Holmstrom Localization

Would like to clarify all different notions of localization, for example the relation between localization in number theory and in homotopy theory.

Gabriel-Zisman in Htpy th folder: Simplicial sets, Localization (in the sense of inverting some morphisms)

Dwyer-Spalinski in the homotopy theory folder: Model categories, Homotopy limits brief intro, localization wrt a homology theory: very brief intro on p. 54.

nLab

http://ncatlab.org/nlab/show/category+of+fractions

http://ncatlab.org/nlab/show/local+object

http://ncatlab.org/nlab/show/localization+of+an+(infinity,1)-category

http://ncatlab.org/nlab/show/localization+of+a+simplicial+model+category

http://ncatlab.org/nlab/show/Ore+localization

http://nlab.mathforge.org/nlab/show/Bousfield+localization

http://ncatlab.org/nlab/show/simplicial+localization

http://www.ncatlab.org/nlab/show/homotopy+localization

The localization of any category wrt any class of morphisms is briefly described in Toen: Essen talk. Also ref to Dwyer-Kan hammock localization. See also remark 2.4.4 and preceeding pragraphs for a coneptual discussion of different notions of localization. Section 2.6 treats homotopical localization, and left Bousfield loc.

See also Bousfield localization

For localization in the context of DG-categories, and also a discussion of Gabriel-Zisman, see Toen: Lecture on DG-categories. File Toen web unpubl swisk.pdf.

Dwyer (2006): Noncommutative localisation in homotopy theory

Ravenel: Localization and periodicity in homotopy theory (1987)

Something by Kahn and Sujatha

Perhaps the book by Dwyer-Hirschhorn-Kan treats localisation in a good way?

Gabriel-Zisman

Section 12 in Baues: Homotopy types. For a generalized homology theory $k_*$, there is a somewhat unclear def of $k_*$-equivalence, $k_*$-local space, and a map being a $k_*$-localization. Equiv of cats between the localization of $spaces$ wrt $k_*$-eqiuvalences, and the homotopy cat of $k_*$-local spaces, sending a CW space to its $k_*$-localization. Can view $k_*$-equivalences as WEs in a “cofibration cat”. More related to localization, omitted here, e.g. Postnikov stuff.

Localisation of CW complexes

Let $P$ be an arbitrary set of primes. We say that an abelian group is $P$-local if multiplication by $ell$ is an isomorphism for all $ell$ NOT in $P$. A homomorphism is a $P$-localisation if the target is $P$-local and satisfies the obvious universal property.

Consider the category $CW_1$ of 1-connected CW complexes. An $X$ in this category is called $P$-local if $\pi_n(X)$ is $P$-local for all $n$. A morphism is a $P$-localisation if the target is $P$-local and is universal with respect to this property. (The universal property here refers to a bijection of homotopy classes of maps, so we are really in the homotopy category.)

Fact: Any $X \in CW_1$ has a $P$-localisation, unique up to homotopy. $P$-localisation is a functor (I think).

Thm: TFAE, for a morphism $f$ in $CW_1$:

• $f$ is a $P$-localisation
• $\pi_n(f)$ is a $P$-localisation for all $n \geq 2$
• $H_n(f)$ is a $P$-localisation for all $n \geq 2$

Example: Let $P$ consist of a single prime $p$. Then homotopy and homology groups (and maps between them) are obtained by tensoring with $\mathbb{Z}_{(p)}$.

Example: Let $P$ consist of all primes except $p$. Then homotopy and homology groups (and maps between them) are obtained by tensoring with $\mathbb{Z}[\frac{1}{p}]$

In Kono-Tamaki, there are also further compatibilities, and a construction of the $P$-localisation.

A Generalized cohomology theory $h^*$, and a set $P$ of primes, define a generalized cohomology theory $h^*( - )_{P}$, because $P$-localisation is exact on $Ab$. If the coefficient group of $h^*$ is $P$-local, then the natural transformation from $h^*$ to its $P$-localisation is an isomorphism.

Theorem: Isomorphism on $H_*( - , \mathbb{Z}_P)$ implies isomorphism on $\tilde{h}^*(-) \otimes \mathbb{Z}_P$

nLab page on Localization