Our aim is to translate the order-theoretic notion of a continuous poset into a category theory setting. The generalisation is based on "way-below morphisms", which replace the well-known way-below relation. The emphasis of this note is on motivation; hence, most proofs are omitted. The ideas presented here are special cases of a more general construction which will appear in [K]. However, it seems useful to extract the aspects interesting to an audience mainly oriented towards computer science.
After comparing the familiar order-theoretic notions of finiteness and algebraicity with well-known categorical concepts, we show that way-below morphisms arise as a very natural generalization of the way-below relation for posets. (The "wavy arrows" of Johnstone and Joyal [JJ] present a different generalisation.) Next, following the order-theoretic ideas of Erné [E0], we try to eliminate the cocompleteness requirements on the categories. This essentially amounts to defining categorical analogues of Dedekind cuts.
There are basically two ways of generalizing the notion of an ideal from lattices to partially ordered sets. Either one can consider order ideals (which are just directed lower sets) or ideals in the sense of Frink [F] (which are lower sets containing the Dedekind cuts generated by each of their finite subsets). These lead to different generalizations of the concepts of algebraicity and continuity for posets, originally defined for complete lattices.
In order to translate these order-theoretical notions into categorical terms, we recall that the principal ideals are supremum-dense in the complete lattice of lower sets and how different kinds of lower sets can be characterized in terms of principal ideals. The lower sets of a poset (X,≤) via characteristic functions can be interpreted as monotone functions from (X,≤)op into the two-element chain 2. If one considers posets as categories "over 2" (in the sense of V-categories), it seems natural to study the contravariant SET-valued functors for an ordinary category X instead. Then, in particular, representable functors will correspond to principal ideals.
Johnstone and Joyal [JJ] in their approach to continuous categories have used a categorical version of order ideals, namely directed colimits of representable functors, to play the rôle of ideals. Frink ideals on the other hand should correspond to directed colimits of the categorical analogues of Dedekind cuts. We will investigate the latter approach.
KeywordsComplete Lattice Principal Ideal Order Ideal Continuous Lattice Dual Notion
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