Oxford Scholarship Online 2018
DOI: 10.1093/oso/9780198758914.003.0003
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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 conside… Show more

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Cited by 9 publications
(37 citation statements)
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“…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.…”
Section: Free Structures and Syntactic Categoriesmentioning
confidence: 99%
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“…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.…”
Section: Free Structures and Syntactic Categoriesmentioning
confidence: 99%
“…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.…”
Section: Free Structures and Syntactic Categoriesmentioning
confidence: 99%
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