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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?

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