We briefly review some standard material about the category of vector spaces. The discussion includes the kernel, cokernel, image, coimage of a linear map, the duality functor, the internal hom for the tensor product, and idempotent operators.

In this chapter, we study further the Hadamard product on species. The Hadamard product of two free monoids is again free. Similarly, the Hadamard product of two free commutative monoids is again free commutative. In either case, we give an explicit formula for a basis of the Hadamard product in terms of bases of its two factors. It involves the meet operation on faces and flats, respectively. We also show that the Hadamard product of bimonoids is free as a monoid if one of two factors is free as a monoid. We study in detail the Hadamard product of the free bimonoid on a comonoid with the cofree bimonoid on a monoid. It is neither commutative nor cocommutative, so Borel-Hopf does not apply. This bimonoid is both free and cofree. Interestingly, we prove this using Loday-Ronco (which is a theorem about 0-bimonoids). We also give a cancelation-free formula for its antipode. We give an explicit description of its primitive part, and more generally, its primitive filtration. An illustrative example of this construction is the bimonoid of pairs of chambers. We give a parallel discussion for a commutative counterpart where we take the Hadamard product of the free commutative bimonoid on a cocommutative comonoid with the cofree cocommutative bimonoid on a commutative monoid. (Since this bimonoid is bicommutative, it can also be tackled using Leray-Samelson.)

The definition of a bimonoid in species makes use of the Tits monoid of the hyperplane arrangement. The latter is a monoid structure on the set of faces.On the other hand, there is the bimonoid of faces, which is itself built out of faces.This double occurrence of faces acquires formal meaning now. Elements of the bimonoid of faces give rise to characteristic operations on any bimonoid.This yields a morphism from the bimonoid of faces to the biconvolution bimonoid associated to the given bimonoid. Further, when the given bimonoid is commutative or cocommutative, each face-component map of this morphism is an algebra antimap or an algebra map, with the Tits product on the former and composition product on the latter. The above story has a simpler commutative analogue. Bicommutative bimonoids can be formulated using the Birkhoff monoid. The latter is a monoid structure on the set of flats. On the other hand, there is the bimonoid of flats, which is itself built out of flats. Formally, elements of the bimonoid of flats give rise to commutative characteristic operations on any bicommutative bimonoid. This yields a morphism from the bimonoid of flats to the biconvolution bimonoid associated to the given bimonoid. Further, each flat-component of this morphism is an algebra map. There are more general operations one can consider on bimonoids by working with bifaces instead of faces. We call these the two-sided characteristic operations. The role of the Tits algebra is now played by the Janus algebra. More generally, one can also consider q-bimonoids whose face-components are acted upon by the q-Janus algebra.

We review the notion of internal hom for a monoidal category. The discussion includes the endomorphism monoid, the convolution monoid, the internal hom for functor categories (which includes the category of modules over a monoid algebra). We also discuss the enriched counterpart of the tensor-hom adjunction, which gives rise to the notion of power and copower.

In this chapter, we forge a connection between bimonoids in species and representation theory of monoid algebras. More precisely, the category of cocommutative bimonoids is equivalent to the category of left modules over the Tits algebra. Similarly, commutative bimonoids relate to right modules over the Tits algebra, bicommutative bimonoids to modules over the Birkhoff algebra, arbitrary bimonoids to modules over the Janus algebra, and more generally, q-bimonoids to modules over the q-Janus algebra. Moreover, these equivalences are compatible with duality and base change and also have signed analogues. Some illustrative examples are as follows. The (bicommutative) exponential bimonoid corresponds to the trivial module over the Birkhoff algebra. Both are self-dual in an appropriate sense. The (bicommutative) bimonoid of flats corresponds to the Birkhoff algebra viewed as a module over itself. Similarly, the (cocommutative) bimonoid of chambers corresponds to the left module of chambers over the Tits algebra, while the (cocommutative) bimonoid of faces corresponds to the Tits algebra viewed as a left module over itself. The bimonoid of bifaces corresponds to the Janus algebra viewed as a left module over itself.We approach these results through characteristic operations on bimonoids. They can also be derived by computing the Karoubi envelopes of the Birkhoff monoid, Tits monoid, Janus monoid, and using the interpretation of bimonoids as functor categories.

The main purpose of this chapter is to discuss groups generated by reflexions, concentrating here on the finite and discrete infinite groups in euclidean spaces. While establishing notation and conventions, there are surveys of the algebraic and metrical properties of euclidean spaces, and a treatment of the main features of convex sets that are appealed to subsequently. The classification of the finite and discrete infinite reflexion groups in euclidean spaces is a core feature; the initial part of the treatment is novel. There is then a brief description of subgroup relationships among these groups. Certain angle-sum relations for polytopes and cones are employed to find the orders of the finite Coxeter groups by purely elementary geometric methods; these are established here for polyhedral sets in general. The lower-dimensional spaces are somewhat special. The finite rotation groups in three dimensions are classified, and are shown to be subgroups of reflexion groups. Finally, there is an introduction to quaternions, which provide an alternative approach to finite orthogonal groups in 4-dimensional space; these are needed to describe certain regular polyhedra in that space.

This chapter is devoted to completing the classification of the regular polytopes and apeirotopes of full rank, by treating the non-classical cases. Naturally, it appeals to the listing of the classical regular polytopes and apeirotopes in the previous chapter. First treated are the cases derived from the four infinite families of regular polytopes and honeycombs which occur in all dimensions; this includes eliminating some initially plausible examples. There are two interesting non-polytopes in dimensions six and eight, the first particularly so since its facet is genuinely polytopal. The exceptional cases are to be found in dimensions two, three and four only. First considered are the remaining regular polyhedra in three dimensions, since the apeirohedra in the plane need little comment. The other examples are in four dimensions; one of these shows that the full range of possible mirror vectors can occur.

Just as the exceptional regular polytopes of full rank are only of dimension at most four, so the exceptions of nearly full rank have dimension at most eight. The remaining regular polytopes and apeirotopes of nearly full rank are treated in this chapter, which completes their classification. The ‘gateway’ dimension five is crucial to the investigation, since there is a severe restriction on the possible symmetry groups, and hence on the corresponding (finite) regular polytopes. This dimension is first looked at only in general terms, since the polytopes not previously described fall naturally into families that are considered in later sections. However, one case is dealt with in full detail: there is a sole regular polytope in five dimensions (and none in higher dimensions) whose symmetry group consists only of rotations. The new families of regular polytopes of nearly full rank are closely related to the Gosset–Elte polytopes, so these are briefly described here. There are three families, which are dealt with in turn; however, a fourth putative family is shown to degenerate.

The concept of rigidity of regular polytopes is relatively new; it asks to what extent does the specification of certain geometric data about a regular polytope determine it. After a little historical background to the subject, the central notion of fine Schläfli symbol is described, and then the formal concepts of shape (or similarity class) and rigidity are defined, and some familiar examples are introduced. Finally, general criteria for rigidity are given.

The monograph ‘Abstract Regular Polytopes’ described the rich abstract theory, of which some basics are needed here. A new recursive definition is given, which corresponds more closely than that of the monograph to one’s intuitive idea of what a polytope should be. Regularity of abstract polytopes and the central idea of string C-groups are then introduced, and it is shown that the two concepts are equivalent. The intersection property defines a C-group; various conditions on a group are established that ensure it, in particular some quotient criteria. Presentations of the groups of regular polytopes are treated next, including the circuit criterion, and some related general concepts and notation are introduced. Maps or polyhedra, the polytopes of rank three, form an important class of regular polytopes; some of their properties and some useful examples are described. There is a brief discussion of amalgamation (constructing polytopes with given facets and vertex-figures) and universality. Finally, there is a treatment of certain special properties of regular polytopes, such as central symmetry, flatness and collapsibility.

A regular polytope is called locally toroidal if its minimal infaces which are not spherical are toroidal. This chapter treats certain locally toroidal regular polytopes with not too many vertices, especially those which were subjects of ‘Abstract Regular Polytopes’; it includes some geometric descriptions and realizations. In the course of the investigation, some hitherto undescribed families of universal locally toroidal regular polytopes are presented.