Holmstrom Homotopy limit

Shulman survey on homotopy limits and colimits

http://mathoverflow.net/questions/454/references-for-homotopy-colimit

See maybe also Weak limit and Weighted limit

A brief intro is in Toen: Essen talk, p 13.

A discussion can be found in Hovey p. 189.

The original source is probably Bousfield and Kan: Homotopy limits, completions, and localizations (Springer Lecture notes vol 304).

A good modern reference in probably the book of Hirschhorn. See also the book of Dwyer-Hirschhorn-Kan. Hirschhorn et al: Homotopy Limit Functors on Model Categories and Homotopical Categories (AMS)

Thomason: Homotopy colimits in the category of small categories.

Thomason: The homotopy limit problem

Vogt: Homotopy limits and colimits (1973)

nLab. See also limit in quasi-categories

Shulman: Homotopy limits and colimits and enriched homotopy theory arXiv

Baez on weighted colimits

A related idea might possibly be terminal object in a quasi-category

http://mathoverflow.net/questions/17425/homotopy-limits-over-fibered-categories

Dwyer: Localizations. In the Axiomatic, enriched, motivic book.

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

Goerss-Jardine p 128 introduces the holim functor as the total left derived functor $\\Ho(S^I) \to Ho(S)$ of the usual limit functor. Here $S$ is a cofibrantly generated simplicial model cat I think. There is also a coend formula for the holim, but this should be treated in more detail elsewhere in the book.

Very brief notes from LNM0304

Bousfield and Kan: Homotopy limits, localization and completion

Want: Functorial notion of R-completion of a space X s.t.

• For $Z/p$, get under finiteness assumptions and up to homotopy, the p-profinite completion of Quillen and Sullivan
• For a subring of the rationals, get, up to htpy, the loclizations of Sullivan, Quillen and others.

Can define R-completion for arbitrary spaces. Need separate results on towers of fibs, cosimplicial spaces, and homotopy limits.

Usefulness: Can fracture homotopy theory into mod-p components, and construct new spaces

More on all this.

General advantages:

• Up to homotopy, R-completion preserves fibrations whenever the fundamental gp of the base acts nilpotently on the R-homology of the fibre
• Many spaces are R-good, i.e. the canonical map from X to its R-completion preserves R-homology, and is, up to homotopy, terminal among such maps
• The mod/R spectral sequence relates the R-homology of a spaces with the homotopy groups of its R-completion.
• more on Malcev completion

Now a summary of the book: See LNM304 pp4 for full version. Some highlights: A map induces iso on reduced R-homology iff it induces a homotopy equivalence between the R-completions.

Chapter 1

The R-completion is defined by first constructing a cosimplicial diagram of spaces RX, then associating with this a tower of fibrations \\{ R_s X \\} and finally taking the inverse limit of this tower. The R-completion comes with a natural map from X. Can think of this as the "Artin-Mazur like completion" in two different ways. The tower of fibs give us a spectral sequence, useful. Recover the homotopy spectral sequence with coeffs in R, as well as the unstable Adams spectral sequence, and the primitive elements in the rational cobar ss.

Up to homotopy, the R-completion commutes with disjoint unions and finite products, and preserves multiplicative structures.

It might be the case that the homotopy groups of the R-completion are the reduced homology groups of the space, with coeffs in R. This gives an alternative view on the Hurewicz homomorphism.

Note: The R-completion of a space is fibrant because surjections are sent to fibrations.

Cont on p24

nLab page on Homotopy limit