Revisions to Some intuition behind the five lemma?

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edit approved Jul 14, 2013 at 20:00

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Slightly simplified, the five lemma states that if we have a commutative diagram (in, say, an abelian category)


A1 -> A2 -> A3 -> A4 -> A5
|     |     |     |     |
v     v     v     v     v
B1 -> B2 -> B3 -> B4 -> B5

where$$\require{AMScd} \begin{CD} A_1 @>>> A_2 @>>> A_3 @>>> A_4 @>>> A_5\\ @VVV @VVV @VVV @VVV @VVV\\ B_1 @>>> B_2 @>>> B_3 @>>> B_4 @>>> B_5 \end{CD} $$ where the rows are exact and the maps A_i -> B_i$A_i \to B_i$ are isomorphisms for i=1,2,4,5$i=1,2,4,5$, then the middle map A₃ -> B₃$A_3\to B_3$ is an isomorphism as well.

This lemma has been presented to me several times in slightly different contexts, yet the proof has always been the same technical diagram chase and no further intuition behind the statement was provided. So my question is: do you have some intuition when thinking about the five lemma? For instance, particular choices of the A_i, B_i$A_i, B_i$ which make it more transparent why the result should be true? Some analogy, heuristic, ...?

Slightly simplified, the five lemma states that if we have a commutative diagram (in, say, an abelian category)


A1 -> A2 -> A3 -> A4 -> A5
|     |     |     |     |
v     v     v     v     v
B1 -> B2 -> B3 -> B4 -> B5

where the rows are exact and the maps A_i -> B_i are isomorphisms for i=1,2,4,5, then the middle map A₃ -> B₃ is an isomorphism as well.

This lemma has been presented to me several times in slightly different contexts, yet the proof has always been the same technical diagram chase and no further intuition behind the statement was provided. So my question is: do you have some intuition when thinking about the five lemma? For instance, particular choices of the A_i, B_i which make it more transparent why the result should be true? Some analogy, heuristic, ...?

Slightly simplified, the five lemma states that if we have a commutative diagram (in, say, an abelian category)

$$\require{AMScd} \begin{CD} A_1 @>>> A_2 @>>> A_3 @>>> A_4 @>>> A_5\\ @VVV @VVV @VVV @VVV @VVV\\ B_1 @>>> B_2 @>>> B_3 @>>> B_4 @>>> B_5 \end{CD} $$ where the rows are exact and the maps $A_i \to B_i$ are isomorphisms for $i=1,2,4,5$, then the middle map $A_3\to B_3$ is an isomorphism as well.

This lemma has been presented to me several times in slightly different contexts, yet the proof has always been the same technical diagram chase and no further intuition behind the statement was provided. So my question is: do you have some intuition when thinking about the five lemma? For instance, particular choices of the $A_i, B_i$ which make it more transparent why the result should be true? Some analogy, heuristic, ...?