The salamander lemma is a lemma in homological algebra from which a number of theorems quickly drop out, some of the more famous ones include the snake lemma, the five lemma, the sharp 3×3 lemma (generalized nine lemma), etc. However, the only proof I've ever seen of this lemma is by a diagram chase after reducing to R-mod by using mitchell's embedding theorem. Is there an elementary proof of this lemma by universal properties in an abelian category (I don't know if we can weaken the requirements past an abelian category)?
If you haven't heard of the salamander lemma, here's the relevant paper.
And here's an article on it by our gracious administrator, Anton Geraschenko: Click!
Also, small side question, but does anyone know a good place to find some worked-out diagram-theoretic proofs that don't use mitchell and prove everything by universal property? It's not that I have anything against doing it that way (it's certainly much faster), but I'd be interested to see some proofs done without it, just working from the axioms and universal properties.
PLEASE NOTE THE EDIT BELOW
EDIT: Jonathan Wise posted an edit to his answer where he provided a great proof for the original question (doesn't use any hint of elements!). I noticed that he's only gotten four votes for the answer, so I figured I'd just bring it to everyone's attention, since I didn't know that he'd even added this answer until yesterday. The problem is that he put his edit notice in the middle of the text without bolding it, so I missed it entirely (presumably, so did most other people).
Best Answer
There's a proof of the snake lemma without elements (a non-elementary proof?) on my (old) website.
http://math.stanford.edu/~jonathan/papers/snake.pdf
Edit: I added a section about the salamander lemma.
Much later edit: As Charles Rezk points out below, my proof of the salamander lemma is correct only in a special case. I will correct the proof when I find the tex file.
What makes working with elements in an abelian category easier than working with objects is that elements of the target of an epimorphism can be lifted to the source. If your abelian category has enough projectives, then a proof with elements can usually be adapted to one without elements by replacing each element by a surjection from a projective object. If you don't have enough projectives, you can still get by without elements. You have to replace the concept of "element" with "epimorphism from something"; then every "element" can still be lifted by passing to a more refined epimorphism.
This is just code for working locally in the topology generated by epimorphisms. (That it is a topology is implied by AB2.) Since there are always enough injective sheaves of abelian groups, this gives an exact embedding of any abelian category in an abelian category with enough injectives (or, if we work with "cotopologies", a category with enough projectives). This permits one to apply the simpler approach outlined above (using projectives) rather than "pro-epimorphisms".
Once one has an embedding in a Grothendieck abelian category (the category of sheaves of abelian groups always is one), it is not much further to a proof of Mitchell's embedding theorem anyway.