Mal'cev, Protomodular, Homological and Semi-Abelian Categories

Preface Metatheorems 0.1 The Yoneda embedding 0.2 Pointed categories 1 Intrinsic centrality 1.1 Spans and relations 1.2 Unital categories 1.3 Cooperating and central morphisms 1.4 Commutative objects 1.5 Symmetrizable morphisrns 1.6 Regular unital categories 1.7 Associated abelian object 1.8 Strongly unital categories 1.9 Gregarious objects 1.10 Linear and additive categories 1.11 Antilinear and antiadditive categories 1.12 Complemented subobjects 2 Mal'cev categories 2.1 Slices, coslices and points 2.2 Mal'cev categories 2.3 Abelian objects in Mal'cev categories 2.4 Naturally Mal'cev categories 2.5 Regular Mal'cev categories 2.6 Connectors in Mal'cev categories 2.7 Connector and cooperator 2.8 Associated abelian object and commutator 2.9 Protoarithmetical categories 2.10 Antilinear Mal'cev categories 2.11 Abelian groupoids 3 Protomodular categories 3.1 Definition and examples 3.2 Normal subobjects 3.3 Couniversal property of the product 3.4 Groupoids, protomodularity and normality 4 Homological categories 4.1 The short five lemma 4.2 The nine lemma 4.3 The Noether isomorphism theorems 4.4 The snake lemma 4.5 The long exact homology sequence 4.6 Examples of homological categories 5 Semi-abelian categories 5.1 Definition and examples 5.2 Semi-direct products 5.3 Semi-associative Mal'cev varieties 6 Strongly protomodular categories 6.1 Centrality and normality 6.2 Normal subobjects in the fibres 6.3 Normal functors 6.4 Strongly protomodular categories 6.5 A counterexample 6.6 Connector and cooperator 7 Essentially affine categories 7.1 The fibration of points 7.2 Essentially affine categories 7.3 Abelian extensions Appendix A.1 Algebraic theories A.2 Internal relations A.3 Internal groupoids A.4 Variations on epimorphisms A.5 Regular and exact categories A.6 Monads A.7 Fibrations Classification table of the fibration of points Bibliography Index of symbols Index of definitions