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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
’nothing but’?
That must be it. I’ll add the ’but’.
Thanks. Yes, I must have meant “nothing but”. Maybe this is a phrase that would better be removed? It doesn’t really add information.
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.)
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).
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.
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
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
I have uploaded a new pdf-version at differential cohomology in a cohesive topos (schreiber): now including
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)
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;
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 $SynthDiff \infty Grpd$
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
I have uploaded a new pdf-version at differential cohomology in a cohesive topos (schreiber): now including
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
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
I have uploaded a new pdf-version at differential cohomology in a cohesive topos (schreiber): now including
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