How can I prove that sheaffication functor is exact? (equally it preserves kernels, because it's left adjoint) Every possible source I can find states somethink like ,,proving that sheaffication functor preserves kernels requires more knowledge than proving that it preserves cokernels and thus the proof is omitted." Thank you for all your answers.
2026-03-30 20:36:32.1774902992
Sheaffication functor is exact
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The sheafification functor, over a small site, preserves finite limits (in particular kernels) as a result of being given by small filtered colimits, and because filtered colimits commute with finite limits. See Theorem 1 in section III.5 of Mac Lane and Moerdijk's Sheaves in Geometry and Logic.