# Homotopy Limits, Completions and Localizations

## Homotopy Limits, Completions and Localizations

From this it follows Some general facts about localizations of categories are moved to section 4 to not interrupt the discussion. That section should be read alongside with the pres I would like to thank all participants of the summer school for the nice week, and for their very useful questions, comments and remarks. Documents: Advanced Search Include Citations. DMCA F.

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Citation Context Powered by:. Model categories and their localizations, volume 99 - Hirschhorn - Show Context Citation Context Homotopy limit functors on model categories and homotopical categories, volume - Dwyer, Hirschhorn, et al. Homological algebra. Princeton Landmarks in Mathematics - Cartan, Eilenberg - Methods of homological algebra.

## Homotopy Limits, Completions and Localizations

Le localisateur fondamental minimal - Cisinski. This seems to be no problem when the system is countable since then there is a short exact sequence from section IX.

Bousfield and D. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics , Springer, What about uncountable limits? Also, how could one translate the corresponding result into a computation of morphisms in a simplicial proper?

Check out page 34 of May-Ponto "More Concise Algebraic Topology" for a more modern treatment of the homotopy limit of a sequence of spaces. In particular, Proposition 2.

So it seems this proof generalizes completely to the non-countable case. Incidentally, it's not that uncommon for statements from before or so to have only been proven for the countable case, but often very similar proofs work far more generally.

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A good example is Neeman's book on triangulated categories, which introduces the notion of a well-generated triangulated category. For your second question, on model categories, the reference I use is Hirschhorn Chapters Chapter 18 is for simplicial model categories, chapter 19 is in full generality, using framings. For simplicity I'll focus on the material in Chapter 18, but everything has a corresponding generalization in chapter On page Hirschhorn gives the general definition for homotopy limits in model categories. On page , right after Theorem Another is that it's very hard for a model category to say that a map is surjective, but it can say something is a weak equivalence.

So it can be phrased as looking for an isomorphism.

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## phodeverga.cfaic topology - Homotopy groups of filtered homotopy limits - MathOverflow

The closest thing in Hirschhorn I can find is Theorem There's a lot of stuff in Hirschhorn, so if this doesn't sound like the model category version of the statement for Top it's very possible a different result of his will be a better fit. I encourage people to leave comments if they think there's a better model category analogue. Sign up to join this community. The best answers are voted up and rise to the top.

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