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 $A_{S.}$

Now, as I gather from nLab's article on sheafification and Kashiwara and Schapiro's Categories and Sheafs, if the category $\mathcal C$ is such that:

  • Small projective and small inductive limits exist,
  • Small filtrant limits are exact,
  • The IPC property holds,

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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.