Skip to main content
1 of 4
Armin Straub
  • 1.4k
  • 2
  • 18
  • 24

Some intuition behind the five lemma?

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 Ai -> Bi 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 Ai, Bi which make it more transparent why the result should be true? Some analogy, heuristic, ...?

Armin Straub
  • 1.4k
  • 2
  • 18
  • 24