An absolute characterisation of locally determined omega-colimits

Characterising colimiting ω-cocones of projection pairs in terms of least upper bounds of their embeddings and projections is important to the solution of recursive domain equations. We present a universal characterisation of this local property as ω-cocontinuity of locally continuous functors. We present a straightforward proof using the enriched Yoneda embedding. The proof can be generalised to Cattani and Fiore’s notion of locality for adjoint pairs. DOMAINS XI WORKSHOP CONTRIBUTED TALK In the category theoretic solution of recursive domain equations [SP82], several technical results hinge upon the fact that the universality of ω-cocones of projection pairs can be characterised locally in terms of least upper bounds (lubs) of their embeddings and projections. To fix terminology and notation, consider an O-category K . Let KPR be the Ocategory consisting of projection pairs f : A → B given by f = 〈 f : A → B, f : B → A 〉 where f ◦ f = idA and f ◦ f ≤ idB . Definition ([SP82, Definition 8]). We say that a cocone 〈C, c〉 for an ω-chain of projection pairs is locally determined when ∨ n∈N c L n ◦ c R n = idC . When all colimiting ω-cocones of projection pairs are locally determined, we say that the O-category has locally determined ω-colimits of projection pairs. For example, the category ωCPO of (not necessarily pointed) ω-cpos and continuous functions has locally determined ω-colimits. The importance of these cocones lies in the fact that every locally determined cocone is colimiting. As any locally continuous functor F : K → L gives a continuous functor FPR : KPR → LPR, given by FPRf ≔ 〈 Ff, Ff 〉 , and locally determined ω-cocones are preserved by these functors. Our contribution is to show the converse: Theorem. An ω-colimiting cocone of projection pairs is locally determined if and only if it is preserved by every locally continuous functor. Let K̂ be the O-category of O-presheaves, namely locally continuous functors and natural transformations from K to ωCPO. Let y : K → K̂ be the enriched Yoneda embedding yx ≔ ωCPO(−, x). Then, following from general principles [Kel82, Section 2.4], y is locally continuous and fully faithful. As is well-known, lubs and colimits in O-functor categories are given pointwise. The same argument shows that ω-colimits of projection pairs are also given componentwise in O-functor categories. Therefore: Proposition. If K , L are O-categories and L has locally determined ω-colimits of projection pairs, then so does the Ofunctor category L . In particular, every O-presheaf category K̂ has locally determined ω-colimits. We complete the proof of our theorem. Let 〈C, c〉 be any colimiting cocone that is preserved (in particular) by the locally continuous Yoneda embedding. As K̂ has locally determined ω-colimits: