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# Homology, Homotopy and Applications

## Volume 18 (2016)

### Number 2

### Duality for diagram chasing a la Mac Lane in non-abelian categories

Pages: 85 – 106

DOI: https://dx.doi.org/10.4310/HHA.2016.v18.n2.a5

#### Author

#### Abstract

In this paper we provide an axiomatic analysis of the classical diagram chasing method of Mac Lane, that relies on duality and chasing elements of pointed sets, which allows one to generalize this method from abelian categories to non-abelian ones. Among the examples where the generalized method can be used are all modular semiexact categories in the sense of Grandis (which include all Puppe–Mitchell exact categories) and sequentiable categories in the sense of Bourn (which include all semi-abelian categories, and in particular, the categories of group-like structures). The method turns out to be closely related to the essential features of these categories. At the same time, in some sense, it simplifies the usual proofs of some of the standard diagram lemmas in them.

#### Keywords

$3 \times 3$ lemma, abelian category, diagram lemma, exact sequence, five lemma, homological algebra, homological category, modular semiexact category, pointed set, pointed subobject functor, protomodular category, regular category, sequentiable category, subobject chasing

#### 2010 Mathematics Subject Classification

18A20, 18A32, 18C99, 18D99, 18E10, 18G50, 20J15, 55U30

Published 29 November 2016