一句话：提出 codensity lifting，将底范畴上的单子提升到纤维化全范畴，并给出对偶与代数操作提升。
本文研究如何把底范畴上的单子沿纤维化提升到全范畴，以支持比先前的 categorical TT-lifting 更广泛的纤维化。作者提出 codensity lifting：通过余密度构造逐纤维地提升单子，并给出对偶的 density lifting 用于余单子。文中展示了沿 Preorder、Top、扩展伪度量空间到 Set 的纤维化，以及可测空间间二元关系纤维化等例子。进一步，作者研究代数运算到 codensity lifting 的提升，并将具有纤维化小交的 posetal 纤维化上的单子提升刻画为某大图的极限。结果主要体现为范畴构造、示例与刻画定理。
We introduce a method to lift monads on the base category of a fibration to its total category. This method, which we call codensity lifting, is applicable to various fibrations which were not supported by its precursor, categorical TT-lifting. After introducing the codensity lifting, we illustrate some examples of codensity liftings of monads along the fibrations from the category of preorders, topological spaces and extended pseudometric spaces to the category of sets, and also the fibration from the category of binary relations between measurable spaces. We also introduce the dual method called density lifting of comonads. We next study the liftings of algebraic operations to the codensity liftings of monads. We also give a characterisation of the class of liftings of monads along posetal fibrations with fibred small meets as a limit of a certain large diagram.Comment: Extended version of the paper presented at CALCO 2015, accepted for publication in LMCS
@article{paperbot364,
title = {Codensity Lifting of Monads and its Dual},
author = {Shin-ya Katsumata and Tetsuya Sato and Tarmo Uustalu},
journal = {Logical Methods in Computer Science},
volume = {Volume 14, Issue 4},
year = {2018},
doi = {10.23638/lmcs-14(4:6)2018}
}