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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
  • $\begingroup$ I can't believe you're doing this without pictures, at least in your writeup there are no pictures. Category theory results are highly visual. The app I'm writing is highly needed at this point in time. $\endgroup$ – BananaCats Category Theory App May 16 at 2:54

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