Holmstrom Homotopy type

Essay topic Grojnowski

This essay is an introduction to homological and homotopical algebra. The first aim is to understand differential graded commutative algebras, which appear everywhere in characteristic zero geometry, topology, and algebra. For example, given a manifold, one can consider the algebra of differential forms on it; the main theorem of ”rational homotopy theory” is the statement that this tells you everything about the homotopy type of the manifold if you neglect torsion. The second, more ambitious goal is to understand torsion. The language of dga’s is inadequate, and the correct (homotopy invariant) notion is of ”algebras for the E1-operad”. Begin with two computations (both of which will require you learning a lot!) (1) Compute the rational homotopy of spheres, (2) Compute the homology of Eilenberg-MacLane spaces. Then understand how the deRham functor gives an equivalence of categories between topological spaces localised at Q and connected cdga’s; that is learn the rational homotopy theory of Quillen and Sullivan. This encodes the computation of (1); you can find it in various textbooks and papers. If any time remains you can aim to understand Mandell’s theorem, which is the characteristic p analogue. This requires a lot more background on homotopical algebra; a basic computational input is (2) above. References Many textbook expositions exist, now. The original papers are: [1] D. Quillen, Rational homotopy theory. Ann. of Math. 90, 1969, 205–295. [2] D. Sullivan, Infinitesimal computations in topology. Inst. Hautes Etudes Sci. Publ. Math. No. 47, (1977), 269–331. [3] M. Mandell, E1 algebras and p-adic homotopy theory, Topology 40 (2001), 43–94.

Notes from Baues: Homotopy types (e)

Homotopy types can be modeled by polyhedra, simplicial complexes, simplicial sets, singular sets of topological spaces, nerves of small cats, nerves of posets, CW complexes, simplicial groups, free simplicial groups.

n-types (vanishing higher homotopy groups), Postnikov functor from the homotopy cat of (CW?) spaces to n-types. Get sequence of functors $1-types \leftarrow 2-types \leftarrow 3-types \ldots$

1-types are the same as $K(\pi, 1)$‘s (Eilenberg-MacLane spaces), so can be viewed as groups. More precisely, $\pi_1$ is an equiv of cats from 1-types to groups. Similarly for 2-types and (the homotopy cat of) crossed modules, and for 3-types and quadratic modules.

Other algebraic models for homotopy types. One interesting example: The Grothendieck construction for n-types with vanishing $\pi_i$ for $1 i n$, using group cohomology.

Simply connected homotopy types and $H \pi$-duality: Have homotopy decompositions, using fiber sequences and Eilenberg-MacLane spaces, which constructs a homotopy type as a inverse limit. Dually, homology decomposition, using Moore spaces, cofiber sequences, and a direct limit. The homotopy decomposition can also be obtained via the Postnikov functors. Dwyer, Kan and Smith constructed a space which parameterizes all homotopy decompositions associated to a graded abelian group $A_*$. Its path components corresponds to the set of homotopy types associated to this graded group. The n-th homotopy group is an equiv of cats from the relevant cat of E-M spaces to abelian groups, and there is a functor in the other direction (iterated classifying space). $H_n$ is not an eq of cats on Moore spaces, but is a detecting functor. Classes of maps between E-M spaces are by def cohomology operations; these are computable in some sense. Classes of maps between Moore spaces include homotopy groups of spheres as special examples. Def: Homotopy and cohomology groups with coeffs in a group $A$. k-invariants = Postnikov invariants, defined as certain elements in $H^{n+1}(P_{n-1}X, \pi_n(X) )$. Also dual notion of boundary invariant. Def of pseudo-homology, using Moore spaces, dual to singular chain def of cohomology.

The Hurewicz homomorphism, coming from applying $\pi_n$ to the map from $X$ to the infinite symmetric product of $X$ (whose homotopy is the homology of $X$ by Dold-Thom). Relation to homotopy types. Related notion of $\Gamma$-groups (with coefficients), fitting the Hurewicz homomorphisms into a long exact sequence.

Use of Postnikov and boundary invariants to classify homotopy types.

Infinite symmetric product, Dold-Thom iso.

Stable homotopy types. Properties of suspension. Algebraic models.

Some explicit examples and applications, for example allowing us to compute generalized (co)homology of 4-dimensional polyhedra in terms of the values on a small number of “prime factors”.

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Homotopy type

Abstract: This is an expended and revised version of the preprint “Schematization of homotopy types”.

The purpose of this work is to introduce a notion of \emph{affine stacks}, which is a homotopy version of the notion of affine schemes, and to give several applications in the context of algebraic topology and algebraic geometry. As a first application we show how affine stacks can be used in order to give a new point of view (and new proofs) on rational and $p$-adic homotopy theory. This gives a first solution to A. Grothendieck’s \emph{schematization problem} described in [gr]. We also use affine stacks in order to introduce a notion of \emph{schematic homotopy types}. We show that schematic homotopy types give a second solution to the schematization problem, which also allows us to go beyound rational and $p$-adic homotopy theory for spaces with arbitrary fundamental groups. The notion of schematic homotopy types is also used in order to construct various homotopy types of algebraic varieties corresponding to various cohomology theories (Betti, de Rham, $l$-adic, …), extending the well known constructions of the various fundamental groups. Finally, as algebraic stacks are obtained by gluing affine schemes we define \emph{$\infty$-geometric stacks} as a certain gluing of affine stacks. Example of $\infty$-geometric stacks in the context of algebraic topology (moduli spaces of dga structures up to quasi-isomorphisms) and Hodge theory (non-abelian periods) are given.

Martin Olsson: F-isocrystals and homotopy types

We study a positive characteristic analogue of the nonabelian Hodge structure constructed by Katzarkov, Pantev, and Toen on the homotopy type of a complex algebraic variety. Given a proper smooth scheme X over a perfect field of characteristic p and a Tannakian category Click to view the MathML source of isocrystals on X, we construct an object Click to view the MathML source in a suitable homotopy category of simplicial presheaves whose category of local systems is equivalent to Click to view the MathML source in a manner compatible with cohomology. We then study F-isocrystal structure on these simplicial presheaves. As applications of the theory, we prove a p-adic analogue of a result of Hain on relative Malcev completions, a generalization to the level of homotopy types of a theorem of Katz relating p-adic étale local systems and F-isocrystals, as well as a p-adic version of the formality theorem in homotopy theory. We have also included a new proof based on reduction modulo p of the formality theorem for complex algebraic varieties.

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Homotopy type

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Homotopy type

AT (Algebraic topology), AG (Algebraic geometry), CT (Category theory)?

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Homotopy type

Pure

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Homotopy type

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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Homotopy type

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: http://www.math.uiuc.edu/K-theory/0675

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Homotopy type

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.

nLab page on Homotopy type