
A Foundation for PROPs, Algebras, and Modules
American Mathematical Society (Publisher)
Will be published approx. on 30. July 2015
Book
Hardback
311 pages
978-1-4704-2197-7 (ISBN)
Description
PROPs and their variants are extremely general and powerful machines that encode operations with multiple inputs and multiple outputs. In this respect PROPs can be viewed as generalizations of operads that would allow only a single output. Variants of PROPs are important in several mathematical fields, including string topology, topological conformal field theory, homotopical algebra, deformation theory, Poisson geometry, and graph cohomology. The purpose of this monograph is to develop, in full technical detail, a unifying object called a generalized PROP. Then with an appropriate choice of pasting scheme, one recovers (colored versions of) dioperads, half-PROPs, (wheeled) operads, (wheeled) properads, and (wheeled) PROPs.
Here the fundamental operation of graph substitution is studied in complete detail for the first time, including all exceptional edges and loops as examples of a new definition of wheeled graphs. A notion of generators and relations is proposed which allows one to build all of the graphs in a given pasting scheme from a small set of basic graphs using graph substitution. This provides information at the level of generalized PROPs, but also at the levels of algebras and of modules over them. Working in the general context of a symmetric monoidal category, the theory applies for both topological spaces and chain complexes in characteristic zero.
This book is useful for all mathematicians and mathematical physicists who want to learn this new powerful technique.
Here the fundamental operation of graph substitution is studied in complete detail for the first time, including all exceptional edges and loops as examples of a new definition of wheeled graphs. A notion of generators and relations is proposed which allows one to build all of the graphs in a given pasting scheme from a small set of basic graphs using graph substitution. This provides information at the level of generalized PROPs, but also at the levels of algebras and of modules over them. Working in the general context of a symmetric monoidal category, the theory applies for both topological spaces and chain complexes in characteristic zero.
This book is useful for all mathematicians and mathematical physicists who want to learn this new powerful technique.
More details
Series
Language
English
Place of publication
Providence
United States
Target group
Professional and scholarly
Dimensions
Height: 254 mm
Width: 178 mm
Weight
748 gr
ISBN-13
978-1-4704-2197-7 (9781470421977)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Persons
Donald Yau, The Ohio State University at Newark, OH, USA.
Mark W. Johnson, Pennsylvania State University Altoona, PA, USA.
Mark W. Johnson, Pennsylvania State University Altoona, PA, USA.
Content
Wheeled graphs and pasting schemes: Wheeled graphs
Special sets of graphs
Basic operations on wheeled graphs
Graph groupoids
Graph substitution
Properties of graph substitution
Generators for graphs
Pasting schemes
Well-matched pasting schemes
Generalized PROPs, algebras, and modules: Generalized PROPs
Biased characterizations of generalized PROPs
Functors of generalized PROPs
Algebras over generalized PROPs
Alternative descriptions of generalized PROPs
Modules over generalized PROPs
May modules over algebras over operads
Bibliography
Index
Special sets of graphs
Basic operations on wheeled graphs
Graph groupoids
Graph substitution
Properties of graph substitution
Generators for graphs
Pasting schemes
Well-matched pasting schemes
Generalized PROPs, algebras, and modules: Generalized PROPs
Biased characterizations of generalized PROPs
Functors of generalized PROPs
Algebras over generalized PROPs
Alternative descriptions of generalized PROPs
Modules over generalized PROPs
May modules over algebras over operads
Bibliography
Index