Applied Categorical Structures – Official Blog

by Dikranjan, Dikran; Peinador, Elena Martín; Tarieladze, Vaja

We study various degrees of completeness for a Tychonoff space X. One of them plays a central role, namely X is called a Conway space if X is sequentially closed in its Stone–Čech compactification β
X (a prominent example of Conway spaces is provided by Dieudonné complete spaces). The Conway spaces constitute a bireflective subcategory of the category of Tychonoff spaces. Replacing sequential closure by the general notion of a closure operator C, we introduce analogously the subcategory

_C
of C-Conway spaces, that turns out to be again a bireflective subcategory of . We show that every bireflective subcategory of can be presented in this way by building a Galois connection between bireflective subcategories of and closure operators of finer than the Kuratowski closure. Other levels of completeness are considered for the (underlying topological spaces of) topological groups. A topological group G is sequentially complete if it is sequentially closed in its Raĭkov completion ${ \ifmmode\expandafter\tilde\else\expandafter\~\fi{G}}$. The sequential completeness for topological groups is stronger than Conway’s property, although they coincide in some classes of topological groups, for example: free (Abelian) topological groups, pseudocompact groups, etc.

DOI: 10.1007/s10485-007-9073-8
Online Date: 4/27/2007
Print publication date: 12/1/2007
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by Grandis, Marco

DOI: 10.1007/s10485-007-9086-3
Online Date: 4/27/2007
Print publication date: 8/1/2007
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by Kolesnikov, Pavel

In this note, we introduce a class of algebras that are in some sense related to conformal algebras. This class (called TC-algebras) includes Weyl algebras and some of their (associative and Lie) subalgebras. By a conformal algebra we generally mean what is known as H-pseudo-algebra over the polynomial Hopf algebra $H = \Bbbk [T_{1} , \ldots ,T_{n} ]$. Some recent results in structure theory of conformal algebras are applied to get a description of TC-algebras.

DOI: 10.1007/s10485-007-9077-4
Online Date: 4/26/2007
Print publication date: 4/1/2008
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by Martinez, Jorge

DOI: 10.1007/s10485-007-9089-0
Online Date: 4/19/2007
Print publication date: 4/1/2007
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by Porst, Hans-E.

We study the various categories of corings, coalgebras, and comodules from a categorical perspective. Emphasis is given to the question which properties of these categories can be seen as instances of general categorical resp. algebraic results. However, we also obtain new results concerning the existence of limits and of factorizations of morphisms.

DOI: 10.1007/s10485-007-9075-6
Online Date: 4/17/2007
Print publication date: 4/1/2008
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by Droste, Manfred; Holland, W. Charles

Aut(Ω) denotes the group of all order preserving permutations of the totally ordered set Ω, and if e ≤ u ∈ Aut(Ω), then B
_u
Aut(Ω) denotes the subgroup of all those permutations bounded pointwise by a power of u. It is known that if Aut(Ω) is highly transitive, then Aut(Ω) has just five normal subgroups. We show that if Aut(Ω) is highly transitive and u has just one interval of support, then B
_u
Aut(Ω) has $2^{2^{\aleph_0}}$ normal subgroups, and there is a certain ideal ${\cal Z}$ of the lattice of subsets of ($\mathbb{Z}$), the power set of the integers, such that the lattice of normal subgroups of every such Aut(Ω) is isomorphic to ${\cal Z}$.

DOI: 10.1007/s10485-007-9060-0
Online Date: 4/17/2007
Print publication date: 4/1/2007
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by Moreno, Yolanda

We obtain new categorical proofs that generalize the diagonal principles introduced in Castillo and Moreno (Israel J. Math. 140:253–270, 2004) to study the automorphic and partially automorphic character of Banach spaces. We then introduce and study the automorphy index $\mathfrak a(\cdot)$ for a Banach space, showing that $\mathfrak a(l_\infty)= \aleph_0$ while $\mathfrak a(C[0,1])=\aleph_1$.

DOI: 10.1007/s10485-007-9074-7
Online Date: 4/11/2007
Print publication date: 10/1/2008
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