Proof that objects are colimits of generators - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T13:37:39Z http://mathoverflow.net/feeds/question/27219 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/27219/proof-that-objects-are-colimits-of-generators Proof that objects are colimits of generators Joey Hirsh 2010-06-06T02:57:40Z 2010-11-21T16:53:06Z <p>Suppose C is a category with small colimits, and G $\in$ ob(C) is a strong generator. (This means that $f: X \to Y$ is an isomorphism iff $f_{*}: Hom(G,X) \to Hom(G,Y)$ is a bijection.)</p> <blockquote> <p>How does one prove that every object X is a colimit of G (ie, a quotient of coproducts of G)?</p> </blockquote> <p>I think X $\simeq$ colim<sub>G --> X</sub> G. I can see there is a natural map e: colim<sub>G --> X</sub> G $\to$ X, and that the induced map $e_{*}: Hom(G,colim<sub>G --> X</sub> G) \to Hom(G, X)$ is surjective, but I cannot see that it map is injective.</p> http://mathoverflow.net/questions/27219/proof-that-objects-are-colimits-of-generators/27287#27287 Answer by Mike Shulman for Proof that objects are colimits of generators Mike Shulman 2010-06-06T23:19:49Z 2010-06-06T23:19:49Z <p>I think there are actually <em>three</em> possible things that you might be asking, but the answer to all of them is no. Suppose that G is a strong generator in a cocomplete category C. Then you can ask:</p> <ol> <li><p>Is every object X of C the colimit of G over the canonical diagram of shape $(G\downarrow X)$? (If so, then G is called <em>dense</em> in C.)</p></li> <li><p>Is every object <em>some</em> colimit of a diagram all of whose vertices are G? (If so, then G is called <em>colimit-dense</em> in C.)</p></li> <li><p>Is C the smallest subcategory of itself containing G and closed under colimits? (If so, then G is a <em>colimit-generator</em> of C.)</p></li> </ol> <p>The category of compact Hausdorff spaces is a counterexample to the first two. It is monadic over Set (the monad is the ultrafilter monad, aka Stone-Cech compactification of the discrete topology), and hence cocomplete, and the one-point space is a strong generator. But any colimit of a diagram consisting entirely of 1-point spaces must be in the image of the free functor from Set (the one-point space being its own Stone-Cech compactification), and hence (since that functor is a left adjoint and preserves colimits) must be the free object on some set. However, not every compact Hausdorff space is the Stone-Cech compactification of a discrete set.</p> <p>As Todd pointed out in the comments, though, CptHaus is the colimit-closure of the one-point space, since every object is a coequalizer of maps between free ones (because the category is monadic over Set); thus it isn't a counterexample to the third question. Counterexamples to the third question are actually much harder to come by, and in fact if you assume additionally that C has finite limits and is "extremally well-copowered", then it is true that any strong generator is a colimit-generator. I think there's a proof of this somewhere in Kelly's book "Basic concepts of enriched category theory," and a brief version can be found <a href="http://www.math.uchicago.edu/~shulman/exposition/generators/generators.pdf" rel="nofollow">here</a>. However, without these assumptions, one can cook up ugly and contrived counterexamples, such as example 4.3 in the paper "Total categories and solid functors" by Borger and Tholen.</p> http://mathoverflow.net/questions/27219/proof-that-objects-are-colimits-of-generators/46832#46832 Answer by Buschi Sergio for Proof that objects are colimits of generators Buschi Sergio 2010-11-21T16:53:06Z 2010-11-21T16:53:06Z <p>See cor. 4.4 in "Strong regular and dense generators" Tholen , Borger:</p> <p><a href="http://www.numdam.org/numdam-bin/fitem?id=CTGDC_1991__32_3_257_0" rel="nofollow">http://www.numdam.org/numdam-bin/fitem?id=CTGDC_1991__32_3_257_0</a></p>