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E-raamat: Colored Operads

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The subject of this book is the theory of operads and colored operads, sometimes called symmetric multicategories. A (colored) operad is an abstract object which encodes operations with multiple inputs and one output and relations between such operations. The theory originated in the early 1970s in homotopy theory and quickly became very important in algebraic topology, algebra, algebraic geometry, and even theoretical physics (string theory). Topics covered include basic graph theory, basic category theory, colored operads, and algebras over colored operads. Free colored operads are discussed in complete detail and in full generality.

The intended audience of this book includes students and researchers in mathematics and other sciences where operads and colored operads are used. The prerequisite for this book is minimal. Every major concept is thoroughly motivated. There are many graphical illustrations and about 150 exercises. This book can be used in a graduate course and for independent study.

Arvustused

Colored Operads has a very low barrier to entry, and so would be suitable even for strong undergraduates. Each chapter has exercises at the end, so this book could form the core of a final-year reading course or project...The organizational aspects of this book are exceptional, with a very thorough List of Notations, a Table of Contents that is fine enough to be very usable without being so long as to discourage perusal, a helpful List of Main Facts giving concise versions of all major theorems, and a good Index. The Bibliography is varied, with good references to a wide literature on operads. Yau's monograph offers a very careful introduction to the theory of operads that would complement any library on the subject. It has a calculational flavor that sets it apart from other texts, and this makes it accessible to both graduate and strong undergraduate studentsit occupies a very interesting space in the operadic literature." - Nick Gurski, Jahresbericht der Deutschen Mathematiker-Vereinigung

"This book is a useful introduction to colored operads or symmetric multicategories, to the destination of students as well as researchers interested in these objects." - Loïc Foissy, Zentralblatt Math

"An introductory undergraduate course in abstract algebra is sufficient as a prerequisite for almost all of the material covered in the book. One impressive feature of the book is the emphasis on motivating new concepts as they are introduced and providing numerous graphical illustrations to clarify their geometric significance; there are also numerous exercises collected at the ends of the chapters. The author also provides a list of references to related literature to assist the reader who wishes to continue the study of operads beyond the topics covered in this book." - Murray R. Bremner, Mathematical Reviews

"The book contains much valuable information and detail, which can potentially save a struggling newcomer into operad land many hours of frustration." - Ittay Weiss, MAA Reviews

Preface xiii
Acknowledgments xxi
List of Notations xxiii
Part
1. Graphs and Trees
Chapter 1 Directed Graphs
3(16)
1.1 Set Notations
3(2)
1.2 Graphs
5(3)
1.3 Directed Graphs
8(2)
1.4 Directed (m, n)-Graphs
10(6)
1.5 Exercises
16(1)
1.6 Notes
17(2)
Chapter 2 Extra Structures on Graphs
19(10)
2.1 Edge Coloring
19(1)
2.2 Vertex Decoration
20(1)
2.3 Input Labeling
21(3)
2.4 Incoming Edge Labeling
24(2)
2.5 Isomorphisms with Extra Structures
26(1)
2.6 Exercises
27(2)
Chapter 3 Rooted Trees
29(14)
3.1 Rooted Trees
30(1)
3.2 Profile of a Vertex
31(1)
3.3 Profile of a Rooted Tree
31(1)
3.4 Exceptional Edge and Corollas
32(2)
3.5 Simple Trees
34(2)
3.6 Level Trees
36(4)
3.7 Linear Graphs
40(1)
3.8 Exercises
40(3)
Chapter 4 Collapsing an Internal Edge
43(10)
4.1 Motivation
43(1)
4.2 Defining the Quotient
44(2)
4.3 Examples
46(2)
4.4 Associativity
48(1)
4.5 Compatibility with Extra Structures
48(3)
4.6 Exercises
51(2)
Chapter 5 Grafting of Rooted Trees
53(22)
5.1 Motivation
54(1)
5.2 Defining Grafting
55(2)
5.3 Examples
57(3)
5.4 Unity
60(2)
5.5 Horizontal Associativity
62(2)
5.6 Vertical Associativity
64(2)
5.7 Grafting Decomposition of Rooted Trees
66(6)
5.8 Exercises
72(3)
Chapter 6 Grafting and Extra Structures
75(14)
6.1 Edge Coloring
75(1)
6.2 Vertex Decoration
76(1)
6.3 Input Labeling
77(2)
6.4 Incoming Edge Labeling
79(1)
6.5 Canonical Vertex Labeling
80(2)
6.6 Canonical Input Labeling
82(3)
6.7 Exercises
85(4)
Part
2. Category Theory
Chapter 7 Basic Category Theory
89(32)
7.1 Categories
90(8)
7.2 Functors
98(2)
7.3 Natural Transformations
100(5)
7.4 Equivalence
105(2)
7.5 Coproducts
107(2)
7.6 Products
109(2)
7.7 Adjoint Functors
111(5)
7.8 Exercises
116(3)
7.9 Notes
119(2)
Chapter 8 Symmetric Monoidal Categories
121(20)
8.1 Motivation for Monoidal Categories
121(1)
8.2 Monoidal Categories
122(3)
8.3 Monoidal Functors
125(3)
8.4 Mac Lane's Theorem
128(2)
8.5 Symmetry
130(2)
8.6 Permuting Iterated Tensor Products
132(3)
8.7 Symmetric Monoidal Closed Categories
135(2)
8.8 Standing Categorical Assumptions
137(2)
8.9 Exercises
139(1)
8.10 Notes
140(1)
Chapter 9 Colored Symmetric Sequences and Objects
141(20)
9.1 Colors and Profiles
141(2)
9.2 Permutation Category
143(4)
9.3 Colored Symmetric Sequences
147(6)
9.4 Colored Objects
153(3)
9.5 Exercises
156(2)
9.6 Notes
158(3)
Part
3. Operads and Algebras
Chapter 10 Motivation for Colored Operads
161(12)
10.1 Categories via Linear Graphs
161(3)
10.2 Colored Operads via Trees
164(9)
Chapter 11 Colored Operads
173(30)
11.1 Block Permutations and Block Sums
173(2)
11.2 Defining Colored Operads
175(6)
11.3 Examples of the Axioms
181(4)
11.4 Initial and Terminal Colored Operads
185(5)
11.5 Changing the Base Category
190(3)
11.6 1-Colored Operads
193(4)
11.7 Colored Non-Symmetric Operads
197(1)
11.8 Exercises
198(2)
11.9 Notes
200(3)
Chapter 12 Operads in Arity 1
203(12)
12.1 Motivation for Monoids
203(2)
12.2 Monoids
205(3)
12.3 Small Enriched Categories
208(4)
12.4 Exercises
212(3)
Chapter 13 Algebras over Colored Operads
215(22)
13.1 Motivation
215(2)
13.2 Defining Algebras
217(2)
13.3 Examples of the Axioms
219(2)
13.4 Visualizing Algebras
221(1)
13.5 Algebras over a 1-Colored Operad
222(2)
13.6 Algebras over a Colored Non-Symmetric Operad
224(1)
13.7 Motivation for Endomorphism Operads
225(1)
13.8 Colored Endomorphism Operads
226(5)
13.9 Algebra as a Map
231(1)
13.10 Exercises
232(3)
13.11 Notes
235(2)
Chapter 14 Examples of Algebras
237(28)
14.1 Initial and Terminal Algebras
237(2)
14.2 Operad for Monoids
239(7)
14.3 Colored Operad for Monoid Maps
246(7)
14.4 Colored Operad for Colored Monoids
253(5)
14.5 Exercises
258(7)
Chapter 15 Motivation for Partial Compositions
265(10)
15.1 Simplifying the Operadic Composition
265(2)
15.2 Operadic Composition to Partial Composition
267(3)
15.3 Partial Composition to Operadic Composition
270(1)
15.4 Associativity
271(1)
15.5 Unity
272(1)
15.6 Equivariance
273(2)
Chapter 16 Colored Pseudo-Operads
275(46)
16.1 Partial Compositions of Profiles
275(3)
16.2 Defining Colored Pseudo-Operads
278(7)
16.3 Examples of the Axioms
285(4)
16.4 Colored Pseudo-Operads Are Colored Operads
289(4)
16.5 1-Colored Pseudo-Operads
293(3)
16.6 Colored Non-Symmetric Pseudo-Operads
296(2)
16.7 Algebras via Partial Compositions
298(3)
16.8 Rooted Trees Operad
301(4)
16.9 Little Square Operad
305(4)
16.10 Exercises
309(8)
16.11 Notes
317(4)
Part
4. Free Colored Operads
Chapter 17 Motivation for Free Colored Operads
321(12)
17.1 Free Monoids
321(6)
17.2 Heuristic Free Colored Non-Symmetric Operads
327(3)
17.3 Heuristic Free Colored Operads
330(2)
17.4 Exercises
332(1)
Chapter 18 General Operadic Composition
333(18)
18.1 Decoration by a Colored Object
333(7)
18.2 Defining General Operadic Composition
340(7)
18.3 Associativity
347(3)
18.4 Exercises
350(1)
Chapter 19 Free Colored Non-Symmetric Operads
351(30)
19.1 Entries and Structure Maps
352(5)
19.2 Colored Non-Symmetric Operad Structure
357(4)
19.3 The Adjunction
361(9)
19.4 Motivation for Tree Operads
370(2)
19.5 Colored Non-Symmetric Operad Generated by a Tree
372(7)
19.6 Exercises
379(2)
Chapter 20 Free Colored Operads
381(28)
20.1 Colored Non-Symmetric Operads to Colored Operads
382(7)
20.2 Symmetrization
389(5)
20.3 Free Colored Operad of a Colored Object
394(10)
20.4 Colored Operad Generated by a Tree
404(4)
20.5 Exercises
408(1)
Further Reading 409(4)
Bibliography 413(6)
List of Main Facts 419(4)
Index 423
Donald Yau, The Ohio State University at Newark, OH, USA.