**Mathematical Surveys and Monographs**

Volume: 203;
2015;
311 pp;
Hardcover

MSC: Primary 18;
Secondary 55; 13; 81

Print ISBN: 978-1-4704-2197-7

Product Code: SURV/203

List Price: $110.00

Individual Member Price: $88.00

**Electronic ISBN: 978-1-4704-2347-6
Product Code: SURV/203.E**

List Price: $110.00

Individual Member Price: $88.00

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#### Supplemental Materials

# A Foundation for PROPs, Algebras, and Modules

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*Donald Yau; Mark W. Johnson*

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.

#### Readership

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

#### Table of Contents

# Table of Contents

## A Foundation for PROPs, Algebras, and Modules

- Cover Cover11
- Title page iii4
- Contents vii8
- Preface xv16
- List of notations xxv26
- Chapter 1. Wheeled graphs 336
- Chapter 2. Special sets of graphs 2154
- Chapter 3. Basic operations on wheeled graphs 3568
- Chapter 4. Graph groupoids 4780
- Chapter 5. Graph substitution 5790
- Chapter 6. Properties of graph substitution 75108
- Chapter 7. Generators for graphs 107140
- Chapter 8. Pasting schemes 133166
- Chapter 9. Well-matched pasting schemes 147180
- Chapter 10. Generalized PROPs 159192
- Chapter 11. Biased characterizations of generalized PROPs 191224
- Chapter 12. Functors of generalized PROPs 223256
- Chapter 13. Algebras over generalized PROPs 239272
- Chapter 14. Alternative descriptions of generalized PROPs 263296
- Chapter 15. Modules over generalized PROPs 273306
- Chapter 16. May modules over algebras over operads 291324
- Bibliography 303336
- Index 307340
- Back Cover Back Cover1346