162 research outputs found
Thin fillers in the cubical nerves of omega-categories
It is shown that the cubical nerve of a strict omega-category is a sequence
of sets with cubical face operations and distinguished subclasses of thin
elements satisfying certain thin filler conditions. It is also shown that a
sequence of this type is the cubical nerve of a strict omega-category unique up
to isomorphism; the cubical nerve functor is therefore an equivalence of
categories. The sequences of sets involved are in effect the analogues of
cubical T-complexes appropriate for strict omega-categories. Degeneracies are
not required in the definition of these sequences, but can in fact be
constructed as thin fillers. The proof of the thin filler conditions uses chain
complexes and chain homotopies.Comment: Revised version to appear in Theory and Applications of Categories;
changed terminology; additional figures, examples and references; 27 page
Omega-categories and chain complexes
There are several ways to construct omega-categories from combinatorial
objects such as pasting schemes or parity complexes. We make these
constructions into a functor on a category of chain complexes with additional
structure, which we call augmented directed complexes. This functor from
augmented directed complexes to omega-categories has a left adjoint, and the
adjunction restricts to an equivalence on a category of augmented directed
complexes with good bases. The omega-categories equivalent to augmented
directed complexes with good bases include the omega-categories associated to
globes, simplexes and cubes; thus the morphisms between these omega-categories
are determined by morphisms between chain complexes. It follows that the entire
theory of omega-categories can be expressed in terms of chain complexes; in
particular we describe the biclosed monoidal structure on omega-categories and
calculate some internal homomorphism objects.Comment: 18 pages; as published, with minor changes from version
Simple omega-categories and chain complexes
The category of strict omega-categories has an important full subcategory
whose objects are the simple omega-categories freely generated by planar trees
or by globular cardinals. We give a simple description of this subcategory in
terms of chain complexes, and we give a similar description of the opposite
category, the category of finite discs, in terms of cochain complexes. Berger
has shown that the category of simple omega-categories has a filtration by
iterated wreath products of the simplex category. We generalise his result by
considering wreath products of categories of chain complexes over the simplex
category.Comment: 14 pages; v2 has minor corrections and a little additional materia
Harmonic equiangular tight frames comprised of regular simplices
An equiangular tight frame (ETF) is a sequence of unit-norm vectors in a
Euclidean space whose coherence achieves equality in the Welch bound, and thus
yields an optimal packing in a projective space. A regular simplex is a simple
type of ETF in which the number of vectors is one more than the dimension of
the underlying space. More sophisticated examples include harmonic ETFs which
equate to difference sets in finite abelian groups. Recently, it was shown that
some harmonic ETFs are comprised of regular simplices. In this paper, we
continue the investigation into these special harmonic ETFs. We begin by
characterizing when the subspaces that are spanned by the ETF's regular
simplices form an equi-isoclinic tight fusion frame (EITFF), which is a type of
optimal packing in a Grassmannian space. We shall see that every difference set
that produces an EITFF in this way also yields a complex circulant conference
matrix. Next, we consider a subclass of these difference sets that can be
factored in terms of a smaller difference set and a relative difference set. It
turns out that these relative difference sets lend themselves to a second,
related and yet distinct, construction of complex circulant conference
matrices. Finally, we provide explicit infinite families of ETFs to which this
theory applies
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