A faithfully flat functor from an abelian category to another abelian category is an additive preserves and reflects (short) exact sequences. What is an example of a functor that reflects (short) exact sequences but does not preserve them? Also what is such a functor called? (A faithful functor means something else - injective on Hom sets.)
2026-03-27 16:52:42.1774630362
A non-flat functor that reflects exact sequences
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You could take the direct sum of a functor that preserves and reflects exact sequences and a functor that is not exact.
For example, take the functor $$F(A) = A\oplus\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}/2\mathbb{Z},A)$$ from abelian groups to abelian groups.