Does anyone understand the notation in this equation for the sheafification of a presheaf on a site? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T00:35:56Z http://mathoverflow.net/feeds/question/71319 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/71319/does-anyone-understand-the-notation-in-this-equation-for-the-sheafification-of-a Does anyone understand the notation in this equation for the sheafification of a presheaf on a site? Mario Carrasco 2011-07-26T14:21:23Z 2011-07-26T15:41:27Z <p>Hi there, I'm trying to sheafify a constant presheaf on a site, I went to <a href="http://ncatlab.org/nlab/show/sheafification" rel="nofollow">http://ncatlab.org/nlab/show/sheafification</a>, but can't understand the notation in the equation for W (in the proof for existence)</p> <p>W = {S(U_i) := lim... }. </p> <p>Can anyone help me figure out the notation? With respect to what is the limit taken? I'm still trying to figure out the whole sheafification process but so far I have like 2 or 3 books on the subject and none of them has been exactly helpful or straightforward for my liking, I need this result for a proof I'm working on and am not exactly an expert in Topos theory, I just need this result, hope you guys can help.</p> http://mathoverflow.net/questions/71319/does-anyone-understand-the-notation-in-this-equation-for-the-sheafification-of-a/71324#71324 Answer by Urs Schreiber for Does anyone understand the notation in this equation for the sheafification of a presheaf on a site? Urs Schreiber 2011-07-26T15:41:27Z 2011-07-26T15:41:27Z <p>The $\lim_\to$ in that formula denotes the <em>colimit</em> over that diagram of two parallel morphisms right after it. So it's the <em>coequalizer</em> of these two morphisms. This is just a very explicit way (or maybe a very implicit way? :-) to write out the <em>sieve</em> that corresponds to a given <em>cover</em> in the site (it's the presheaf that sends any test object to the set of ways of homming that test object into the object being covered that factor through the cover)</p> <p>As the text around the formula meant to indicate (I have now tried to clarify this further in the entry) the details for this are discussed (in quite some pedestrian detail, I think) at the entry on sieves</p> <p><a href="http://ncatlab.org/nlab/show/sieve" rel="nofollow">http://ncatlab.org/nlab/show/sieve</a></p> <p>and that on sheaves</p> <p><a href="http://ncatlab.org/nlab/show/sheaf" rel="nofollow">http://ncatlab.org/nlab/show/sheaf</a></p> <p>Probably such questions on nLab entries are best posted (and most welcome) over at the nForum</p> <p><a href="http://www.math.ntnu.no/~stacey/Mathforge/nForum/" rel="nofollow">http://www.math.ntnu.no/~stacey/Mathforge/nForum/</a></p>