The Lter Construction Revisited

The lter construction, as an endo-functor on the category of small coherent categories, was used extensively by A. Pitts in a series of papers in the 80's to prove completeness and interpolation results. Later I. Moerdijk and E. Palmgren used the lter construction to construct non-standard models of Heyting arithmetic. In this paper we describe lter construction as a left-adjoint: applied to a left-exact category it is simply the completion of subobject semi-lattices under ltered (and thus all) meets. We study ltered coherent logic which is coherent logic extended to arbitrary meets and the rules that existential quantiication and binary disjunction distribute over ltered meets. This logic is sound and complete for interpretations in Pitts' ltered coherent categories, and conservative over coherent logic. Restricting further to rst-order logic we show that the minimal models of Heyting arithmetic described by Moerdijk and Palmgren are nothing but generic models of Heyting arithmetic in which truth of sequents V I 'i) V J j for 'i; j nitary rst-order formulae is governed by rst-order variants of the rules described above. Filter constructions have been described in many cases, 11] seems to be the rst example. A. Blass 2] studies two categories of lters and their relation. The rst is the category Sub=2 where Sub is the contravariant subobject functor Set op ! ^-SLat from sets to meet-semi lattices, and the other is the category FSet of lters on subobjects of sets, and morphisms equivalence classes of local continuous maps (see Section 2 for the precise deenition). In a series of papers 18, 19, 20] A. Pitts used this second lter construction on coherent categories to prove completeness and interpolation results in categorical logic. Closely related is the construction of taking ideals of lters, which is (the object part of) a contravariant embedding of the category of distributive lattices into the category of locales. This functor restricts to an embedding of the category of Heyting algebras into the category of complete Heyting algebras, see 18] for details. I. Moerdijk 15], motivated by models of synthetic diierential geometry, used a lter construction to get a non-standard model of Heyting arithmetic. This model is further explained in 16]. Our paper exhibits the universal property of the lter construction: in Section 2, we describe it as a functor F: Lex ! V lt:-Lex from small categories with nite limits to those small categories with nite limits such …