Do h-coequalizers and coproducts give all h-colimits? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T04:51:06Z http://mathoverflow.net/feeds/question/13174 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/13174/do-h-coequalizers-and-coproducts-give-all-h-colimits Do h-coequalizers and coproducts give all h-colimits? Chris Schommer-Pries 2010-01-27T19:58:26Z 2010-01-27T20:27:17Z <p>It is well known that if a category has all coequalizers and all (small) coproducts then in fact it has all (small) colimits. More important is the proof which shows that every colimit can be built by using coproducts and coequalizers. This implies that if a functor commutes with coproducts and coequalizers, then it must commute with all (small) colimits as well. </p> <blockquote> <p>Is there homotopical analog of this? If I have a functor which commutes with all small (homotopy) coproducts and all homotopy coequalizers, does it necessarily commute with all homotopy colimits in general?</p> </blockquote> <p>This question makes sense for general model categories, but I am particularly interested in the usual model structure on spaces. </p> http://mathoverflow.net/questions/13174/do-h-coequalizers-and-coproducts-give-all-h-colimits/13177#13177 Answer by Reid Barton for Do h-coequalizers and coproducts give all h-colimits? Reid Barton 2010-01-27T20:18:51Z 2010-01-27T20:27:17Z <p>There is an analogue, but one should replace coequalizers by geometric realizations (homotopy colimits over &Delta;<sup>op</sup>). If $F : I \to M$ is a diagram in a model category, one has</p> <p><code>$$\operatorname{hocolim}_I F = \operatorname{hocolim}_{k \in \Delta^{\operatorname{op}}} \coprod_{i_0 \to \cdots \to i_k \in I} F(i_0).$$</code></p> <p>For instance, see section 2 of <a href="http://www.math.uchicago.edu/~eriehl/hocolimits.pdf" rel="nofollow">http://www.math.uchicago.edu/~eriehl/hocolimits.pdf</a> for the simplicial model category case.</p>