Topos-theoretic background

Abstract

This chapter provides the topos-theoretic background necessary for understanding the contents of the book; the presentation is self-contained and only assumes a basic familiarity with the language of category theory. The chapter begins by reviewing the basic theory of Grothendieck toposes, including the fundamental equivalence between geometric morphisms and flat functors. Then it presents the notion of first-order theory and the various deductive systems for fragments of first-order logic that will be considered in the course of the book, notably including that of geometric logic. Further, it discusses categorical semantics, i.e. the interpretation of first-order theories in categories possessing ‘enough’ structure. Lastly, the key concept of syntactic category of a first-order theory is reviewed; this notion will be used in Chapter 2 for constructing classifying toposes of geometric theories.

Priestley-type dualities for partially ordered structures

“…As remarked in [6], the theory of syntactic categories can be profitably applied to the problem of constructing structures presented by generators and relations. In fact, any syntactic category of a given theory T can be regarded, in a sense that we shall not make precise in the present paper, as a structure presented by a set of 'generators', given by the sorts in the signature of the theory T, subject to 'relations' expressed by the axioms of the theory T. Conversely, to any structure C one can attach a canonical signature Σ C to express 'relations' holding in the structure, consisting of one sort ⌜c⌝ for each element c of C and possibly function or relation symbols whose canonical interpretation in C coincide with specified functions or subsets in C in terms of which the designated 'relations' holding in C can be formally expressed; over such a canonical signature one can then write down axioms possibly involving generalized connectives and quantifiers so to obtain a Stheory (in the sense of section 8 of [5]) T whose S-syntactic category C S T can be identified with 'the free structure on C subject to the relations R', meaning that the S-structure D in which the relations R are satisfied naturally correspond to the S-homomorphism C S T → D, in a way which can be concretely described as follows.…”

“…In fact, any syntactic category of a given theory T can be regarded, in a sense that we shall not make precise in the present paper, as a structure presented by a set of 'generators', given by the sorts in the signature of the theory T, subject to 'relations' expressed by the axioms of the theory T. Conversely, to any structure C one can attach a canonical signature Σ C to express 'relations' holding in the structure, consisting of one sort ⌜c⌝ for each element c of C and possibly function or relation symbols whose canonical interpretation in C coincide with specified functions or subsets in C in terms of which the designated 'relations' holding in C can be formally expressed; over such a canonical signature one can then write down axioms possibly involving generalized connectives and quantifiers so to obtain a Stheory (in the sense of section 8 of [5]) T whose S-syntactic category C S T can be identified with 'the free structure on C subject to the relations R', meaning that the S-structure D in which the relations R are satisfied naturally correspond to the S-homomorphism C S T → D, in a way which can be concretely described as follows. To any S-structure D we can canonically associate a S-homomorphism C → D, assigning to any element c of C the interpretation of ⌜c⌝ in D; in particular we have a canonical S-morphism i ∶ C → C S T , in terms of which the universal property of C S T can be expressed by saying that any S-homomorphism f ∶ C → D to a S-structure D in which the relations R are satisfied can be extended, uniquely up to isomorphism, along the canonical morphism i, to a S-homomorphism C S T → D.…”

“…[6] for a comprehensive treatment of this context); in this paper we shall in particular be concerned with the construction of Boolean algebras presented by generators and relations.…”

“…As we already remarked in [6], the methods of Topos Theory can be profitably exploited to obtain such descriptions starting from logical descriptions of the given structures; indeed, any topos which can be naturally attached to the given structure in such a way that the structure can be recovered from it up to isomorphism (for example, a topos from which the structure can be recovered as its full subcategory of C-compact objects for a topos-theoretic invariant C -cf. [6] for many examples of such situations), admits in general many different representations, which can be obtained by using a variety of topos-theoretic methods; any such representation (of algebraic, resp. geometric or topological) nature will thus produce a corresponding representation (of algebraic, resp.…”

“…As it is wellknown, topologically the duality is based on the patch topology construction, while algebraically the Boolean algebra of clopen sets of the Priestley space associated to a distributive lattice can be characterized as the free Boolean algebra on it. The unification between the algebraic and the topological formulations of the duality is conveniently provided by the notion of topos; in fact, the toposes involved in Priestley-type dualities admit, on one hand, an algebraic representation (as categories of sheaves on a preordered structure with respect to an appropriate Grothendieck topologies on it) and on the other hand a topological one (as categories of sheaves on natural spectra of the structures, as provided by the techniques of [6]). …”