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Is the inclusion of the category of sheaves into the category of presheaves monadic? If not, then maybe it preserves directed colimits?

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  • $\begingroup$ Ok. I think the answer is no. Sorry for disturbing. $\endgroup$
    – thinker
    Commented Dec 18, 2011 at 16:17
  • $\begingroup$ The answer to the second question is "not in general", but it is true that it will always be accessible, i.e. preserve sufficiently-highly-filtered colimits. For some purposes that is good enough. $\endgroup$
    – Mike Shulman
    Commented Dec 20, 2011 at 1:17

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The answer (to the first question) is yes: reflections are always monadic, and the associated monad is idempotent.

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  • $\begingroup$ Thanks. It is nice to know such general result. I still think that the answer for second question is no. $\endgroup$
    – thinker
    Commented Dec 18, 2011 at 20:19

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