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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 27th 2012
   • (edited Feb 27th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexprompted by [this](https://plus.google.com/u/0/103404025783539237119/posts/YSKRCdz3RKw) G+ post by David Roberts, I have started an entry _[[arithmetic Chow group]]_ .

   prompted by this G+ post by David Roberts, I have started an entry arithmetic Chow group .

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeFeb 27th 2012
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexSo is a general search for differential cohomological situations likely to yield new examples of cohesion? How [about](http://www.math.wayne.edu/research/seminars/topabstr/12.02.14.pdf) this? >We report on a joint project with Mike Hopkins and introduce Hodge filtered complex cobordism groups for complex manifolds. These groups combine integral topological and complex geometric information in analogy to *differential cohomology groups* for smooth manifolds. By taking the Hodge filtration into account these groups form a natural generalization of Deligne cohomology and become especially interesting for Kahler manifolds.

   So is a general search for differential cohomological situations likely to yield new examples of cohesion? How about this?

   We report on a joint project with Mike Hopkins and introduce Hodge filtered complex cobordism groups for complex manifolds. These groups combine integral topological and complex geometric information in analogy to differential cohomology groups for smooth manifolds. By taking the Hodge filtration into account these groups form a natural generalization of Deligne cohomology and become especially interesting for Kahler manifolds.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeFeb 27th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItex> So is a general search for differential cohomological situations likely to yield new examples of cohesion? In this generality I have to reply like this: just as I cannot prevent anyone from calling "cohomology" something that is not computed by homs in an oo-topos, so I cannot prevent anyone from calling "differential cohomology" something that is not computed in a cohesive oo-topos. But in both cases I expect that a "good" definition will make this true. But more concretely, in the case that you point to: establishing cohesion for _complex geometry_ should be pretty straightforward. If I had a single second to spare (which I don't), I would try to work that out in more detail. Alternatively, that would be the kind of topic I might hand to a student (success on immediate questions guaranteed, with a good supply of arbitrarily advanced questions already visible).

   So is a general search for differential cohomological situations likely to yield new examples of cohesion?

   In this generality I have to reply like this: just as I cannot prevent anyone from calling “cohomology” something that is not computed by homs in an oo-topos, so I cannot prevent anyone from calling “differential cohomology” something that is not computed in a cohesive oo-topos. But in both cases I expect that a “good” definition will make this true.

   But more concretely, in the case that you point to:

   establishing cohesion for complex geometry should be pretty straightforward. If I had a single second to spare (which I don’t), I would try to work that out in more detail. Alternatively, that would be the kind of topic I might hand to a student (success on immediate questions guaranteed, with a good supply of arbitrarily advanced questions already visible).