The closure of chains of embedding-projection pairs (ep-pairs) under bilimits in some categories of predomains and domains is standard and well-known. For instance, Scott's $D_\infty$ construction is well-known to produce directed bilimits of ep-pairs in the category of directed-complete partial orders, and de Jong and Escard\'o have formalized this result in the constructive domain theory of a topos. The explicit construcition of bilimits for categories of predomains and partial maps is considerably murkier as far as constructivity is concerned; most expositions employ the constructive taboo that every lift-algebra is free, reducing the problem to the construction of bilimits in a category of pointed domains and strict maps. An explicit construction of the bilimit is proposed in the dissertation of Claire Jones, but no proof is given so it remained unclear if the category of dcpos and partial maps was closed under directed bilimits of ep-pairs in a topos. We provide a (Grothendieck)-topos-valid proof that the category of dcpos and partial maps between them is closed under bilimits; then we describe some applications toward models of axiomatic and synthetic domain theory.
翻译:在某些类别的前域和域中,在双界限下封闭嵌入式投影配对(ep-pairs)链(ep-pairs)是标准且众所周知的。例如,斯科特的$D ⁇ infty$建造公司众所周知,在定向-完整部分订单类别中直接设定双界限,而德钟和Escard\'o则正式确定了这一结果,这是具有建设性的托词理论。就建构而言,对预设和部分地图类别的双界限的明确解释相当模糊;多数解释采用了建设性的禁忌,即每个升升升数布拉都是免费的,将问题降低到在某一特定领域和严格地图类别中设定双界限。Claire Jones的解析提议明确构建了双界限,但是没有提供证据,因此仍然不清楚,如果在指定双界限和部分地图类别中,在指定的双界限下关闭,那么,我们提供了一张(Grothendimical-algebrale)的建设性禁忌,在当时的模型中提供了部分版图的封闭式图和部分域域图的封闭式图。