Sheafification of Arens-Michael algebra-valued presheaves - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T02:54:34Z http://mathoverflow.net/feeds/question/75007 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/75007/sheafification-of-arens-michael-algebra-valued-presheaves Sheafification of Arens-Michael algebra-valued presheaves Rodrigo Vargas 2011-09-09T15:11:00Z 2011-09-09T15:11:00Z <p>Let $\mathcal A$ be the category of Arens-Michael algebras, that is, projective limits of Banach algebras. Since $\mathcal A$ is a concrete category, an $\mathcal A$-valued presheaf $A$ admits a set-valued sheafification $A_{S.}$ I would like to know if there is a good way to associate an $\mathcal A$-valued sheaf to $A_{S.}$ </p> <p>Now, as I gather from nLab's article on sheafification and Kashiwara and Schapiro's <em>Categories and Sheafs,</em> if the category $\mathcal C$ is such that:</p> <ul> <li>Small projective and small inductive limits exist,</li> <li>Small filtrant limits are exact,</li> <li>The IPC property holds,</li> </ul> <p>then $\mathcal C$-valued presheaves admit a $\mathcal C$-valued sheafification. Thus, my question is: does the category $\mathcal A$ of Arens-Michael algebras have these properties? And, in the negative case, is there a category of topological algebras, containing $\mathcal A$ as a subcategory, having these properties?</p>