Towards Stone duality for topological theories

Dirk Hofmann, Isar Stubbe
2011 Topology and its Applications  
furthermore preserve weighted limits and T-weighted colimits. By construction, we have a canonical forgetful functor T-Frm −→ V-Cont. Earlier we already explained that V ∈ T-Cat and that the representable functor T-Cat(−, V) : T-Cat op −→ Ord lifts to a functor V − : T-Cat op −→ V-Cont. Now we can prove: Proof. For each T-functor f : X −→ Y , the underlying V-graph of the T-graph V X is a complete V-category, and V f : V Y −→ V X is a T-graph morphism which preserves all weighted limits and all
more » ... T-weighted colimits. Furthermore A = V X satisfies the distributivity axiom in Definition 2.9 since the presheaf V-category P (S X) is completely distributive, and A is closed in P (S X) under weighted limits and T-weighted colimits. 2 But there is more: Proof. This is done by putting on T-Frm(X, V) the largest T-category structure that makes all evaluation maps ev X, Note how, in the two previous corollaries, V plays the role of a dualising object: it is on the one hand an object of T-Cat, and as such represents the functor Ω : T-Cat op −→ T-Frm; but it is also an object of T-Frm, and as such represents the functor pt : (T-Frm) op −→ T-Cat. Next we observe: Proposition 2.12. There is a natural transformation η : Id ⇒ pt ·Ω with components We do not know whether η X is always fully faithful, but we do have the following result (recall that E is the unit for the tensor in T-Cat): Theorem 2.13. For any X ∈ T-Cat, pt(Ω( X)) has the same objects as the Cauchy completionX of X . In fact, we have an isomorphism Map(T-Dist)(E, X) −→ T-Frm(Ω(X), V) of ordered sets, making the diagram commute. Hence X is Cauchy complete if and only if η X is surjective. The proof of the theorem above is the combination of the results below. Lemma 2.14. Let X = (X, a) be a T-category and ϕ : X −→ V be a T-functor. Then the representable V-functor Φ = [ϕ, −] : Ω(X) −→ V is also a T-graph morphism and preserves infima and cotensors. Moreover, if ψ ϕ in T-Dist, then Φ preserves also tensors and T-suprema. Proof. Being a representable V-functor, Φ preserves infima and cotensors. To see that Φ is a T-graph morphism, recall first that ϕ, ϕ = x∈ X hom ϕ(x), ϕ (x) . Since : V X D −→ V (with X D = (X, e X ) being the discrete T-category) is a T-graph morphism, it is enough to show that is a T-graph morphism. But Ψ is just the mate of the composite
doi:10.1016/j.topol.2011.01.010 fatcat:mlq7p4grtjggrckhwlkyi3wh5m