Homotopy colimits of cyclic spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T09:58:32Z http://mathoverflow.net/feeds/question/24487 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/24487/homotopy-colimits-of-cyclic-spaces Homotopy colimits of cyclic spaces Jeffrey Giansiracusa 2010-05-13T10:19:57Z 2010-05-13T13:56:11Z <p>Let $\Lambda$ denote Connes's cyclic category. It is an extension of the simplex category $\Delta$ (of nonempty finite linearly ordered sets) obtained by adding an automorphism of order $n+1$ to the object $\textbf{n}$.</p> <p><strong>Question:</strong> Suppose <code>$X: \Lambda^{op} \to Top$</code> is a cyclic space. What is a description of the homotopy colimit of this functor?</p> <p>Just to put this in a bit of context, if $Y: \Delta^{op} \to Top$ is a simplicial space then it has a geometric realisation $|Y|$. One can also take the homotopy colimit of $Y$, and under some reasonable hypotheses there will be an equivalence <code>$\mathrm{hocolim}_{\Delta^{op}} Y \simeq |Y|$</code>.</p> <p>There is an inclusion $\Delta \to \Lambda$, so a cyclic space $X$ can be considered as a simplicial space and one can thus make a geometric realisation $|X|$. This space is supposed to have a circle action. I suppose my question should be:</p> <p>How is the hocolim of $X$ over $\Delta^{op}$ related to the hocolim of $X$ over $\Lambda^{op}$.</p> http://mathoverflow.net/questions/24487/homotopy-colimits-of-cyclic-spaces/24507#24507 Answer by David Ben-Zvi for Homotopy colimits of cyclic spaces David Ben-Zvi 2010-05-13T13:56:11Z 2010-05-13T13:56:11Z <p>The homotopy theory of cyclic spaces is equivalent to that of spaces over $BS^1$ (Dwyer-Hopkins-Kan). The colimit over the simplicial category is as you say a space $X$ with $S^1$ action, and the colimit over the cyclic category is the quotient (Borel construction) $X/S^1$ as a space over $BS^1$.</p>