Spectra and localizations of the category of topological spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T21:45:54Z http://mathoverflow.net/feeds/question/16224 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/16224/spectra-and-localizations-of-the-category-of-topological-spaces Spectra and localizations of the category of topological spaces Dmitri Pavlov 2010-02-24T03:21:39Z 2010-03-01T00:46:07Z <p>Can we construct the category of spectra (or maybe just its homotopy category) from the category of pointed topological spaces using some kind of localization combined with other categorical constructions?</p> <p>[The first part of the original question was wrong for a trivial reason pointed out by Reid Barton.]</p> http://mathoverflow.net/questions/16224/spectra-and-localizations-of-the-category-of-topological-spaces/16231#16231 Answer by Reid Barton for Spectra and localizations of the category of topological spaces Reid Barton 2010-02-24T04:25:46Z 2010-02-24T06:16:43Z <p>[Removed a paragraph relating to an earlier version of the question]</p> <p>You can construct Spectra categorically by adjoining an inverse to the endofunctor &Sigma; of Top as a presentable (&infin;,1)-category. Inverting an endofunctor is a very different operation than inverting maps! It's like the difference between forming &#8484;[1/p] and &#8484;/(p).</p> <p>Here is one way to verify the claim. To invert the endomorphism &Sigma; of Top we should form the colimit, in the (&infin;,1)-category Pres of presentable categories and colimit-preserving functors, of the sequence Top &rarr; Top &rarr; ... where all the functors in the diagram are &Sigma;. A basic fact about Pres is that we can compute such a colimit by forming the diagram (on the opposite index category) formed by the right adjoints of these functors, and taking its <em>limit</em> as a diagram of underlying (&infin;,1)-categories [HTT 5.5.3.18]. The functors in the limit cone will have left adjoints which are the functors to the colimit in Pres. In our case we obtain the sequence Top &larr; Top &larr; ... where the functors are &Omega;, and the limit of this sequence is precisely the classical definition of (&Omega;-)spectrum: a sequence of spaces X<sub>n</sub> with equivalences X<sub>n</sub> &rarr; &Omega;X<sub>n+1</sub>.</p> http://mathoverflow.net/questions/16224/spectra-and-localizations-of-the-category-of-topological-spaces/16731#16731 Answer by Jeff Strom for Spectra and localizations of the category of topological spaces Jeff Strom 2010-03-01T00:46:07Z 2010-03-01T00:46:07Z <p>I don't know the answer, but I have a related question. What if we let $f$ be the wedge of the maps $X\to \Omega \Sigma X$ (representing suspension) for all countable CW complexes X, and then apply Bousfield/Farjoun localization $L_f$?</p> <p>It seems to me that, for the purposes of mapping in finite complexes, we have inverted the suspension operation. </p>