AI 中文总结
研究任意范畴上任意余单子的拓扑空间理论，以密度余单子为核心方法，刻画了拓扑空间，证明了相关范畴的完备性等性质，还通过“晕”找到拓扑直观并给出反例附录。
AI 中文摘要
集合上的余单子同时推广了范畴和拓扑空间。基于加纳关于离子单子的工作，我们为任意范畴上的任意余单子发展拓扑空间理论。我们的方法以密度余单子为中心，它提供了子基的抽象。我们从密度余单子角度研究子基和基，从余代数范畴间的函子角度研究余单子的连续映射，其定义恢复了拓扑空间的常见概念。我们将拓扑空间刻画为集合子集图的密度余单子，表明集合上的每个余单子都有一个基础拓扑空间和一个基础小范畴，还证明了集合上带连续映射的所有余单子的范畴是完备的，其可及余单子的全子范畴是余完备的。连续映射和普通余单子态射构成一个双范畴。我们用“晕”为这些概念找到拓扑直观，“晕”是点的无穷小邻域的抽象，定义为邻域系统的形式极限。我们还包含了一个反例长附录，许多适用于一般（余）单子理论。
英文摘要
Comonads on Set generalize both categories and topological spaces. Expanding upon Garner's work on ionads, we develop aspects of the theory of topological spaces for arbitrary comonads on arbitrary categories. Our approach is centered around density comonads, which provide an abstraction of subbases. We study subbases as well as bases in terms of density comonads, and we study continuous maps of comonads in terms of functors between coalgebra categories, with definitions that recover the usual notions for topological spaces. Whereas Ahman and Uustalu characterized categories as precisely the polynomial comonads on Set, we characterize topological spaces as precisely the density comonads of diagrams of subsets of a set, which are familiar as topological subbases. We show that every comonad on Set has an underlying topological space, and that this construction is a reflection with respect to continuous maps; similarly, every comonad on Set has an underlying small category, and this construction is a coreflection. We also show that the category of all comonads on Set with continuous maps is complete, and that its full subcategory of accessible comonads is cocomplete. Continuous maps and ordinary comonad morphisms form a double category, which, in the case of the polynomial comonads on Set, recovers the double category of functors and retrofunctors of Clarke and Di Meglio. We find topological intuition for these concepts in terms of "halos", an abstraction of infinitesimal neighborhoods of points, defined as formal limits of neighborhood systems. We include a long appendix of counterexamples, many applicable to general (co)monad theory rather than the particular concerns of this text.
