nLab diagram chasing

Idea

Diagram chasing is a common technique in homological algebra for proving properties of and constructing morphisms in abelian categories, where one traces elements in various ways around commutative diagrams.

Examples

Many basic lemmas in homological algebra, such as the five lemma, the 3x3 lemma and the snake lemma, are typically proven by diagram chases. See for instance the proof at five lemma or any book on homological algebra.

The salamander lemma can sometimes be used to give more conceptual proofs.

Diagram chases in general abelian categories

There are at least five approaches to performing diagram chases in general abelian categories (not assumed to be concrete like Ab or $R$Mod):

1. Directly use the universal properties of the objects involved (kernels, cokernels etc.).
2. Use equivalence classes of generalized elements to mimic naive (element-based) proofs, as detailed at element in an abelian category. These can only be used to show properties of already-given morphisms and not to construct morphisms (for example the connecting homomorphism of a short exact sequence) by giving their action on elements.
3. Use genuine generalized elements, see element in an abelian category. With these, the naive way of constructing morphisms $A \to B$ (first showing that for every $a \in A$ there exists a certain $b \in B$, then showing that this is independent of any choices made) carries over to the general setting.
4. Interpret constructive element-based proofs written in the fragment of regular logic using categorical semantics, see internal logic. The naive way of constructing morphisms works because the “axiom of unique choice” is valid in abelian categories (see Johnstone, Prop. D1.3.12).
5. Use the Freyd-Mitchell embedding theorem.

Related concepts

• salamander lemma

• element in an abelian category

• internal language

• Freyd-Mitchell embedding theorem

• diagram chasing lemmas - contents

References

• George Bergman, A note on abelian categories – translating element-chasing proofs, and exact embedding in abelian groups (1974) [pdf, pdf]

• Peter Johnstone, Sketches of an Elephant (2002)

• Daniel Murfet, Diagram Chasing in Abelian Categories (2006) (pdf)

• George Bergman, On diagram-chasing in double complexes, Theory and Applications of Categories 26 (2012) 60-96 [arXiv:1108.0958, tac:26-03, pdf]

  (introducing the salamander lemma)