Categorical Duality Theory: With Applications to Domains, Convexity, and the Distribution Monad
Categorical Duality Theory: With Applications to Domains, Convexity, and the Distribution Monad
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分类对偶理论：及其在定义域、凸性和分布单子中的应用
DOI：
10.4230/lipics.csl.2013.500
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
Y. Maruyama
中科院分区：
文献类型：
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作者：
Y. Maruyama
利用和扩展的概念，从范畴拓扑和代数，我们设计了一个适度的一般理论之间的对偶代数，点自由空间和集理论，点集空间，其中包括无限斯通对偶，如著名的对偶框架之间（又名。locales）和拓扑空间，以及\sigma-完全布尔代数和可测空间之间的对偶，以及经典的有限Stone，Gelfand和Pontryagin对偶。在我们的理论的不同应用中，我们特别关注域-凸性对偶：从理论中我们导出Scott连续格和凸性空间之间的对偶，并利用所得的见解内在地确定分布单子代数的对偶伴随的对偶等价部分;巴特·雅各布斯揭示了对偶附加，但没有给出我们在这里给出的诱导等价的特征。在附录中，我们将范畴二重性置于更广泛的背景下，并阐明二重性的哲学基础。
Utilising and expanding concepts from categorical topology and algebra, we contrive a moderately general theory of dualities between algebraic, point-free spaces and set-theoretical, point-set spaces, which encompasses infinitary Stone dualities, such as the well-known duality between frames (aka. locales) and topological spaces, and a duality between \sigma-complete Boolean algebras and measurable spaces, as well as the classic finitary Stone, Gelfand, and Pontryagin dualities. Among different applications of our theory, we focus upon domain-convexity duality in particular: from the theory we derive a duality between Scott's continuous lattices and convexity spaces, and exploit the resulting insights to identify intrinsically the dual equivalence part of a dual adjunction for algebras of the distribution monad; the dual adjunction was uncovered by Bart Jacobs, but with no characterisation of the induced equivalence, which we do give here. In the Appendix, we place categorical duality in a wider context, and elucidate philosophical underpinnings of duality.