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Categorical generalization of a universal domain. (English) Zbl 0937.18002
The program started by J. Adámek [Math. Struct. Comput. Sci. 7, No. 5, 419-443 (1997; Zbl 0884.18006)] of transfering computationally interesting aspects of order theory, specifically domain theory, to a general categorical setting is continued here. Building on Adámek’s accessible SC-categories, which generalize Scott domains, the author introduces categorical counterparts of complete upper semilattices (CUSLs, these arise as sub-posets of compact elements of Scott domains), which he calls “finitely consistently cocomplete” (FCC), and of embedding-projection pairs. He then lifts Scott’s theorem [D. S. Scott, in: Proc. ICALP 1982, Lect. Notes Comput. Sci. 140, 577-613 (1982; Zbl 0495.68025)] which essentially asserts the existence of a weak terminal object in the category of countable CUSLs with embeddings as morphisms, to the FCC-setting. The proof is an application of V. Trnková’s embedding theorem [Comment. Math. Univ. Carolin. 7, 447-456 (1966; Zbl 0163.01502)]. While the resulting universal FCC-category turns out to be “too large” to have an accessible category (and hence an SC-category) as completion under directed colimits, for every inaccessible cardinal \(\kappa\) there is a “\(\kappa\)-small” version of the main result. In case of \(\kappa=\aleph_0\) one obtains a new proof of Scott’s original theorem.
18B15 Embedding theorems, universal categories
18B35 Preorders, orders, domains and lattices (viewed as categories)
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