Homotopy type

• General Introduction

Strictly speaking, it is wrong to call this a cohomology theory. However, the idea of homotopy type seem to have a strong bearing on the understanding of cohomology, and every sensible cohomology theory should factor through the homotopy type functor.

In algebraic topology, the notion of homotopy type is elementary and well-known. In algebraic geometry, it is a fairly recent and still poorly understood concept. The idea seems to have its roots in the mind of Grothendieck. Here is a very interesting note by Toen explaining some of this. Here is a longer preprint in which these ideas are developed further, giving some kind of answer to the Schematization problem posed by Grothendieck in Pursuing Stacks (-83).

For now, the notion of Homotopy type is placed in the "Pure" chapter in the book project. This might change.

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• Online References

Toen: Affine stacks

Toen: Champs affines. File Toen web publ chaff.pdf. Among other things, contains constructions of schematic homotopy types for the classical cohomologies (Betti, de Rham, Hodge, crystalline, l-adic), extending the classical notions of fundamental group. Also stuff on rational homotopy theory and p-adic homotopy theory, nonabelian Abel-Jacobi map and nonabelian period, and much more on schematic homotopy types and stacks.

On the etale homotopy type of Voevodsky's spaces , by Alexander Schmidt: K0675

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• Some Research Articles

There is a lot of work on Baues on algebraic models, I think.

arXiv:1101.4818 An algebraic model for free rational G-spectra from arXiv Front: math.AT by J. P. C. Greenlees, B. E. Shipley We show that for any compact Lie group \$G\$ with identity component \$N\$ and component group \$W=G/N\$, the category of free rational \$G\$-spectra is equivalent to the category of torsion modules over the twisted group ring \$H^*(BN)[W]\$. This gives an algebraic classification of rational \$G\$-equivariant cohomology theories on free \$G\$-spaces and a practical method for calculating the groups of natural transformations between them.

This uses the methods of arXiv:1101.2511, and some readers may find the simpler context of the present paper highlights the main thread of the argument.

[arXiv:1101.2511] An algebraic model for rational torus-equivariant spectra from arXiv Front: math.AT by J. P. C. Greenlees, B. Shipley We show that the category of rational G-spectra for a torus G is Quillen equivalent to an explicit small and practical algebraic model, thereby providing a universal de Rham model for rational G-equivariant cohomology theories. The result builds on the first author's Adams spectral sequence, the second author's functors making rational spectra algebraic.

There are several steps, some perhaps of wider interest (1) isotropy separation (replacing G-spectra by modules over a diagram of ring G-spectra) (2) change of diagram results (3) passage to fixed points on ring and module categories (replacing diagrams of ring G-spectra by diagrams of ring spectra) (4) replacing diagrams of ring spectra by diagrams of differential graded algebras (5) rigidity (replacing diagrams of DGAs by diagrams of graded rings). Systematic use of cellularization of model categories is central.

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