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