3 × 3 lemma for star-exact sequences
(2012) Homology, Homotopy and Applications — Vol. 14, n° 2, p. 1-22 (2012)
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- Abstract
- A regular category is said to be normal when it is pointed and every regular epimorphism in it is a normal epimorphism. Any abelian category is normal, and in a normal category one can define short exact sequences in a similar way as in an abelian category. Then, the corresponding 3 × 3 lemma is equivalent to the so-called subtractivity, which in universal algebra is also known as congruence 0-permutability. In the context of non-pointed regular categories, short exact sequences can be replaced with “exact forks” and then, the corresponding 3 × 3 lemma is equivalent, in the universal algebraic terminology, to congruence 3-permutability; equivalently, regular categories satisfying such 3 × 3 lemma are precisely the Goursat categories. We show how these two seemingly independent results can be unified in the context of star-regular categories recently introduced in a joint work of A. Ursini and the first two authors.
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Gran, M., Janelidze, Z., & Rodelo, D. (2012). 3 × 3 lemma for star-exact sequences. Homology, Homotopy and Applications, 14(2), 1-22. https://doi.org/10.4310/HHA.2012.v14.n2.a1 (Original work published 2012)