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Mathematics - Algebra | From Objects to Diagrams for Ranges of Functors

From Objects to Diagrams for Ranges of Functors

Series: Lecture Notes in Mathematics, Vol. 2029

Gillibert, Pierre, Wehrung, Friedrich

2011, X, 158 p. 19 illus.

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  • About this book

  • The book is centered on two statements: namely, CLL, and its main precursor, the Armature Lemma, which are results of category theory, with hard proofs, which appear here for the first time. Most of the book is aimed at applications outside category theory, and is thus written as a toolbox.
  • The results of the book illustrate how certain representation problems have counterexamples of different cardinalities such as aleph zero, one, two, and explain why.
  • CLL and the Armature Lemma have a wide application range, which we illustrate with examples in lattice theory, universal algebra, and ring theory. We also give pointers to solutions, made possible by our results, to previously intractable representation problems, with respect to various functors.
This work introduces tools from the field of category theory that make it possible to tackle a number of representation problems that have remained unsolvable to date (e.g. the determination of the range of a given functor). The basic idea is: if a functor lifts many objects, then it also lifts many (poset-indexed) diagrams.

Content Level » Research

Keywords » 18A30; 18A25; 18A20; 18A35 - Category - condensate - larder - lifter

Related subjects » Algebra - Mathematics

Table of contents 

1 Background.- 2 Boolean Algebras Scaled with Respect to a Poset.- 3 The Condensate Lifting Lemma (CLL).- 4 Larders from First-order Structures.- 5 Congruence-Preserving Extensions.- 6 Larders from von Neumann Regular Rings.- 7 Discussion.

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