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Working through Weibel's Introduction to Homological Algebra, I am frequently unsure when it is acceptable to prove results using diagram-chasing and elements, and when Weibel has in mind a category-theoretic proof.

For example, Weibel 1.3.1 asks for a proof of the claim:

If $0 \rightarrow A_{\bullet} \xrightarrow{f} B_{\bullet} \xrightarrow{g} C_{\bullet} \rightarrow 0$ is a short exact sequence of chain complexes, then, whenever two of the three complexes $A_{\bullet}$,$B_{\bullet}$,$C_{\bullet}$ are exact, so is the third.

A complete solution using diagram chasing and elements is available here. However, the wording of the question, and the fact that Exercise 1.3.2 applies the result in a general Abelian category, suggest that a categorical proof is desired. How would such a proof go?

I understand that Freyd-Mitchell's Embedding Theorem implies that any theorems of a certain form which can be proven for R-modules using diagram chasing automatically hold in any Abelian category (Rotman calls this the Metatheorem). I suspect that this embedding should be "constructive" with respect to proofs.

My question is: Does there exist a concrete set of rules for lifting a proof by chasing elements into one by the equivalent category-theoretic properties? For example, how would one go about converting the following small excerpts in the diagram-chasing solution into language suitable for any Abelian category?

Assume $B, C$ are exact and let $x \in \ker d \subseteq A_n$. By commutative square we have $$d(f_n(x)) = f_{n-1}(d(x)) = f_{n-1}(0) = 0$$ Thus $f_n(x) \in \ker d \subseteq B_n$, which is exactly $\operatorname{im} d \subseteq B_n$ by exactness of $B$, and there is some $b \in B_{n+1}$ such that $d(b) = f_n(x)$.

. . .

For $c \in C_{n+2}$ with $d(c) = g_{n+1}(b)$, since $g_{n+2}$ is surjective, there is some $b' \in B_{n+2}$ such that $g_{n+2}(b') = c$.

Consider $b - d(b')$. We have by commutative square that $b - d(b') \in \ker g_{n+1}$: $$g_{n+1}(b - d(b')) = g_{n+1}(b) - g_{n+1}(d(b')) = g_{n+1}(b) - d(g_{n+2}(b')) = g_{n+1}(b) - g_{n+1}(b) = 0$$

I think I understand how some small parts of this conversion would work. For example, you can replace statements like $\ker d = \operatorname{im} d$ with an isomorphism of their embeddings as submodules. But the step which constructs a difference of elements doesn't obviously seem to have a categorical equivalent.

Edit: As Pedro mentioned in a comment, the topic of converting element-proofs in $R\mathbf{-mod}$ or $\mathbf{Ab}$ into categorical proofs is discussed in these notes of Bergman, where he develops exactly this kind of dictionary of terms. The result seems somewhat more involved than I might hope for 1.3.1, so I'll leave open the restricted version of the question: is there an elementary categorical proof of Weibel 1.3.1 (stated above).

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  • $\begingroup$ You can prove the snake lemma and then the existence of the LES in any abelian category, and then prove the 5 lemma, etc. $\endgroup$ – Pedro Tamaroff Apr 17 '18 at 19:48
  • $\begingroup$ You are surely right, but Weibel places this exercise before any of that, which I suspect means it has a direct categorical proof. More generally, I'm curious about whether there exists a ``dictionary" between proof statements involving elements and their categorical counterparts. $\endgroup$ – C.D. Apr 17 '18 at 19:51
  • $\begingroup$ See here for that. In particular the notes of Bergman :) $\endgroup$ – Pedro Tamaroff Apr 17 '18 at 20:04
  • $\begingroup$ A first step would be to replace all inclusions with canonical morphisms making such and such diagrams commute ("thus $\mathrm{ker}d \subset B_n$" replaced with "we get a canonical morphism $\mathrm{ker}d \to B_n$ such that blabla commutes") - this will get you a long way $\endgroup$ – Max Apr 18 '18 at 15:46

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