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nLab > Latest Changes: differential cohomology in an (∞,1)-topos -- survey

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  1.  
    • 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∞GrpdSmooth \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∞GrpdSmooth \infty Grpd;

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

    • section 3.3.9 , ∞\infty-Chern-Weil homomorphism in Smooth∞GrpdSmooth \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∞GrpdSynthDiff \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 ∞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

1 to 18 of 18

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