Monoidal functors, species and Hopf algebras [book draft]

836 Pages·2011·7.566 MB·English

by  Marcelo Aguiar, Swapneel Mahajan

Monoidal Functors, Species and Hopf Algebras
Marcelo Aguiar
Swapneel Mahajan
Department of Mathematics
Texas A&M University
College Station, TX 77843-3368
USA
E-mail address: [email protected]
URL: http://www.math.tamu.edu/~maguiar
Department of Mathematics
Indian Institute of Technology Mumbai
Powai, Mumbai 400 076
India
E-mail address: [email protected]
URL: http://www.math.iitb.ac.in/~swapneel
2010 Mathematics Subject Classification. Primary 05A30, 16T30,18D10, 18D35,
20B30, 81R50; Secondary 05A18, 05B35, 05C25, 05E05,05E45,06A11, 06A15,
16T25, 18D05, 18D20, 18D25, 18D50, 18G30, 18G35,20F55, 51E24, 81S05
Key words and phrases. bilax monoidal functor; Fock functor; Hopf algebra; Hopf
monoid; monoidal category; 2-monoidal category; operad; species; symmetric
group; symmetric functions
Contents
List of Tables ix
List of Figures xi
Forewordby Kenneth Brown and Stephen Chase xiii
Forewordby Andr´e Joyal xv
Introduction xxi
Acknowledgements li
Part I. Monoidal Categories 1
Chapter 1. Monoidal Categories 3
1.1. Braided monoidal categories 3
1.2. Hopf monoids 7
1.3. The internal Hom functor 16
1.4. Coherence 17
Chapter 2. Graded Vector Spaces 21
2.1. Graded vector spaces 21
2.2. The Schubert statistic 26
2.3. q-Hopf algebras 34
2.4. Multigraded vector spaces and Q-Hopf algebras 39
2.5. The norm map 42
2.6. The tensor algebra and its relatives 45
2.7. Chain complexes 51
2.8. Graded vector spaces with creation-annihilation operators 55
2.9. N-complexes 56
Chapter 3. Monoidal Functors 61
3.1. Bilax monoidal functors 61
3.2. Examples of bilax monoidal functors 67
3.3. Composites of bilax monoidal functors 72
3.4. A comparison of bimonoids and bilax monoidal functors 75
3.5. Normal bilax monoidal functors 80
3.6. Bistrong monoidal functors 84
3.7. Hopf lax monoidal functors 88
3.8. An alternative description of bilax monoidal functors 97
3.9. Adjunctions of monoidal functors 100
3.10. The contragredientconstruction 107
iii
iv CONTENTS
3.11. The image of a morphism of bilax monoidal functors 111
Chapter 4. Operad Lax Monoidal Functors 119
4.1. Other types of monoids and monoidal functors 119
4.2. Types of monoid: the general case 124
4.3. Types of monoidal functor: the general case 127
4.4. Composites of monoidal functors and transformation of monoids 131
Chapter 5. Bilax Monoidal Functors in Homological Algebra 137
5.1. The simplicial category and simplicial modules 138
5.2. Topological spaces and simplicial sets 141
5.3. Alexander–Whitney and Eilenberg–Zilber 142
5.4. The chain complex functors 145
5.5. The (co)homology functors. Cup and Pontrjaginproducts 150
5.6. A q-analogue of the chain complex functor 154
Chapter 6. 2-Monoidal Categories 161
6.1. The basic theory of 2-monoidal categories 162
6.2. Coherence 167
6.3. Braided monoidal categories as 2-monoidal categories 170
6.4. Examples of 2-monoidal categories 172
6.5. Bimonoids and double (co)monoids 178
6.6. Modules and comodules over a bimonoid 184
6.7. Examples of bimonoids and double monoids 185
6.8. Bilax and double (co)lax monoidal functors 189
6.9. Examples of bilax and double (co)lax monoidal functors 195
6.10. The free monoid functor as a bilax monoidal functor 198
6.11. 2-monoidal categories viewed as pseudomonoids 199
6.12. Contragredience for 2-monoidal categories 203
Chapter 7. Higher Monoidal Categories 207
7.1. 3-monoidal categories 207
7.2. Symmetric monoidal categories as 3-monoidal categories 212
7.3. Constructions of 3-monoidalcategories 213
7.4. Monoids in 3-monoidal categories 216
7.5. Monoidal functors between 3-monoidal categories 217
7.6. Higher monoidal categories 220
7.7. Monoids in higher monoidal categories 222
7.8. Monoidal functors between higher monoidal categories 224
7.9. Higher monoidal categories viewed as pseudomonoids 228
7.10. Contragredience for higher monoidal categories 230
Part II. Hopf Monoids in Species 233
Chapter 8. Monoidal Structures on Species 235
8.1. Species 235
8.2. Monoids and comonoids in species 239
8.3. Bimonoids and Hopf monoids in species 243
8.4. Antipode formulas for connected bimonoids 247
8.5. The simplest Hopf monoids 249
CONTENTS v
8.6. Duality in species 252
8.7. Set species and linearized species 254
8.8. Bimonoids as bilax monoidal functors 262
8.9. Connected and positive species 265
8.10. Primitive elements and the coradical filtration 269
8.11. Derivatives and internal Hom 270
8.12. Species with up-down operators 272
8.13. The Hadamard product and an interchange law on species 275
Chapter 9. Deformations of Hopf Monoids 283
9.1. q-Hopf monoids 283
9.2. Connected 0-bimonoids 284
9.3. The signed exponential species 287
9.4. The Hadamard and signature functors 288
9.5. The q-Hopf monoids of linear orders 290
9.6. Cohomology of linearized comonoids in species 294
9.7. The Schubert and descent cocycles 301
Chapter 10. The Coxeter Complex of Type A 305
10.1. Partitions and compositions 306
10.2. Faces, chambers, flats and cones 311
10.3. The Coxeter complex of type A 314
10.4. Tits projection maps and the monoid of faces 316
10.5. The gallery metric and the gate property 320
10.6. Shuffle permutations 323
10.7. The descent and global descent maps 324
10.8. The action of faces on chambers and the descent algebra 328
10.9. Directed faces and directed flats 330
10.10. The dimonoid of directed faces 335
10.11. The break and join maps 337
10.12. The weighted distance function 341
10.13. The Schubert cocycle and the gallery metric 346
10.14. A bilinear form on chambers. Varchenko’s result 353
10.15. Bilinear forms on directed faces and faces 357
Chapter 11. Universal Constructions of Hopf Monoids 363
11.1. The underlying species for the universal objects 364
11.2. The free monoid and the free Hopf monoid 365
11.3. The free commutative Hopf monoid 370
11.4. The cofree comonoid and the cofree Hopf monoid 372
11.5. The cofree cocommutative Hopf monoid 377
11.6. The norm transformation and the abelianization 379
11.7. The deformed free and cofree Hopf monoids 382
11.8. Antipode formulas 388
11.9. Primitive elements and related functors 393
11.10. The free and cofree 0-Hopf monoids 397
Chapter 12. Hopf Monoids from Geometry 401
12.1. Bases 402
12.2. The q-Hopf monoid of chambers 403
vi CONTENTS
12.3. The q-Hopf monoid of pairs of chambers 406
12.4. The q-Hopf monoids of faces 414
12.5. The q-Hopf monoids of directed faces 419
12.6. The Hopf monoids of flats 426
12.7. The Hopf monoids of directed flats 429
12.8. Relating the Hopf monoids 433
Chapter 13. Hopf Monoids from Combinatorics 443
13.1. Posets 443
13.2. Simple graphs 450
13.3. Rooted trees and forests 452
13.4. Relations 460
13.5. Combinatorics and geometry 463
13.6. Set-graded posets 465
13.7. Set-balanced simplicial complexes 473
13.8. Closures, matroids, convex geometries, and topologies 477
13.9. The Birkhoff transform 481
Chapter 14. Hopf Monoids in Colored Species 487
14.1. Colored species 487
14.2. Q-Hopf monoids 490
14.3. The colored exponential species 494
14.4. The colored Hadamard and signature functors 496
14.5. The colored linear order species 499
14.6. The colored free and cofree Hopf monoids 502
14.7. Colored Hopf monoids from geometry 509
Part III. Fock Functors 517
Chapter 15. From Species to Graded Vector Spaces 519
15.1. The Fock bilax monoidal functors 519
15.2. From Hopf monoids to Hopf algebras: Stover’s constructions 526
15.3. Values of Fock functors on particular Hopf monoids 528
15.4. The norm transformation between full Fock functors 532
15.5. The Fock functors and commutativity 536
15.6. The Fock functors and primitive elements 540
15.7. The full Fock functors and dendriform algebras 543
Chapter 16. Deformations of Fock Functors 547
16.1. Deformations of the full Fock functors 548
16.2. The deformed norm transformation 552
16.3. The fermionic and anyonic Fock bilax monoidal functors 555
16.4. The deformed full Fock functor and commutativity 558
16.5. Deformations of Hopf algebras arising from species 561
Chapter 17. From Hopf Monoids to Hopf Algebras: Examples 565
17.1. Shifting and standardization 566
17.2. The Hopf algebra of permutations 568
17.3. Quasi-symmetric and noncommutative symmetric functions 571
17.4. Symmetric functions 575
CONTENTS vii
17.5. Combinatorial Hopf algebras 577
Chapter 18. Adjoints of the Fock Functors 581
18.1. The right adjoint of K 582
18.2. The right adjoint of K 586
18.3. The left adjoint of K 587
18.4. Nonexistence of certain adjoints 590
18.5. The right adjoints of K and K on comonoids 591
Chapter 19. Decorated Fock Functors and Creation-Annihilation 599
19.1. Decorated Fock functors 600
19.2. The decorated norm transformation 605
19.3. Classical creation-annihilationoperators 607
19.4. The generalized Fock spaces of Gu¸t˘a and Maassen 610
19.5. Creation-annihilation on generalized bosonic Fock spaces 616
19.6. Species with balanced operators 620
19.7. Deformations of decorated Fock functors 625
19.8. Deformations related to up-down and creation-annihilation 627
19.9. Yang–Baxter deformations of decorated Fock functors 631
Chapter 20. Colored Fock Functors 635
20.1. The colored Fock functors 636
20.2. The colored norm transformation and the anyonic Fock functor 642
20.3. The colored full Fock functor and commutativity 646
20.4. Colors and decorations 651
20.5. Quantum objects 652
Appendices 655
Appendix A. Categorical Preliminaries 657
A.1. Products, coproducts and biproducts 657
A.2. Adjunction and equivalence 659
A.3. Colimits of functors 661
A.4. Kan extensions 665
A.5. Comma categories 667
Appendix B. Operads 669
B.1. Positive operads 669
B.2. Positive cooperads 675
B.3. Hereditary species. Schmitt’s construction of positive cooperads 678
B.4. General operads and cooperads 680
B.5. Modules over operads and monoids in species 689
B.6. Hopf operads 691
B.7. Nonsymmetric operads 695
Appendix C. Pseudomonoids and the Looping Principle 697
C.1. 2-categories and bicategories 697
C.2. Pseudomonoids 701
C.3. Enrichment 706
C.4. The looping principle 708
viii CONTENTS
C.5. Bipartite graphs, spans, and bimodules 710
Appendix D. Monoids and the Simplicial Category 713
D.1. Mac Lane’s simplicial category 713
D.2. Monoids, lax monoids, and homotopy monoids 715
D.3. Lax monoidal categories 717
D.4. The convolution homotopy monoid 720
References 725
Bibliography 727
Notation Index 741
Author Index 763
Subject Index 767
List of Tables
1.1 Categories of “monoids” in monoidal categories. 7
2.1 Monoidal structures on graded vector spaces. 22
2.2 Categories of “monoids” in (graded) vector spaces. 24
2.3 Monoidal categories related to graded vector spaces. 57
3.1 Bilax monoidal functors. 68
3.2 Bistrong monoidal functors. 68
3.3 Morphisms between bilax monoidal functors. 69
3.4 Self-dual functors. 111
3.5 Self-dual natural transformations. 111
4.1 Operad-monoids and familiar types of monoids. 127
4.2 Operad-lax functors and familiar types of functors. 130
5.1 Monoidal categories related to k-modules. 138
5.2 Simplicial objects in various categories. 139
6.1 Coherence for 2-monoidal categories. 167
6.2 Categories of different types of monoids. 183
6.3 2-categories related to Cat. 200
6.4 More 2-categories related to Cat. 203
8.1 Monoidal structures on species. 238
8.2 Categories of (co)monoids in species. 240
8.3 Categories of bimonoids in species. 243
11.1 Universal objects. 363
11.2 Universal objects. Deformed and signed versions. 364
12.1 Hopf monoids. 401
12.2 Universal maps. 438
13.1 Hopf monoids from combinatorics. 444
13.2 Hopf monoids from relations. 460
13.3 Combinatorics and geometry. 463
13.4 Analogies: from numbers to sets. 473
14.1 Categories of colored (co, bi, Hopf) monoids in species. 490
14.2 Universal Q-objects. 502
ix
x LIST OF TABLES
14.3 Q-Hopf monoids. 510
15.1 The Fock functors. 519
16.1 The deformed Fock functors. 547
18.1 Adjoints of K and K. 581
18.2 Adjoints of K∨ and K∨. 581
19.1 Decorated Fock functors. 599
19.2 Fock spaces. 600
19.3 Categories with +1 and −1 operators. 610
20.1 Colored Fock functors. 635
C.1 Pseudomonoids in various monoidal 2-categories. 706