On the relationship between compact regularity and Gentzen's cut rule

AbstractThe patch topology on a stably compact space, generalizing the Lawson topology on a domain, is a coreflection of stably compact spaces in compact regular spaces. This paper investigates compact regularity and the patch coreflection in multilingual sequent calculus (MLS), which can be regarded as a category of predicative representations of stably compact spaces. An object of MLS is a certain sort of generalization of the positive fragment of Gentzen's sequent calculus. We show that an object of MLS represents a compact regular space if and only if every sequent arises as an instance of Gentzen's cut rule with complete freedom to choose the placement of the cut formula.The relationship between compact regularity and Gentzen's cut rule is further explicated by the patch coreflection in MLS. The construction is a universal solution (up to a certain equivalence of tokens) to the problem of adding opposites to a logic, i.e., tokens that obey Gentzen's rules for negation. In the spectral case, this is equivalent to adding Boolean complements. The paper closes by considering the full subcategory of MLS consisting of objects with opposites. By taking contrapositives of sequents, we obtain an anti-involution on morphisms making this category equivalent to the Freyd/Scedrov allegory of compact regular spaces and closed binary relations. Moreover, the category of “maps” of this allegory is predicatively equivalent to the image of the patch functor