Lawvere Completion and Separation Via Closure

Abstract

Abstract. For a quantale V, first a closure-theoretic approach to completeness and separation in V-categories is presented. This approach is then generalized to T-categories, where T is a topological theory that entails a set monad Ì and a compatible Ì-algebra structure on V.

Injective spaces via adjunction

“…In [19] we observed already that the down-closure as well as the up-closure of a Zariski-closed set is again Zariski-closed. A presheaf ψ ∈X can be identified with the Zariski-closed and down-closed subset A = ψ −1 (1) ⊆ UX , and we consider The Yoneda map y X : X −→X is given by y X (x) = {x ∈ UX | x → x}.…”

“…(1) can be found in [19], (2) follows immediately from (1) for all x ∈ TX and y ∈ Y . Such a function g : Y −→ X is necessarily a T -functor.…”

“…For each filter f on X we can consider f ∩ τ ∈ F 0 X and f ∩ σ ∈ F 1 X , and f is determined by this restriction precisely if f has a basis of open respectively closed sets. In [19] we showed that f =  …”

“…Hence, f ∼ = g if and only if f * = g * , which in turn is equivalent to f * = g * . We call a T -category X L-separated (see [19] for details) whenever f ∼ = g implies f = g, for all T -functors f , g : Y −→ X with codomain X . One easily verifies that it is enough to consider here the case Y = G = (1, e • 1 ).…”

“…A different proof of this property of V can be found in [19,Lemma 3.18]. Note that also in the proof of [19] the condition T 1 = 1 is crucial.…”