Applied Categorical Structures – Official Blog

by Schauerte, Anneliese

The category of coherent biframes is shown to be equivalent to that of the coupled lattices, and dually equivalent to the spectral bispaces. Stably continuous biframes arise as the retracts of the coherent biframes. The coherent and the stably continuous biframes are coreflective in all biframes. Weak local compactness is introduced, and in conjunction with regularity, is shown to be sufficient for the construction of smallest compactifications.

DOI: 10.1007/s10485-006-9023-x
Online Date: 7/29/2006
Print publication date: 6/1/2006
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by Ball, Richard N.; Hager, Anthony W.

$W$ is the category of archimedean $l$-groups with distinguished weak order unit, with $l$-group homomorphisms which preserve unit. This category includes all rings of continuous functions $C(X)$ and all rings of measurable functions modulo null functions, with ring homomorphisms. The authors, and others, have studied previously the epimorphisms (right-cancellable morphisms) in $W$. There is a rich theory. In this paper, we describe a topological approach to the analysis of these epimorphisms. On each $W$– object$B$, we define a topology $\tau^{B}$ and a convergence $\mathop{\longrightarrow}\limits^{B}$. These have the same closure operator, and this closure “captures epics” in the sense: a divisible subobject $A$ of $B$ is dense iff $A$ is epically embedded. The topology is $T_{1}$, but only sometimes Hausdorff or an $l$-group topology. The convergence is a Hausdorff $l$-group convergence, but only sometimes topological. The associations of $B$ to $\tau^{B}$, and to $\mathop{\longrightarrow}\limits^{B}$, are functorial.

DOI: 10.1007/s10485-006-9021-z
Online Date: 7/21/2006
Print publication date: 4/1/2007
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by Grandis, Marco

We have introduced, in a previous paper, the fundamental lax 2-category of a ‘directed space’ $X$. Here we show that, when
$ X $
has a
$T_1$-topology, this structure can be embedded into a larger one, with the same objects (the points of $ X$), the same arrows (the directed paths) and the same cells (based on directed homotopies of paths), but a larger system of comparison cells. The new comparison cells are absolute, in the sense that they only depend on the arrows themselves rather than on their syntactic expression, as in the usual settings of lax or weak structures. It follows that, in the original structure, all the diagrams of comparison cells commute, even if not constructed in a natural way and even if the composed cells need not stay within the old system.

DOI: 10.1007/s10485-006-9017-8
Online Date: 7/19/2006
Print publication date: 6/1/2006
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by Dube, Themba; Walters-Wayland, Joanne

A frame homomorphism is coz-onto if it maps the cozero part of its domain surjectively onto that of its codomain. This captures the notion of a z-embedded subspace of a topological space in a point-free setting. We give three different types of characterizations of coz-onto homomorphisms. The first is in terms of elements, the second in terms of quotients, and the last in terms of ideals. As an application of properties of coz-onto homomorphisms developed herein, we present some characterizations of $F$- and $F^{\prime }$-frames.

DOI: 10.1007/s10485-006-9022-y
Online Date: 7/12/2006
Print publication date: 4/1/2007
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by Zeibig, G.

Let $(A,\mu^A, \eta^A)$ and $(B, \mu^B, \eta^B)$ be two monoids (algebras) in a monoidal category $${\left( {{\user1{\mathcal{V}}}{\text{,}}\,{\user1{\square }},\,e} \right)}$$. Further let $\iota: B\Box A \to A\Box B$ be a distributive law in the sense of [J. Beck, Lect. Notes Math., 80:119–140, 1969]; $\iota$ naturally yields a monoid $(A\Box B, \eta, \mu)$. Consider a word $W’$ in the symbols $A$, $B$, and $e$. The first coherence theorem proved in this paper asserts that all morphisms $W’ \to A\Box B$ coincide in $${\user1{\mathcal{V}}}$$, provided they arise as composites of morphisms which are $\Box$-products of $${\user1{\mathcal{V}}}$$’s ‘canonical’ structure morphisms, and of $1_A$, $1_B$, $1_e$, $\eta^A$, $\mu^A$, $\eta^B$, $\mu^B$, and $\iota$. Assume now that an object $X$ is endowed with both an ${\left( {A,\mu ^{A} ,\eta ^{A} } \right)}$ -object structure $(X,\nu^A)$, and an ${\left( {B,\mu ^{B} ,\eta ^{B} } \right)}$ -object structure $(X,\nu^B)$. Further assume that these two structures are compatible, in the sense that they naturally yield an ${\left( {A\Box B,\mu ,\eta } \right)}$-object $(A\Box B, \nu)$. Let $W”$ be a word in $A$, $B$, $e$, and $X$, which contains a single instance of $X$, in the rightmost position. The second coherence theorem states that all morphisms $W” \to X$ coincide in $${\user1{\mathcal{V}}}$$, provided they arise as composites of morphisms which are $\Box$-products of $${\user1{\mathcal{V}}}$$’s ‘canonical’ structure morphisms, and of $1_A$, $1_B$, $1_e$, $\eta^A$, $\mu^A$, $\eta^B$, $\mu^B$, $\iota$, $\nu^A$, and $$\nu ^{B} $$.

DOI: 10.1007/s10485-006-9015-x
Online Date: 7/8/2006
Print publication date: 6/1/2006
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