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# Homology, Homotopy and Applications

## Volume 14 (2012)

### Number 2

### 3 × 3 lemma for star-exact sequences

Pages: 1 – 22

DOI: https://dx.doi.org/10.4310/HHA.2012.v14.n2.a1

#### Authors

#### 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.

#### Keywords

3 × 3 lemma, subtractive category, Goursat category, homological category, star-regular category, ideal of null morphisms, quasi-pointed category, sequentiable category, non- abelian homological algebra

#### 2010 Mathematics Subject Classification

08B05, 18Cxx, 18E10, 18G50

Published 4 December 2012