Bimonoids for Hyperplane Arrangements (Encyclopedia of Mathematics and its Applications, Band 173)

Contents
Preface
Introduction
Part I. Species and operads
	Chapter 1. Hyperplane arrangements
		1.1. Faces, bifaces, flats
		1.2. Nested faces and lunes
		1.3. Partial-flats
		1.4. Minimal galleries, distance functions, Varchenko matrices
		1.5. Incidence algebras, and zeta and M¨obius functions
		1.6. Bilune-incidence algebra
		1.7. Descent, lune, Witt identities
		1.8. Noncommutative Zaslavsky formula
		1.9. Birkhoff algebra, Tits algebra, Janus algebra
		1.10. Takeuchi element
		1.11. Orientation space and signature space
		1.12. Lie and Zie elements
		1.13. Braid arrangement
		Notes
	Chapter 2. Species and bimonoids
		2.1. Species
		2.2. Monoids, comonoids, bimonoids
		2.3. (Co)commutative (co)monoids
		2.4. Deformed bimonoids and signed bimonoids
		2.5. Signed (co)commutative (co)monoids
		2.6. Subspecies and quotient species
		2.7. (Co)abelianizations of (co)monoids
		2.8. Generating subspecies of monoids
		2.9. Duality functor on species
		2.10. Op and cop constructions
		2.11. Monoids, comonoids, bimonoids as functor categories
		2.12. Presentation for (co)monoids using covering generators
		2.13. Partially commutative monoids
		2.14. Set-species and set-bimonoids
		2.15. Bimonoids for a rank-one arrangement
		2.16. Joyal species and Joyal bimonoids
		Notes
	Chapter 3. Bimonads on species
		3.1. Bimonoids as bialgebras over a bimonad
		3.2. Bicommutative bimonoids as bialgebras over a bimonad
		3.3. Duality as a bilax functor
		3.4. Opposite transformation
		3.5. Lifting of monads to comonoids
		3.6. Monad for partially commutative monoids
		3.7. Bimonad for set-species
		3.8. Symmetries, braidings, lax braidings
		3.9. LRB species
		3.10. Mesablishvili–Wisbauer
		Notes
	Chapter 4. Operads
		4.1. Dispecies
		4.2. Operads
		4.3. Set-operads
		4.4. Connected and positive operads
		4.5. Commutative, associative, Lie operads
		4.6. May operads
		4.7. Hadamard product. Hopf operads
		4.8. Orientation functor and signature functor
		4.9. Operad presentations
		4.10. Black and white circle products
		4.11. Left modules over operads
		4.12. Bioperads. Mixed distributive laws
		4.13. Incidence algebra of an operad
		4.14. Operads for LRB species
		Notes
Part II. Basic theory of bimonoids
	Chapter 5. Primitive filtrations and decomposable filtrations
		5.1. Cauchy powers of a species
		5.2. Graded and filtered bimonoids
		5.3. Primitive filtrations of comonoids
		5.4. Decomposable filtrations of monoids
		5.5. Trivial (co)monoids and (co)derivations
		5.6. Bimonoid axiom on the primitive part
		5.7. Primitively generated bimonoids and cocommutativity
		5.8. Browder–Sweedler and Milnor–Moore
		Notes
	Chapter 6. Universal constructions
		6.1. Free monoids on species
		6.2. Cofree comonoids on species
		6.3. (Co)free (co)commutative (co)monoids on species
		6.4. (Co)free bimonoids associated to species
		6.5. (Co)free (co)commutative bimonoids associated to species
		6.6. (Co)abelianizations of (co)free (co)monoids
		6.7. Primitive filtrations and decomposable filtrations
		6.8. Alternative descriptions of bimonoids
		6.9. Norm transformation
		6.10. (Co)free graded (co)monoids on species
		6.11. Free partially bicommutative bimonoids
		Notes
	Chapter 7. Examples of bimonoids
		7.1. Species characteristic of chambers
		7.2. Exponential species
		7.3. Species of chambers
		7.4. Species of flats
		7.5. Species of charts and dicharts
		7.6. Species of faces
		7.7. Species of top-nested faces and top-lunes
		7.8. Species of bifaces
		7.9. Lie and Zie species
		Notes
	Chapter 8. Hadamard product
		8.1. Hadamard functor
		8.2. Internal hom for the Hadamard product
		8.3. Biconvolution bimonoids
		8.4. Internal hom for comonoids. Bimonoid of star families
		8.5. Species of chamber maps
		8.6. Universal measuring comonoids
		8.7. Enrichment of the category of monoids over comonoids
		8.8. Internal hom for monoids and bimonoids
		8.9. Hadamard product of set-species
		8.10. Signature functor
		Notes
	Chapter 9. Exponential and logarithm
		9.1. Exp-log correspondences
		9.2. Commutative exp-log correspondence
		9.3. Deformed exp-log correspondences
		9.4. 0-exp-log correspondence
		9.5. Primitive and group-like series of bimonoids
		9.6. Primitive and group-like series of bicomm. bimonoids
		9.7. Comparisons between exp-log correspondences
		9.8. Formal power series. Series of Joyal species
		Notes
	Chapter 10. Characteristic operations
		10.1. Characteristic operations
		10.2. Commutative characteristic operations
		10.3. Two-sided characteristic operations
		10.4. Set-theoretic characteristic operations
		10.5. Idempotent operators on bimonoid components
		Notes
	Chapter 11. Modules over monoid algebras and bimonoids in species
		11.1. Modules over the Tits algebra
		11.2. Modules over the Birkhoff algebra
		11.3. Modules over the Janus algebra
		11.4. Examples
		11.5. Duality and base change
		11.6. Signed analogues
		11.7. A unified viewpoint via partial-biflats
		11.8. Karoubi envelopes
		11.9. Monoid-sets and bimonoids in set-species
		11.10. Bimonoids of h-faces and h-flats
		Notes
	Chapter 12. Antipode
		12.1. Takeuchi formula
		12.2. Interaction with op and cop constructions
		12.3. Commutative Takeuchi formula
		12.4. Logarithm of the antipode
		12.5. Examples
		12.6. Antipodes of (co)free bimonoids
		12.7. Antipodes of (co)free (co)commutative bimonoids
		12.8. Takeuchi element and characteristic operations
		12.9. Set-bimonoids
		Notes
Part III. Structure theory for bimonoids
	Chapter 13. Loday–Ronco, Leray–Samelson, Borel–Hopf
		13.1. Loday–Ronco for 0-bimonoids
		13.2. Leray–Samelson for bicommutative bimonoids
		13.3. Borel–Hopf for cocommutative bimonoids
		13.4. Borel–Hopf for commutative bimonoids
		13.5. Unification using partially bicommutative bimonoids
		13.6. Rigidity of q-bimonoids for q not a root of unity
		13.7. Monad for Lie monoids
		Notes
	Chapter 14. Hoffman–Newman–Radford
		14.1. Free 0-bimonoids on comonoids
		14.2. Free bicomm. bimonoids on cocomm. comonoids
		14.3. Free 0-~-bicommutative bimonoids
		14.4. Free bimonoids on cocommutative comonoids
		14.5. Free q-bimonoids on comonoids
		14.6. Zeta and M¨obius as inverses
		Notes
	Chapter 15. Freeness under Hadamard products
		15.1. Freeness under Hadamard products
		15.2. Product of free and cofree bimonoids
		15.3. Product of free comm. and cofree cocomm. bimonoids
		15.4. Product of bimonoids with one free factor
		15.5. Species of pairs of chambers
		Notes
	Chapter 16. Lie monoids
		16.1. Lie monoids
		16.2. Commutator bracket and primitive part of bimonoids
		16.3. Free Lie monoids on species
		16.4. Lie species and Zie species as Lie monoids
		16.5. Universal enveloping monoids
		16.6. Abelian Lie monoids
		16.7. Signed Lie monoids
		16.8. Lie comonoids
		Notes
	Chapter 17. Poincar´e–Birkhoff–Witt and Cartier–Milnor–Moore
		17.1. Comonoid sections to the abelianization map
		17.2. Poincar´e–Birkhoff–Witt
		17.3. Projecting the free monoid onto the free Lie monoid
		17.4. Solomon operator
		17.5. Cartier–Milnor–Moore
		17.6. Lie monoids for a rank-one arrangement
		17.7. Joyal Lie monoids
		17.8. Lie monoids in LRB species
		Notes
Appendices
	Appendix A. Vector spaces
		A.1. Kernel, cokernel, image, coimage
		A.2. Duality functor on vector spaces
		A.3. Internal hom for the tensor product
		A.4. Linear maps between direct sums of vector spaces
		A.5. Idempotent operators
	Appendix B. Internal hom for monoidal categories
		B.1. Monoidal and 2-monoidal categories
		B.2. Internal hom
		B.3. Powers and copowers
		B.4. Internal hom for functor categories
		B.5. Modules over a monoid algebra
		Notes
	Appendix C. Higher monads
		C.1. Higher monads
		C.2. Higher monad algebras
		C.3. Adjunctions
		Notes
References
List of Notations
List of Tables
Author Index
Subject Index