Homotopy category

Résumé

In mathematics, the homotopy category is a built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed below. More generally, instead of starting with the category of topological spaces, one may start with any and define its associated homotopy category, with a construction introduced by Quillen in 1967. In this way, homotopy theory can be applied to many other categories in geometry and algebra. The naive homotopy category The Top has objects the topological spaces and morphisms the continuous maps between them. The older definition of the homotopy category hTop, called the naive homotopy category for clarity in this article, has the same objects, and a morphism is a homotopy class of continuous maps. That is, two continuous maps f: X → Y are considered the same in the naive homotopy category if one can be continuously deformed to the o

Source officielle

https://en.wikipedia.org/wiki/Homotopy_category

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Kan spectra, group spectra and twisting structures

Marc Stephan

Kan spectra provide a combinatorial model for the stable homotopy category. They were introduced by Dan Kan in 1963 under the name semisimplicial spectra. A Kan spectrum is similar to a pointed simplicial set, but it has simplices in negative degrees as well and all its simplices have infinitely many faces and degeneracies. In the first part of this thesis, we define a functor from the category of Gamma-spaces to the category of Kan spectra without passing through any other category of spectra. We show that the resulting Kan spectrum of a Gamma-space A agrees with the usual object associated to A in the stable homotopy category by comparing it to Bousfield-Friedlander's spectrum construction. In particular, applying our construction to the Gamma-space associated to a symmetric monoidal category provides a combinatorial model of its algebraic K-theory spectrum. For the Gamma-space associated to an abelian group, this yields via the stable Dold-Kan correspondence the unbounded chain complex with the abelian group concentrated in degree zero. The second part of this work concerns group spectra and twisting structures. Group spectra are the group objects in the category of Kan spectra. They provide an algebraic, combinatorial model for the stable homotopy category. We transfer Brown's model structure from the category of Kan spectra to a Quillen equivalent model structure on the category of group spectra. We then construct the analogues of Kan's loop group functor and its right adjoint Wbar together with corresponding classifying bundles, so that the category of Kan spectra becomes a twisted homotopical category in the sense of Farjoun and Hess.

EPFL2015

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Why H Z-algebra Spectra are Differential Graded Algebras?

Varvara Karpova

In homological algebra, to understand commutative rings R, one studies R-modules, chain complexes of R-modules and their monoids, the differential graded R-algebras. The category of R-modules has a rich structure, but too rigid to efficiently work with homological invariants and homotopy invariant properties. It appears more appropriate to operate in the derived category D(R), which is the homotopy category of differential graded R-modules. Algebra of symmetric spectra offers a generalization of homological algebra. In this frame, spectra are objects that take the place of abelian groups; in particular, the analogue of the initial ring Z is the sphere spectrum S. Tensoring over S endows the category of spectra with a symmetric monoidal smash product, analogous to the tensor product of abelian groups. Thus, spectra are S-modules, and ring spectra, which extend the notion of rings, are the S-algebras. To any discrete ring R, one can associate the Eilenberg-Mac Lane ring spectrum HR, which is commutative if R is.

2010

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