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A Foundation for PROPs, Algebras, and Modules

Donald Yau The Ohio State University at Newark, Newark, OH
Mark W. Johnson Pennsylvania State University Altoona, Altoona, PA
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Hardcover ISBN: 978-1-4704-2197-7
Product Code: SURV/203
List Price: $110.00 MAA Member Price:$99.00
AMS Member Price: $88.00 Electronic ISBN: 978-1-4704-2347-6 Product Code: SURV/203.E List Price:$110.00
MAA Member Price: $99.00 AMS Member Price:$88.00
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List Price: $165.00 MAA Member Price:$148.50
AMS Member Price: $132.00 Click above image for expanded view A Foundation for PROPs, Algebras, and Modules Donald Yau The Ohio State University at Newark, Newark, OH Mark W. Johnson Pennsylvania State University Altoona, Altoona, PA Available Formats:  Hardcover ISBN: 978-1-4704-2197-7 Product Code: SURV/203  List Price:$110.00 MAA Member Price: $99.00 AMS Member Price:$88.00
 Electronic ISBN: 978-1-4704-2347-6 Product Code: SURV/203.E
 List Price: $110.00 MAA Member Price:$99.00 AMS Member Price: $88.00 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$165.00 MAA Member Price: $148.50 AMS Member Price:$132.00
• Book Details

Mathematical Surveys and Monographs
Volume: 2032015; 311 pp
MSC: 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, 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.

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. Well-matched 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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Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 2032015; 311 pp
MSC: 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, 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.

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. Well-matched 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
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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