Cech nerve as homotopy colimit? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T09:36:35Z http://mathoverflow.net/feeds/question/87814 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87814/cech-nerve-as-homotopy-colimit Cech nerve as homotopy colimit? dhagbert 2012-02-07T16:42:31Z 2012-02-07T21:52:05Z <p>Given a category <code>$\mathcal{C}$</code> with a notion of covering <code>$\{ U_{i} \rightarrow X \}$</code> for an object <code>$X$</code> (say <code>$\mathcal{C}$</code> is a Grothendieck site), we can form the Cech nerve</p> <p><code>$$\cdots \coprod_{i}{U_{ijk}} \overrightarrow{\overrightarrow{\rightarrow}}\coprod_{i}{U_{ij}} \overrightarrow{\rightarrow} \coprod_{i} U_{i}$$</code></p> <p>(In the notation, I've suppressed degeneracy maps going from right to left.) </p> <p>This can be viewed in two ways. 1. As a simplicial object in simplicial presheaves, by considering each <code>$U_{i}$</code> as a simplicial presheaf constant in the simplicial direction. I'll denote that by <code>$U_{\bullet}$</code>. 2. As a simplicial object in presheaves and hence a simplicial presheaf. I'll denote that by <code>$\check{U}_{\bullet}$</code>. </p> <p>The latter <code>$\check{U}_{\bullet}$</code> can be shown to be level-wise weakly equivalent to <code>$colim(\coprod_{i}{U_{ij}} \overrightarrow{\rightarrow} \coprod_{i} U_{i})$</code>, where we consider this as a simplicial presheaf constant in the simplicial direction. </p> <p>On the other hand, we could compute <code>$hocolim(U_{\bullet})$</code>, and from things I read, this is supposed to be identified with/weakly equivalent to <code>$\check{U}_{\bullet}$</code>. One reason I am having difficulty seeing this is that I don't really understand <code>$hocolim(U_{\bullet})$</code>. Since each object in $U_{\bullet}$ is cofibrant, I would guess I could just take the usual colimit, but this seems to produce something constant in the simplicial direction, which is clearly wrong.</p> <p>So the question is:</p> <p>How to see if there is a weak equivalence <code>$hocolim(U_{\bullet}) \rightarrow \check{U}_{\bullet}$</code>?</p> <p>Probably if I read through the many pages of material suggested in the comments to <a href="http://mathoverflow.net/questions/87658/computing-homotopy-colimits-in-a-nice-simplicial-model-category" rel="nofollow">this question</a>, I'd be able to figure this out. But a more direct answer would make that reading more fruitful for me, I think. At least, pointing out what things I need to know to figure this out would help.</p> http://mathoverflow.net/questions/87814/cech-nerve-as-homotopy-colimit/87824#87824 Answer by David Carchedi for Cech nerve as homotopy colimit? David Carchedi 2012-02-07T17:56:25Z 2012-02-07T21:52:05Z <p>More generally, let $X_\cdot$ be a simplicial presheaf. As such, we can consider it as a simplicial object in presheaves, which in particular may be thought of as a simplicial object in simplicial presheaves $X'_\cdot.$ So we have:</p> <p>$$X_\cdot:\Delta^{op} \to Set^{C^{op}}$$ and <code>$$X'_\cdot =\left( \mspace{3mu} \cdot \mspace{3mu}\right)^{(id)} \circ X_\cdot:\Delta^{op} \to Set_{\Delta}^{C^{op}}$$</code> where $$\left( \mspace{3mu} \cdot \mspace{3mu}\right)^{(id)}:Set^{C^{op}} \to Set_{\Delta}^{C^{op}}$$ is the evident inclusion of presheaves into simplicial presheaves.</p> <p>The homotopy colimit of $X'_\cdot$ in simplicial presheaves is computed "object-wise". Hence, for all <code>$c \in C$</code> we have $$hocolim\left(X'\right)(C)=hocolim \left( X'\left(C\right)\right).$$ The right-hand side is the homotopy colimit of a simplicial object in simplicial sets, and can be computed by taking the diagonal of the corresponding bisimplicial set. But the diagonal of <code>$X'\left(C\right)_\cdot$ is simply $X\left(C\right)_\cdot$</code> since <code>$X'_\cdot$</code> in constant in one simplicial direction. Hence $$hocolim\left(X'\right)=X.$$</p>