Completing Simple Valuations in K-categories
Completing Simple Valuations in K-categories
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DOI：
10.1016/j.topol.2022.108192
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发表时间：
2020-02
期刊：
影响因子：
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通讯作者：
Xiaodong Jia;M. Mislove
中科院分区：
文献类型：
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作者：
Xiaodong Jia;M. Mislove
我们证明了简单赋值单子Vs的Keimel和Lawson的K-完备化Kc在每个象1-范畴K上定义了一个单子Kc ∈ Vs。我们还将Kc ∈ V s的Eilenberg-Moore代数定义为弱局部凸K-锥，并将其代数态射定义为连续线性映射.此外，我们明确地描述了Vs在Kc上的分配律，这使我们能够证明任何局部凸的K-完备化（分别是，弱局部凸，局部线性）拓扑锥是局部凸的（分别地，弱局部凸，局部线性）K-锥。我们还给出了一个例子--有顶的Cantor树--证明了简单赋值的dcpo-完备化不是一般简单赋值的D-完备化，这里D是单调收敛空间和连续映射的范畴.
We prove that Keimel and Lawson's K-completion K c of the simple valuation monad V s defines a monad K c∘ V s on each Image 1-category K. We also characterise the Eilenberg-Moore algebras of K c∘ V s as the weakly locally convex K-cones, and its algebra morphisms as the continuous linear maps. In addition, we explicitly describe the distributive law of V s over K c, which allows us to show that the K-completion of any locally convex (resp., weakly locally convex, locally linear) topological cone is a locally convex (resp., weakly locally convex, locally linear) K-cone. We also give an example–the Cantor tree with a top–that shows the dcpo-completion of the simple valuations is not the D-completion of the simple valuations in general, where D is the category of monotone convergence spaces and continuous maps.