most recent 30 from http://mathoverflow.net 2013-05-21T07:57:18Z http://mathoverflow.net/feeds/question/89568 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/89568/is-sheafification-functor-exact Hiro 2012-02-26T10:06:27Z 2012-02-26T20:41:37Z

<p>I know that sheafification functor from the category of abelian presheaves on $C$ to the category of abelian sheaves on $C$. Here, $C$ is a category with Grothendieck pretopology.</p> <p>My question is:</p> <p>How about the sheafification functor from the category of presheaves of "sets" on $C$ to the category of sheaves of "sets" on $C$?</p> <p>Is this an exact functor? (i.e. preserving finite limits and finite colimits?)</p> <p>If so, how can one prove it?</p> <p>In fact, I want to know whether sheafification functor preserves cartesian products or not.</p> <p>Please give me any advice.</p>

http://mathoverflow.net/questions/89568/is-sheafification-functor-exact/89574#89574 Martin Brandenburg 2012-02-26T10:44:33Z 2012-02-26T10:44:33Z

<p>Preservation of colimits is trivial from the adjunction, preservation of finite limits comes down - using the usual construction - to the fact that finite limits commute with filtered colimits. The latter holds in every algebraic category, in particular in (Set). </p>

http://mathoverflow.net/questions/89568/is-sheafification-functor-exact/89603#89603 Andreas Blass 2012-02-26T20:41:37Z 2012-02-26T20:41:37Z

<p>Martin has probably answered everything Hiro meant to ask, but, since "whether sheafification functor preserves cartesian products or not" didn't explicitly say <em>finite</em> products, let me add that sheafification will not in general preserve infinite products. Intuitively, the reason is that a section of a product of sheafifications is a family of locally defined sections of the original presheaves, and there might not be a single covering over which all those local sections are simultaneously available.</p>