We discuss the free monoid and the cofree comonoid on a species (relative to a fixed hyperplane arrangement). In addition, we discuss the free bimonoid on a comonoid, and dually the cofree bimonoid on a monoid. More generally, for any scalar q, we have the free q-bimonoid on a comonoid and the cofree q-bimonoid on a monoid. An important special case is when the starting (co)monoid has trivial (co)product. We employ the terms concatenation and q-(quasi)shuffle for the products, and deconcatenation and q-de(quasi)shuffle for the coproducts. For q = 1, the q-(quasi)shuffle product is commutative, while the q-de(quasi)shuffle coproduct is cocommutative. The concatenation product and deconcatenation coproduct do not depend on q, and do not satisfy any commutativity property. In addition, we also discuss the free commutative monoid and the cofree cocommutative comonoid on a species and related constructions. These have signed analogues. We discuss the q-norm map between free and cofree q-bimonoids. It is an isomorphism when q is not a root of unity. Invertibility of the Varchenko matrix associated to the q-distance function plays a critical role here. We also discuss the (co)free graded (co)monoid on a graded species. Every species can be viewed as a graded species concentrated in degree 1. The free graded monoid on a species has a unique coproduct which turns it into a graded q-bimonoid. This is precisely the q-deshuffle coproduct. Dually, the q-shuffle product is the unique product which turns into a graded q-bimonoid.

This introductory chapter gives a detailed summary of the contents of the book along with pointers to important results.

This chapter provides a categorical framework for the notions of monoids, comonoids, bimonoids in species (relative to a fixed hyperplane arrangement). The usual categorical setting for monoids is a monoidal category. However, that is not the case here; the relevant concept is that of monads and algebras over monads. We construct a monad on the category of species, and observe that algebras over it are the same as monoids in species. Dually, we construct a comonad whose coalgebras are the same as comonoids in species. In addition, we construct a mixed distributive law between this monad and comonad such that bialgebras over the resulting bimonad are the same as bimonoids in species. Moreover, the mixed distributive law can be deformed by a parameter q such that the resulting bialgebras are the same as q-bimonoids. The above monad, comonad, bimonad have commutative counterparts which relate to commutative monoids, cocommutative comonoids, bicommutative bimonoids in species. We briefly discuss the Mesablishvili-Wisbauer rigidity theorem. As a consequence, the category of species is equivalent to the category of 0-bimonoids, as well as to the category of bicommutative bimonoids. These ideas are developed in more detail later. We extend the notion of species from a hyperplane arrangement to the more general setting of a left regular band (LRB).

For any q-bimonoid (relative to a fixed hyperplane arrangement), we define its antipode as a map from the bimonoid to itself whose face-components are defined by taking an alternating sum with each summand being the composite of a coproduct componentfollowed by a product component. We refer to this formula as the Takeuchi formula. Up to signs, it equals the 0-logarithm of the identity map on the bimonoid. There is also a commutative analogue of this formula for bicommutative bimonoids. We study interactions of the antipode with morphisms of bimonoids, the duality functor, bimonoid filtrations, the signature functor. The antipode is intimately related to the antipodal map on the arrangement; this is brought forth by its interaction with op and cop constructions. We compute logarithm of the antipode map using the noncommutative Zaslavsky formula, and moreover relate it to logarithm of the identity map. Understanding the cancelations in the Takeuchi formula for a given bimonoid is often a challenging combinatorial problem. We solve this problem for the exponential bimonoid, the bimonoid of chambers, the bimonoid of faces, the bimonoid of flats, and so on. More generally, we provide cancelation-free formulas for the antipode of any bimonoid which arises from a universal construction. In a similar vein, for any set-bimonoid, this problem can be interpreted as the calculation of the Euler characteristic of a cell complex. The descent and lune identities provide motivating examples of such a calculation. The antipode map is closely related to the Takeuchi elements associated to arrangements. More precisely, the face-component of the antipode map is the characteristic operation by the Takeuchi element of the arrangement over the support of that face. This connection makes it possible to study the antipode using properties of the Takeuchi elements.

We discuss some important rigidity theorems related to universal constructions. They usually take the form of an adjoint equivalence between suitable categories.The Loday-Ronco theorem says that the category of 0-bimonoids is equivalent to the category of species. In particular, 0-bimonoids are both free and cofree. This theorem is a special case of a more general result in which 0-bimonoids are replaced by q-bimonoids with q not a root of unity. We refer to this result as the rigidity of q-bimonoids. Invertibility of the Varchenko matrix associated to the q-distance function on faces plays a critical role here. The Leray-Samelson theorem says that the category of bicommutative bimonoids is equivalent to the category of species. In particular, bicommutative bimonoids are both free commutative and cofree cocommutative. There is also a signed analogue of Leray–Samelson which applies to signed bicommutative signed bimonoids. The Borel–Hopf theorem says that any cocommutative bimonoid is cofree on its primitive part, and dually, any commutative bimonoid is free on its indecomposable part. This result also has a signed analogue which applies to signed (co)commutative signed bimonoids. We present three broad approaches to these rigidity theorems. The first approach is elementary and proceeds by an induction on the primitive filtration of the bimonoid. Here a key role is played by how the bimonoid axiom works on the primitive part.The second approach is more direct and proceeds by constructing an explicit inverse to the appropriate universal map. The universal map is defined using a zeta function and the inverse using a Möbius function. These maps have connections to the exponential and logarithm operators. The third approach is also constructive and employs(commutative, usual or two-sided) characteristic operations by suitable families of idempotents in the (Birkhoff, Tits or q-Janus) algebra, respectively, to decompose the given bimonoid. All results in this chapter are independent of the characteristic of the base field.

We introduce the notion of dispecies relative to a fixed hyperplane arrangement. The category of dispecies carries a monoidal structure which we call the substitution product. Operads are monoids in this monoidal category. We describe the free operad on a dispecies, and then proceed to operad presentations with an emphasis on binary quadratic operads. Apart from the substitution product, the category of dispecies also carries the Hadamard product which turns it into a 2-monoidal category. Hopf operads are bimonoids in this 2-monoidal category. We use these ideas to construct the black and white circle products on binary quadratic operads. We discuss three main examples of operads, namely, commutative, associative, Lie. These are all binary quadratic. Further, under a suitable notion of quadratic duality, the commutative and Lie operads are duals of each other, while the associative operad is self-dual. These can be viewed as extensions of well-known facts from the classical theory of May operads. The category of species is a left module category over the monoidal category of dispecies (under the substitution product). Hence, to each operad, one can associate the category of its left modules. A left module over the associative operad is the same as a monoid in species, over the commutative operad is the same as a commutative monoid in species, over the Lie operad is the same as a Lie monoid in species. To every operad, one can attach an (associative) algebra called its incidence algebra. The incidence algebra of the commutative operad is the flat-incidence algebra, of the associative operad is the lune-incidence algebra, and of the Lie operad is the Tits algebra. The incidence algebra of any connected quadratic operad is elementary and its quiver can be explicitly described. Operads can also be defined in the more general setting of left regular bands. Interestingly, the commutative, associative, Lie operads extend to this setting.

We discuss some basic and important examples of species (relative to a fixed hyperplane arrangement). These include the exponential species, signed exponential species, species of chambers, species of faces, species of flats, species of top-nested faces, species of top-lunes, species of bifaces. They are constructed from well-known objects associated to an arrangement such as faces, flats, chambers, top-nested faces, top-lunes, bifaces. The exponential species is simpler and has all its components equal to the base field. Each of these species carries the structure of a bimonoid. Some of them also admit q-analogues. All these bimonoids arise from universal constructions. We also introduce the Lie species and Zie species. These arise as the primitive part of the bimonoids of chambers and faces, respectively. As a consequence, they carry the structure of a Lie monoid.

To each pair (p,q) of nonnegative integers, we associate a 2-category whose 0-cells are (p,q)-monads, 1-cells are (p,q)-lax functors, 2-cells are morphisms between (p,q)-lax functors. The starting point is the pair (0,0) which yields the 2-category of all categories whose 0-cells are categories, 1-cells are functors, 2-cells are natural transformations. This is level zero. For the pair (1,0), we have the 2-category whose 0-cells are monads, 1-cells are lax functors, and 2-cells are morphisms between lax functors. For the pair (0,1), there is a similar 2-category involving comonads and colax functors. This is level one. The passage from level zero to level one can be formalized via the monad and comonad constructions. Applying these constructions on level one yields the 2-categories on level two, and so on. Double monads (distributive laws), double lax functors, bimonads (mixed distributive laws), bilax functors, and so on, appear on level two. Higher monad algebras are constructed in a straightforward manner from the 2-categories of higher monads. Here we concentrate on the case of algebras, coalgebras, bialgebras which arise from monads, comonads, bimonads, respectively.

We discuss graded and filtered monoids, comonoids, bimonoids in species (relative to a fixed hyperplane arrangement). Every comonoid has a primitive part and more generally a primitive filtration which turns it into a filtered comonoid. Dually, every monoid has a decomposable part and more generally a decomposable filtration which turns it into a filtered monoid. The indecomposable part of a monoid is the quotient by its decomposable part.A map from a species to a comonoid is a coderivation if it maps into the primitive part of that comonoid. Dually, a map from a monoid to a species is a derivation if it factors through the indecomposable part of that monoid. A (co)derivation is the same as a (co)monoid morphism with the species viewed as a (co)monoid with the trivial (co)product. For a q-bimonoid, there is a canonical map from its primitive part to its indecomposable part. For a q-bimonoid for q not a root of unity, this map is bijective. For a bimonoid, this map is surjective iff the bimonoid is cocommutative, injective iff thebimonoid is commutative, and bijective iff the bimonoid is bicommutative.For a q-bimonoid, both the primitive and the decomposable filtrations turn it into a filtered q-bimonoid. Thus, for either filtration, we can consider the corresponding associated graded q-bimonoid. For q = 1, the associated graded bimonoid wrt the primitive filtration is commutative, and wrt the decomposable filtration is cocommutative. These are the Browder-Sweedler and Milnor-Moore (co)commutativity results.

The lune-incidence algebra (of a hyperplane arrangement) acts on the space of all maps from a comonoid to a monoid. In particular, any noncommutative zeta function defines an invertible operator on this space. We call such an operator an exponential. The inverse operator is given by a noncommutative Möbius function and we call it a logarithm. Given a cocommutative comonoid and a bimonoid, any mutually inverse pair of exponential and logarithm sets up inverse bijections between coderivations and comonoid morphisms from the comonoid to the bimonoid. Dually, for a bimonoid and a commutative monoid, there are bijections between derivations and monoid morphisms from the bimonoid to the monoid. As a consequence, any logarithm of the identity map on a bimonoid maps into its primitive part when it is cocommutative, and factors through its indecomposable part when it is commutative. This story has a commutative counterpart with the lune-incidence algebra replaced by the flat-incidence algebra. This can either be deduced from the above via the base-case map or can also be developed independently. In this case, the exponential and logarithm are uniquely defined using the zeta function and Möbius function of the poset of flats. There is a parallel theory for q-bimonoids for q not a root of unity. It employs the two-sided q-zeta function and two-sided q-Möbius function. These are elements of the bilune-incidence algebra. Exp-log correspondences can also be developed using the notion of series of a species. For a comonoid, one can further define primitive series and group-like series. The lune-incidence algebra acts on the space of series of a monoid, and any mutually inverse pair of a noncommutative zeta function and noncommutative Möbius function defines inverse operations onthis space. For the exponential bimonoid, this recovers Möbius inversion in the poset of flats, while for the bimonoid of chambers, this yields a noncommutative version of Möbius inversion. Moreover, for any bimonoid, the exp-log correspondences set up a bijection between its primitive series and group-like series. The passage from this approach to the previous approach involves the convolution monoid, the bimonoid of star families, and the universal measuring comonoid. The classical setting is as follows. The space of formal power series (viewed as a monoid under substitution) acts on the space of series of a Joyal monoid. The usual exponential and logarithmic power series yield the exp-log correspondence on it. This classical picture relates to the one for monoids in species relative to any braid arrangement. More precisely, the action of the exponential and logarithmic power series corresponds to that of the uniform noncommutative zeta function and its inverse noncommutative Möbius function.

This chapter establishes the Poincaré-Birkhoff-Witt and Cartier-Milnor-Moore theorems for Lie monoids in species (relative to a fixed hyperplane arrangement). The Poincaré-Birkhoff-Witt theorem (PBW) says that for any Lie monoid, its universal enveloping monoid is isomorphic to the cofree cocommutative comonoid on its underlying species. The isomorphism is that of comonoids. It depends on the choice of a noncommutative zeta function. Equivalently, for any Lie monoid, PBW defines an idempotent operator on the free bimonoid of its underlying species whose image is the cofree cocommutative comonoid and coimage is the universal enveloping monoid. We call this the Solomon operator. We give two proofs of PBW. The first one is elementary and inductively builds the Solomon operator. The second one starts with an explicit definition of the Solomon operator, and then establishes that it has the correct image and coimage. The Cartier-Milnor-Moore theorem (CMM) says that the universal enveloping and primitive part functors determine an adjoint equivalence between the category of Lie monoids and the category of cocommutative bimonoids. It is a formal consequence of Borel-Hopf and PBW.PBW and CMM also have dual versions. They go along with the Borel-Hopf theorem for commutative bimonoids. Relevant notions are Lie comonoids and their universal coenveloping comonoids. PBW and CMM as well as their dual versions have signed analogues.