![Category theory / Lemmas / Snake lemma / Five lemma / Exact sequence / Natural transformation / Commutative diagram / Spectral sequence / Homology / Abstract algebra / Homological algebra / Mathematics Category theory / Lemmas / Snake lemma / Five lemma / Exact sequence / Natural transformation / Commutative diagram / Spectral sequence / Homology / Abstract algebra / Homological algebra / Mathematics](https://www.pdfsearch.io/img/0776f30cc79fc409c152338b9268a5db.jpg) Date: 2012-02-08 09:45:00Category theory Lemmas Snake lemma Five lemma Exact sequence Natural transformation Commutative diagram Spectral sequence Homology Abstract algebra Homological algebra Mathematics | | Theory and Applications of Categories, Vol. 26, No. 3, 2012, pp. 60–96. ON DIAGRAM-CHASING IN DOUBLE COMPLEXES GEORGE M. BERGMAN Abstract. We construct, for any double complex in an abelian category, certain “short-
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