Not signed in (Sign In)

Not signed in

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

  • Sign in using OpenID

Site Tag Cloud

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 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 nforum 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

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

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 5th 2010
    • (edited Mar 5th 2010)

    brief remark on my personal web on Whitehead systems in a locally contractible (oo,1)-topos.

    So the homotopy fibers of the morphism A \to \mathbf{\Pi}(A)\otimes R that induces the Chern character in an (oo,1)-topos are something like the "rationalized universal oo-covering space": all non-torsion homotopy groups are co-killed, or something like that.

    Is there any literature on such a concept?

    • CommentRowNumber2.
    • CommentAuthorTim_Porter
    • CommentTimeMar 9th 2010

    I am not sure that this is closely related but your constructions do remind me of that question I asked you some time ago. In the simplicial group model it is easiest to see. You can model the n-type by a truncated simplicial group and then there is a natural fibration from the given G to that n-type. (Beware I am not being careful about whether the n-type is the n-1 type of n+1 type or whatever.) I think one can do a 'relative Abelianisation' of the fibre and get something which might correspond to chains on the universal n-covering space.

    I know this best in bottom dimensions. The homotopy types you get have vanishing Whitehead products above a given dimension. The 2-crossed complexes are and example, as are Baues' quadratic complexes. The next stage down is crossed complexes and the functor from 2-crossed complexes to crossed complexes does nothing above dimension 3 and coverts the bottom 2-crossed modules to the corresponding crossed module. At the next (and last) stage the passage from crossed complexes to chain complexes over groupoids involves Fox derivative, universal derived modules etc. Ronnie has a treatment in the new book I believe, and there is a reasonably full sketch in the Menagerie.(Sections 3.2-3.4).

    I have never looked at this from the point of view of the homotopy fibre as in the cases I have considered that is just the kernel.

    By the way, this all goes across to simplicial commutative algebras and is related to the cotangent complex construction of Illusie.