a cohesive topos (schreiber): now including

• section 3.3.6 – Exponentiated $\infty$-Lie algebras in $Smooth \infty Grpd$;

• section 3.3.7 – Universal curvature characteristics in terms of exponentiated Lie $n$-algebras;

• section 3.3.9 , $\infty$-Chern-Weil homomorphism in $Smooth \infty Grpd$ (the first definitions and propositions)

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeFeb 6th 2011
• (edited Feb 6th 2011)

I have uploaded a new pdf-version at differential cohomology in a cohesive topos (schreiber): now including

• section 4 – Applications

• section 4.1 – Fractional differential characteristic classes

• section 4.2 – Higher differential spin structures;

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeFeb 8th 2011
• (edited Feb 8th 2011)

I have uploaded a new pdf-version at differential cohomology in a cohesive topos (schreiber): now including

• section 3.4 – Synthetic differential oo-groupoids

• section 3.4.1 Cohomology in $SynthDiff \infty Grpd$

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeFeb 23rd 2011
• (edited Feb 23rd 2011)

I have uploaded a new pdf-version at differential cohomology in a cohesive topos (schreiber): now including

• in section 3.2.3 and section 3.3.4 a refined discussion of the preservation of homotopy fibers by the intrinsic fundamental $\infty$-groupoid functor;

• building on that in section 4.1 an expanded and streamlined discussion of fractional characteristic classes and their differential refinement

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeFeb 28th 2011

I have uploaded a new pdf-version at differential cohomology in a cohesive topos (schreiber): now including

• section 4.3: $\infty$-Chern-Simons functionals
• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeApr 14th 2011

I have uploaded a new pdf-version at differential cohomology in a cohesive topos (schreiber): now including

• in section 2.4: an expanded discussion of formal cohesive $\infty$-groupoids

• in section 3.4 statements and proofs characterizing $L_\infty$-algebroids as formal cohesive $\infty$-groupoids

• in section 4.2 a discussion of the supergravity C-field (“M-theory 3-form”) by $\infty$-Chern-Weil theory

• at the very very end, in section 4.3, higher dimensional supergravity by $\infty$-Chern-Weil theory

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeApr 22nd 2011
• (edited Apr 22nd 2011)

I have uploaded a new pdf-version at differential cohomology in a cohesive topos (schreiber): now including

• section 3.3.5 twisted bundles and torsion-twisted K-theory

• section 3.5: the rudiments of super cohesive $\infty$-groupoids

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeApr 25th 2011
• (edited Apr 25th 2011)

I have uploaded a new pdf-version at differential cohomology in a cohesive topos (schreiber): now including

• in section 3.3.10 some basics of the differential cohomology of orientifolds