Semantic Domains, Injective Spaces and Monads (extended Abstract)
Semantic Domains, Injective Spaces and Monads (extended Abstract)
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发表时间：
2007
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通讯作者：
R. Flagg
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作者：
R. Flagg
许多类别的语义域可以考虑从秩序理论的观点，并从拓扑的观点通过斯科特拓扑。拓扑的观点是特别富有成效的考虑可计算性的经典空间，如欧几里德真实的线。当一个拓扑空间嵌入到域中，一个要求的斯科特连续映射之间的主机域完全捕获的连续映射之间的客人拓扑空间。宿主域的这种性质称为注入性。例如，连续Scott域被刻画为稠密子空间嵌入上的内射空间（Dana Scott，1972，1980）。从第三种观点来看，连续的Scott域是作为单子的代数而出现的（Wyler，1985）。内射性的拓扑刻画是从代数刻画和一般范畴理论（Escardd o 1998）中得出的。本文系统地研究了在语义和拓扑中出现的单子，得到了新的证明，并利用内射性发现了语义域和拓扑空间的新刻画。
Many categories of semantic domains can be considered from an order-theoretic point of view and from a topological point of view via the Scott topology. The topological point of view is particularly fruitful for considerations of computability in classical spaces such as the Euclidean real line. When one embeds topological spaces into domains, one requires that the Scott continuous maps between the host domains fully capture the continuous maps between the guest topological spaces. This property of the host domains is known as injectivity. For example, the continuous Scott domains are characterized as the injective spaces over dense subspace embeddings (Dana Scott, 1972, 1980). From a third point of view, the continuous Scott domains arise as the algebras of a monad (Wyler, 1985). The topological characterization by injectivity turns out to follow from the algebraic characterization and general category theory (Escardd o 1998). In this paper we systematically consider monads that arise in semantics and topology, obtaining new proofs and discovering new characterizations of semantic domains and topological spaces by injectivity.