Hardcover ISBN:  9781470421977 
Product Code:  SURV/203 
List Price:  $110.00 
MAA Member Price:  $99.00 
AMS Member Price:  $88.00 
Electronic ISBN:  9781470423476 
Product Code:  SURV/203.E 
List Price:  $110.00 
MAA Member Price:  $99.00 
AMS Member Price:  $88.00 

Book DetailsMathematical Surveys and MonographsVolume: 203; 2015; 311 ppMSC: Primary 18; Secondary 55; 13; 81;
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, halfPROPs, (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.ReadershipGraduate students and research mathematicians interested in algebraic geometry, algebraic topology, and category theory.

Table of Contents

Part 1. Wheeled graphs and pasting schemes

Chapter 1. Wheeled graphs

Chapter 2. Special sets of graphs

Chapter 3. Basic operations on wheeled graphs

Chapter 4. Graph groupoids

Chapter 5. Graph substitution

Chapter 6. Properties of graph substitution

Chapter 7. Generators for graphs

Chapter 8. Pasting schemes

Chapter 9. Wellmatched pasting schemes

Part 2. Generalized PROPs, algebras, and modules

Chapter 10. Generalized PROPs

Chapter 11. Biased characterizations of generalized PROPs

Chapter 12. Functors of generalized PROPs

Chapter 13. Algebras over generalized PROPs

Chapter 14. Alternative descriptions of generalized PROPs

Chapter 15. Modules over generalized PROPs

Chapter 16. May modules over algebras over operads


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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, halfPROPs, (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.
Graduate students and research mathematicians interested in algebraic geometry, algebraic topology, and category theory.

Part 1. Wheeled graphs and pasting schemes

Chapter 1. Wheeled graphs

Chapter 2. Special sets of graphs

Chapter 3. Basic operations on wheeled graphs

Chapter 4. Graph groupoids

Chapter 5. Graph substitution

Chapter 6. Properties of graph substitution

Chapter 7. Generators for graphs

Chapter 8. Pasting schemes

Chapter 9. Wellmatched pasting schemes

Part 2. Generalized PROPs, algebras, and modules

Chapter 10. Generalized PROPs

Chapter 11. Biased characterizations of generalized PROPs

Chapter 12. Functors of generalized PROPs

Chapter 13. Algebras over generalized PROPs

Chapter 14. Alternative descriptions of generalized PROPs

Chapter 15. Modules over generalized PROPs

Chapter 16. May modules over algebras over operads