Mal'cev, Protomodular, Homological and Semi-Abelian CategoriesThe purpose of the book is to take stock of the situation concerning Algebra via Category Theory in the last fifteen years, where the new and synthetic notions of Mal'cev, protomodular, homological and semi-abelian categories emerged. These notions force attention on the fibration of points and allow a unified treatment of the main algebraic: homological lemmas, Noether isomorphisms, commutator theory. The book gives full importance to examples and makes strong connections with Universal Algebra. One of its aims is to allow appreciating how productive the essential categorical constraint is: knowing an object, not from inside via its elements, but from outside via its relations with its environment. The book is intended to be a powerful tool in the hands of researchers in category theory, homology theory and universal algebra, as well as a textbook for graduate courses on these topics. |
Contents
Metatheorems | 1 |
Chapter 1 | 11 |
A | 99 |
Chapter 2 | 125 |
R | 157 |
Diagram 225 | 173 |
Chapter 3 | 229 |
V | 245 |
Kv | 298 |
Qu | 303 |
1 | 305 |
an+1 | 313 |
1 | 314 |
Chapter 5 | 319 |
Chapter 6 | 345 |
Chapter 7 | 371 |
fX ƒX | 251 |
Chapter 4 | 273 |
k | 283 |
1 | 297 |
Appendix | 399 |
Classifying properties of the fibration of points | 466 |
| 467 | |
Other editions - View all
Mal'cev, Protomodular, Homological and Semi-Abelian Categories Francis Borceux,Dominique Bourn No preview available - 2004 |
Mal'cev, Protomodular, Homological and Semi-Abelian Categories Francis Borceux,Dominique Bourn No preview available - 2010 |
Common terms and phrases
abelian group abelian object admit a connector antiadditive antilinear arrow axioms category Ɛ category of sets category with finite central morphism cocomplete coequalizer cokernel composite conditions are equivalent consider diagram cooperator coproduct Corollary d₁ defined definition diagonal Equiv(E equivalence of categories equivalence relation exact sequence Example factorization fibration of points fibre Pty finite limits finitely complete following conditions follows at once forgetful functor given groupoid homological category implies inverse image functor kernel pair Let Ɛ magmas Mal'cev operation metatheorem 0.1.3 monad monomorphism morphism f naturally Mal'cev normal subobjects notion pointed category pointed sets Proposition protomodular category pullback pushout reflects isomorphisms reflexive relation regular category regular epimorphism semi-abelian short five lemma slice category split epimorphisms strongly epimorphic strongly protomodular strongly unital symmetrizable T-algebras Theorem undiscrete unique unital category yields zero morphism zero object
References to this book
Categories in Algebra, Geometry and Mathematical Physics: Conference and ... Alexei Davydov No preview available - 2007 |
Categories in Algebra, Geometry and Mathematical Physics: Conference and ... Alexei Davydov No preview available - 2007 |