Reference request regarding faithfully exact functors between abelian categories

Question

I am looking for a reference for the following result (or any subresult) in any book or notes:

Lemma. Let $F:\mathcal{A}\to\mathcal{B}$ be an exact functor between abelian categories. The following are equivalent:

1. $F$ reflects exact sequences, i.e., for every sequence $A\to B\to C$ in $\mathcal{A}$ such that $F(A)\to F(B)\to F(C)$ is exact, also $A\to B\to C$ is exact,
2. For every complex $A\to B\to C$ in $\mathcal{A}$ such that $F(A)\to F(B)\to F(C)$ is exact, also $A\to B\to C$ is exact,
3. $F$ reflects short exact sequences, i.e., for every sequence $A\to B\to C$ in $\mathcal{A}$ such that $0\to F(A)\to F(B)\to F(C)\to 0$ is exact, also $0\to A\to B\to C\to 0$ is exact,
4. For every complex $A\to B\to C$ in $\mathcal{A}$ such that $0\to F(A)\to F(B)\to F(C)\to 0$ is exact, also $0\to A\to B\to C\to 0$ is exact,
5. $F$ reflects zero objects (i.e., $\operatorname{Ker}F=0$),
6. $F$ reflects zero morphisms,
7. $F$ is faithful,
8. $F$ reflects monomorphisms and epimorphisms, and
9. $F$ reflects isomorphisms (i.e., $F$ is conservative).

The name I know for such a functor is a faithfully exact functor, but I don't know any books or sources to look for and Google doesn't help. (It's not that I don't know how to prove the result, because I do, but rather that I want to cite it and I'm unable to find it anywhere). The most similar thing I found is T. Ishikawa, Faithfully exact functors and their applications to projective modules and injective modules, Theorem 1.1. However, this result is stated only for modules over a ring (one could leverage Mitchell's embedding, but still, there must be an easy direct proof in the literature), and properties 2, 3, 4, 8, and 9 from the lemma above are missing in Ishikawa's Theorem 1.1

Answer 1

I found a reference that proves some of the subresults in the question. I also left two comments below the question about places I searched that did NOT contain this result. The reference I found was K. Shimizu. On unimodular finite tensor categories. Int Math Res Notices (2016). Of course, this result was known much earlier but it's hard to find a reference.

Lemma 5.1 in the arxiv version covers your (5), (7), and (9). I think the result numbering might be different between the arxiv version and the published version.

Another good reference is Mitchell's book Theory of Categories. Theorem II.7.1, which he credits to Freyd, shows that (7) implies (8), (6), and (1)-(4). In Proposition II.7.2, he proves that (5) implies (7). In Exercise 9 on page 77 he proves that (1) implies (7).

I am sorry I couldn't find a reference with all the pieces you asked for. I'm amazed that none of the standard references had this result. I encourage you to write up a proof and include it as an appendix to your next paper. That would do a service to the field and also probably give you lots of citations, as is happening with Shimizu.

EDIT: This thread motivated me to ask a question about how to find Freyd's thesis. And, that quickly revealed the thesis. Freyd calls functors that satisfy (6) "embeddings" and his Theorem 9 proves (6) and (1) are equivalent. His Proposition 12 proves (5) is equivalent to those two as well.