3 × 3 Lemma and Protomodularity

By Dominique Bourn. 2001

115 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ce04069ce9a3ccbb4d3fd09c44603b509d956c9b

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Citations in the corpus

Centrality and normality in protomodular categories (2001, unknown): “when there is an initial object 0 and the unique arrow 0 ֌ 1 is a monomorphism) the normal monomorphisms can be always produced from the equivalence relations [6].”
Higher Hopf formulae for homology via Galois Theory (2007, background): “Yet, already in this context the basic homological diagram lemma’s hold: the Snake Lemma, the 3× 3-Lemma, etc. [7]”
Central extensions in semi-abelian categories (2002, background): “It follows from Proposition 17 in [5] and from Proposition 3.”
Baer invariants in semi-abelian categories II: Homology (2004, unknown): “Recall that a chain complex C is a collection of morphisms (d n : C n , P C n−1 ) n∈Z such that d n @BULLETd n+1 = 0, for all n ∈ Z.”
On the second cohomology group in semi-abelian categories (2005, background): “Several results, which are classical in the category of groups, can be extended to any semi-abelian category: among them, let us mention the 3× 3 Lemma, the Snake Lemma [9], the long exact homology sequence associated with a proper chain complex, or the Stallings-Stammbach five term exact sequence a”
Topological semi-abelian algebras (2005, background): “They constitute a formal context in which all diagram lemmas of universal algebra are valid (see [10]), but also many properties characteristic of non-abelian situations: the Mal’cev axiom (see [15,14]), the protomodularity axiom (see [8]), the theory of commutators (see [11]), of semi-direct produc”