The limit-colimit coincidence theorem for -categories

In 1973 William Lawvere published a paper (reprinted as (Lawvere 2002)) where he explained that partial orders and metric spaces are examples of categories enriched in a closed category. Indeed, preorders are categories enriched over the two-element Boolean algebra 2, while (generalized) metric spaces are categories enriched over ([0,∞],+). Lawvere’s idea has been extremely influential in the forthcoming years. For example it led to the development of a unified categorical/algebraic description of many familiar elementary structures in mathematics: topology, uniformity, order, metric, etc. (Clementino and Hofmann 2003; Clementino and Tholen 2003; Clementino et al 2004; Hofmann 2007). At the same time, the idea was taken up by computer scientists in the hope that since posets have been so successfully used to create denotational semantics of programming languages, generalized metric spaces (gmses) could be useful too, especially for expressing quantitative properties of programs. As a consequence there are many studies of gmses concerned not as much with generalizing metric spaces as with generalizing domains and domain theory: (Rutten 1996; Flagg and Kopperman 1997; Flagg 1997) speak about Alexandroff and Scott topologies for generalized metric spaces; (America and Rutten 1989; Wagner 1994; Flagg and Kopperman 1995) are devoted to solving recursive domain equations in gmses; (Bonsangue et al 1998) proposes powerdomains for gmses; (Vickers 2005) completes a gms using rounded filters of formal balls, (Zhang and Fan 2005; Waszkiewicz 2009) analyze approximation and continuity in Q-posets, etc. Our work contributes to the same line of research: in this paper we present a generalisation of the limit-colimit coincidence theorem from domain theory, which states, in the nomenclature of (Abramsky and Jung 1994), that the category Dcpo has bilimits of expanding sequences. Concretely, we