Species of type B and related graded Hopf Algebras

Abstract iv
Acknowledgements vi
List of tables x
Introduction 1
Chapter I
Background H
1 . 1 General notions of categories 1 1
1 .2 Monoidal categories 15
1 .2. 1 Monoidal functors 17
1.2.2 Composites of monoidal functors 20
1.2.3 Moronisms of monoidal functors 21
1.2.4 Adjunctions of monoidal functors 22
1.2.5 Hopf monoids 23
1.3 Graded vector spaces 28
1.4 Species 29
1.4.1 Substitution of species 34
1 .4.2 Bilax monoidal functors 40
1.5 Cubical species 42
Chapter II
%-SPECIES 44
2.1 H-sets 44
2.2 H-species 51
2.3 Tensor product on ?-species 58
2.3.1 Dual H-species 65
2.3.2 Positive ?-species 67
2.4 From species to ?-species 68
2.5 Functors properties 76
2.6 Free monoid and cofree comonoid on an ?-species 85
2.7 Mixed substitution of species 98
Chapter III
Graded vector spaces from species 103
3.1 A bilax monoidal functor Sp^ -> gVec 103
3.2 Other bilax monoidal functors HO
3.3 Hopf algebras arising from 7^-species 115
3.4 Morphisms of bilax monoidal functors 121
3.5 Graded Hopf algebras arising from mixed substitutions of species 128
Chapter IV
Bigraded Hopf algebras 133
4.1 Bigraded Hopf algebras 133
4.1.1 DQSym 135
4.1.2 SoP(B) 137
4.2 Hopf algebra of set compositions 138
4.3 Hopf algebra of signed set compositions 141
4.4 Combinatorial Hopf species 145
4.5 Combinatorial bigraded Hopf algebras 148
4.5.1 Terminal object 149
4.5.2 Antipode of DQSym 151
Conclusion 153
Bibliography 156