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 3x3 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).

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 3x3 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.

Bounty Edit

PLEASE NOTE THE EDIT BELOW

EDIT:I'm offering a bounty for Jonathan Wise posted an edit to his answer where he provided a great proof of this lemma that does not use generalized elements or mitchell's embedding theorem. It should proceed using only universal properties and for the axioms original question (doesn't use any hint of an abelian categoryelements!). I am noticed that he's only offering gotten four votes for the bounty because I'm interested in seeing how to do this proof without resorting to some kind of diagram chase. I can prove it just fine using elementsanswer, so don't waste your time if you're I figured I'd just going bring it to do everyone's attention, since I didn't know that he'd even added this answer until yesterday.

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 3x3 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.

Bounty Edit: I'm offering a bounty for a proof of this lemma that does not use generalized elements or mitchell's embedding theorem. It should proceed using only universal properties and the axioms of an abelian category. I am only offering the bounty because I'm interested in seeing how to do this proof without resorting to some kind of diagram chase. I can prove it just fine using elements, so don't waste your time if you're just going to do that.

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 3x3 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.

Bounty Edit: I'm offering a bounty for a proof of this lemma that does not use generalized elements or mitchell's embedding theorem. It should proceed using only universal properties and the axioms of an abelian category.

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 3x3 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 moderatoradministrator, 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.

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 3x3 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 moderator, 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. Not ? 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.

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 3x3 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 moderator, 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. 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.