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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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Ok. I think the answer is no. Sorry for disturbing. – thinker Dec 18 '11 at 16:17
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. – Mike Shulman Dec 20 '11 at 1:17

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

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Thanks. It is nice to know such general result. I still think that the answer for second question is no. – thinker Dec 18 '11 at 20:19

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