M. Makkai and G. E. Reyes. First Order Categorical Logic, Lecture Notes in Math. Vol. 611 (Springer-Verlag, Berlin, 1977).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....theory which will be used throughout this thesis, aiming to set up notation for monads, tensorial strengths and let categories. We also provide a few elementary examples of let categories which illustrate their use in computer science. For some background in categorical logic see [Law69] and [MR77]. We review the computational let calculus and motivate both its uses and also the form in which it is usually presented. This leads to a discussion of suitable extensions and in particular to the fixpoint type: see [NPS90] for some background in type theory. In the presence of a fixpoint type ....

M. Makkai and G.E. Reyes. First Order Categorical Logic. Lecture Notes In Mathematics. Springer-Verlag, 1977.

....with S. Awodey, 2] In fact, they give a detailed exposition of one of the completeness result presented there. In case of pure typed calculus, a more detailed exposition can be found in [1] Our overall presentation is in the line of categorical model theory, as was done for geometric logic in [15] and for first order logic in [6] One of the more prominent theories which can be formulated in the logic L is SDG, synthetic differential geometry [13] In contrast to this we do not intend to do proof theory here, as was one of the items in [14] 1 Syntax We begin by describing the syntax ....

M. Makkai and G. E. Reyes. First Order Categorical Logic, Lecture Notes in Mathematics 611. Springer--Verlag, Berlin, 1977.

....as ff geometric sketches. That is, a) A category axiomatizable by an ff geometric theory is sketchable by an ff geometric sketch, and (b) A category sketchable by an ff geometric sketch is axiomatizable by an ff geometric theory. Statement (a) which is, essentially, present already in [MR, Section 8.4] without mentioning sketches) continues to hold for ff = But Statement (b) does not: there are well known examples of finitarily sketchable categories which are not coherently axiomatizable, see also Example 6 below. From our result on the equivalence of finitary and 1 ....

M. Makkai and G. E. Reyes, First Order Categorical Logic, Lecture N. Math. 611, Springer 1977.

.... at all with locales, you ll understand that generators and relations for a frame give propositions and axioms for a propositional geometric theory (see [9] Some Logical Manipulations I shall not attempt to give a full set of proof rules for the logic you can find them in Makkai and Reyes[8]. There are no surprises, because (apart from the infinite disjunctions) the logic is a restriction of standard logic. On a short digression, let me show how to simplify the sequents considerably. Proposition 1 Every geometric formula OE(x) is equivalent to one of the form W i (E i 9y i : V ....

....these generalized models, for the logic appropriate to an elementary topos is not classical but intuitionistic. But this is not just a concession to the generalizing ambitions of constructivists. It has key significance in at least three ways. First, the proof rules of geometric logic (as in [8]) are classically incomplete: that is to say, within a given theory, there may be a sequent that holds in all the classical set theoretic models but is not provable from the axioms using the proof rules. Even in the propositional case, there are theories that have no models at all, but which are ....

Michael Makkai and Gonzalo E. Reyes, First Order Categorical Logic, Lecture Notes in Mathematics 611, Springer-Verlag, 1977.

....mentioned above (Proposition 1.9) the inverse image functor, f , does not preserve joins, and so the category Sh(Q) is neither coherent, nor Heyting (additionally, f does not preserve s and s) although it is regular. Therefore, formulae are not, in general, stable (in the sense of [11]) and the property of substitution is no longer valid here, i.e. x ] C 6= h[ 1 ] C ; n ] C i [ C( We shall return to this point later. De nition 2.7 (1) A sequent in L is an object S = where and are nite sequences of formulae such that has ....

.... [ x] 1 [ x] n [ x] 1 [ x] k [ x] corresponds to (f ) for an appropriate morphism f (depending on the context of interpretation) The property of substitution has the form [ x ] C 0 = f ; therefore, stability (in the sense of [11]) means that (f ) f for every morphism f (with appropriate codomain) De nition 4.2 Given a structure I, let be a scheme in B and f : A B a morphism. We say that is left stable (right stable, resp. for f if (f ) f ( f ) f , resp. is stable for f if ....

M. Makkai, G.E. Reyes, First-Order Categorical Logic, Lecture Notes in Mathematics, Vol. 611, Springer, Berlin, 1977.

....of a Scott topos E under finite limits. This is equivalent to saying that the full subcategory E coh of E , consisting of the coherent objects, is closed under all finite colimits. In turn this is equivalent to the condition E f = E coh (cf. 4] Expose VI, as well as the appendix of [14]) Such toposes are called perfect in [4] and include all the coherent toposes satisfying the noetherian condition on their lattices of subobjects, notably the simplicial topos, and, among others the Zariski and the etale topos over a scheme. Notice that when E = Set K f is a perfect topos ....

M. Makkai, G. Reyes, First order categorical logic, in Lecture Notes in Mathematics 611, Springer (1977)

....a classical result of Makkai and Reyes, stating that any topos with enough points has an open spatial cover. We show that one can achieve in addition that the cover is connected and locally connected. 1 Introduction Sheaf models of infinitary first order theories were studied intensively in [9]. They are models in Grothendieck toposes and subsume Heyting valued models, boolean valued models, Kripke models, and permutation models. Of particular interest are so called geometric theories T (see Definition 2.1) To these theories one can associate a particular topos B(T) together with a ....

....is geometrically definable (functional completeness) iii) The inverse image preserves function types of first order definable types. Part (ii) and (iii) of the theorem extend a well known result by Makkai and Reyes, stating that any topos with enough points has an open spatial cover ([9], Theorem 6.3.3) This result appears here as part (i) of the theorem. Part (iii) of the theorem needs some explanation. Even if the language has already function types there is no guaranty that these function types are interpreted by exponentials of sheaves. But one can always add new function ....

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M. Makkai and G. E. Reyes. First Order Categorical Logic, Lecture Notes in Mathematics 611. Springer--Verlag, Berlin, 1977.

....in 1963. The rst step was to consider the operation of substitution as the composition of morphisms and the quanti ers as adjoints of the substitution. Since then numerous works in the area have appeared due to Reyes, Joyal, Kock and Makkai, among others. In the book of Makkai and Reyes (c.f. [5], the basis of this paper) the conceptual basis for rst order categorical logic is introduced. They consider nite quanti er in nitary logics, usually denoted by L1 , where many sorts are allowed. The basic idea is that theories can be seen as certain categories (called logical categories) and ....

....counterpart, rede ning the logical calculi G 1 TT and G 2 TT of Makkai and Reyes and proving the soundness and completeness theorems. 1. Some De nitions from Category Theory In this section we list some classical de nition and results which will be useful throughout this paper (c.f. [3, 4, 5]) and which can be skipped by the experienced reader. Definition 1.1. Let C be a xed category with pullbacks. 1) Let f : B C be a morphism and R, C a subobject of C. The inverse image of R by f is the subobject f (R) B of B obtained in the pullback of R, C, B f C. 2) A morphism g : ....

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M. Makkai, G.E. Reyes, First-Order Categorical Logic, Lecture Notes in Mathematics 611, Springer-Verlag, 1977.

....system. A model of the information system is defined to be a functor from DC to the category Set and snapshots of a model are described in terms of functors from SC to Set. 1 Introduction In the last three decades the study of syntax and semantics has benefited greatly from category theory [7], 5] A formal theory has associated with it a category (generally a free category in an appropriate sense) We call this category the syntactic category of the formal theory. A model of the theory (i.e. an algebra ) is a functor from the syntactic category to the category Set of sets and ....

....G : X Theta Y Bool, there are corresponding subtypes dGe and d:Ge of X Theta Y . The categories we use have sufficient structure so that they have canonical languages associated with them. That is to say, the categories are coherent categories, which are the same as the logical categories of [7]. We also require our categories to have further structure: notably every object is to have a decidable equality relation and decidable equivalence relations are to have coequalizers. The logic associated to such a category is what is called coherent logic: this is not quite full first order ....

Michael Makkai and Gonzalo E. Reyes, First Order Categorical Logic, Lecture Notes in Mathematics, 611, Springer-Verlag, 1977.

.... determined, as in (i) by the element A 0 2 P(X 0 ) given by A 0 (x 0 ) def , 9x : X : F (x 0 ; x) A(x) Recall that a category E is a logos if it has finite limits, pullback stable images and dual images of subobjects along morphisms, and pullback stable finite joins of subobjects (Makkai and Reyes 1977). Any category E with finite limits determines an E indexed poset Sub E : E op Poset mapping E objects to their posets of subobjects and mapping E morphisms to pullback functions. Well of course the posets involved may actually be poclasses unless one assumes E is well powered, but size is ....

....we can deduce some exactness properties of C[P ] Theorem 3.4. The category C[P ] of partial equivalence relations of a first order hyperdoctrine is a logos. Moreover, all equivalence relations in C[P ] have a quotient, i.e. have a coequalizer whose kernel pair is the equivalence relation (see Makkai and Reyes 1977, Definition 3.3.7) one says that a logos has effective equivalence relations in this case. Proof. Since C[P ] has finite products (Lemma 3.2) and pullbacks of all monomorphisms (Lemma 3.3) it also has equalizers and hence all finite limits. Using the soundness of first order logic for the ....

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Makkai, M. and G. E. Reyes (1977). First Order Categorical Logic, Volume 611 of Lecture Notes in Mathematics. Springer-Verlag, Berlin.

....interpretation in (C; Sub) We have a remark on the interpretation of equality in a logos. On the one hand, we interpret equality using the equality predicate ffi. On the other hand, one can define the value of a formula f(x) g(x) by the equalizer of (the interpretation of) f and g (see e.g. Makkai and Reyes 1977). E e A f g B P. Knijnenburg and F. Nordemann 10 It is easy to see that the following diagram is a pullback. E fe = ge B e Delta A hf; gi B Theta B Now, in Sub(B) 1B . Furthermore, Delta : B B Theta B is monic. Hence, by Lemma 3.15, 9 Delta ( Delta. So, in ....

Makkai, M. and Reyes, G.E. (1977), First Order Categorical Logic, volume 611 of Lecture Notes in Mathematics. Springer Verlag, Berlin.

....the language. More details can be found in [17] The point of presenting a theory S as a category S is that the models of S can be obtained as the functors S Set, preserving the logical structure. This is the essence of functorial semantics [7] and the foundation of categorical model theory [8, 9]. The applications to software engineering are discussed in [17] Related, more direct, but less uniform procedures allow deriving categories from programs. They usually go under the name of operational semantics [18, 25] and come in too many varieties to justify going into any detail here. ....

M. Makkai and G. Reyes, First Order Categorical Logic. Lecture Notes in Mathematics 611 (Springer 1977)

....function spaces. A comparison between the Recursive and the E ective Topos can be found in [Ros86] Throughout the paper, category theory is used in a fairly elementary way (see [BW85] Chapter 1) the only exception is Section 4, which relies on some knowledge of Grothendieck toposes (see [MR77] Joh77] 1 Partial Morphisms When dealing with computable functions or with the semantics of programs partiality arises naturally. In this section we develop an abstract framework for partial functions, namely categories with domains (see Def 4) In de ning partial morphisms we will simply ....

....preserves limits and (partial) function spaces, so Theor 39 is for free. A more elaborate construction, that sometimes makes it possible to preserve even more structure (e.g. colimits) is the topos of sheaves for a subcanonical Grothendieck topology (for details, see [Joh77] Section 0. 3 and [MR77] Section 1.1) The latter approach is used in [Mul81] to de ne the recursive topos R, in which EN is embedded fully and faithfully and the embedding preserves limits, nite colimits and function spaces. The relation between GEN and the recursive topos is investigated in [Ros86] where it ....

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M. Makkai and G. Reyes. First Order Categorical Logic. Springer Verlag, 1977.

....objects are pairs hc 2 C; f : T c di, T # d C is the projection functor (mapping a pair hc; fi to c) and Colim A I : A I A (with I small category) is a functor mapping an I diagram in A to its colimit. The following proposition is a 2 categorical reformulation of Theorem 1.3. 10 of [MR77]. For the sake of simplicity, we use the strict notions of 2 functor and 2 natural transformation, although we should have used pseudo functors and pseudo natural transformations. Proposition 4.11 Let Cat be the 2 category of small categories, CAT the 2 category of locally small categories and : ....

M. Makkai and G. Reyes. First Order Categorical Logic. Springer Verlag, 1977.

....arbitrary meets of formulae. Regular lter logic is obtained by simply deleting all references to binary joins and to the logical constant . Our presentation will be brief, and we assume that the reader has seen the basics of coherent logic and of interpretations in categories. We refer to [4, 5, 11, 13] for details. A signature for coherent lter logic consists of a set of sorts , together with sets of typed constants, function and relation symbols. Typed) terms are de ned as usual. Coherent lter formulae are built up using the grammar fml : j j t 1 = t 2 j R( x) j 1 1 j 1 ....

M. Makkai and G. Reyes. First Order Categorical Logic. Lecture Notes in Mathematics 611. Springer-Verlag, Berlin-New York, 1977.

.... as in (i) by the element A 0 2 P(X 0 ) given by A 0 (x 0 ) def , 9x : X : F (x 0 ; x) A(x) 2 Recall that a category E is a logos if it has finite limits, pullback stable images and dual images of subobjects along morphisms, and pullback stable finite joins of subobjects (Makkai and Reyes 1977). Any category E with finite limits determines an E indexed poset Sub E : E op Gamma Poset mapping E objects to their posets of subobjects and mapping E morphisms to pullback functions. Well of course the posets involved may actually be poclasses unless we assume E is well powered, but size ....

.... logoses with quotients of equivalence relations, it is very natural to consider ones with finite disjoint coproducts as well i.e. Heyting pretoposes (cf. Pitts 1989) The universal solution to realizing P predicates in a Heyting pretopos is the mild generalisation of C[P] implicit in Makkai and Reyes 1977, Part II) whose objects are partial equivalence relations spread over a finite number of C objects : the definition is like that given on pp 45 46 of (Pitts 1989) 4 When is C[P ] a topos Let C be a category with finite products and P a first order hyperdoctrine over it. Suppose that C[P] ....

Makkai, M. and G. E. Reyes (1977). First Order Categorical Logic, Volume 611 of Lecture Notes in Mathematics. Springer-Verlag, Berlin.

....of limit theories, a wider, yet essentially restricted class. The full scope of first order logic was covered by categorical model theory rather slowly, throughout the seventies and eighties, as some parts tend to be technically rather demanding. Good accounts of the more accessible parts are [1, 20, 21]. The main idea of functorial semantics is to present logical theories as classifying categories with structure, so as to obtain their models as structure preserving functors to Set, with homomorphisms between them as natural transformations. The resulting categories of models will always ....

....commutativity conditions. There are several well known frameworks for building suitable classifying categories and developing functorial semantics for general first order theories, the most categorical being probably sketches [3, 20] We shall however work in the setting of coherent categories [21], closest to the original geometric spirit of categorical logic, because they seem to allow the quickest and perhaps the most intuitive approach to the matters presently of interest. 2.2 Coherent categories Let T be a multisorted first order theory with equality. For simplicity, we assume that it ....

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M. Makkai and G. Reyes, First Order Categorical Logic. Lecture Notes in Mathematics 611 (Springer 1977)

....arbitrary meets of formulae. Regular filter logic is obtained by simply deleting all references to binary joins and to the logical constant . Our presentation will be brief, and we assume that the reader has seen the basics of coherent logic and of interpretations in categories. We refer to [4, 5, 12, 14] for details. A signature for coherent filter logic consists of a set of sorts, together with sets of typed constants, function and relation symbols. Typed) terms are defined as usual. Coherent filter formulae are built up using the grammar fml : j j t 1 = t 2 j R( x) j 1 1 j 1 ....

M. Makkai and G. Reyes. First Order Categorical Logic. Lecture Notes in Mathematics 611. Springer-Verlag, Berlin-New York, 1977.

....for the ontologies of greatest interest in computer science, e.g. dependent type theories, Constructions, linear and modal logics. Both the syntax and semantical analysis of new programming languages is being shaped by these semantic frameworks. The reader is urged to consult the work in e.g. [31, 97, 96, 2, 75, 51, 49, 48, 107, 108, 71, 83, 106, 88] and others cited in the appendix, for further details. The author would like to thank Anil Nerode for introducing him to the realizability interpretation of IZF, and encouraging further study of the field, and Peter Freyd for many insights about the effective topos and PERs. Many thanks also to ....

Makkai, M. and G. Reyes [1977], "First order categorical logic", Lecture Notes in Mathematics 611, Springer-Verlag, Berlin.

....of the Dutch NWO, which made possible a visitor s appointment of the second author at Utrecht. x1 Coherent toposes and statement of the main theorem 2. Preliminaries on coherent toposes. We begin by briefly recalling the basic definitions concerning coherent toposes and morphisms ( 1] see also [4,11,7]) A topos E is coherent if E is (equivalent to) the category of sheaves on a finitary site, i.e. a site with finite limits all of whose covering families are finite. Given a coherent topos E, there is always a canonical such site, viz. the full subcategory (pretopos) of coherent ( 4, 7.3.1] ....

....( 4, 7.3.1] objects with the evident topology of finite epimorphic families. Recall also that any pretopos arises in this way, as the category of coherent objects in a coherent topos. Coherent toposes are exactly those toposes which arise as classifying toposes of finitary geometric logic [11]. Permanent address: Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7700, South Africa. 2 A morphism f : F E between coherent toposes is said to be coherent if f sends coherent objects to coherent objects. This is the case if and only if f is ....

M. Makkai and G. Reyes. First Order Categorical Logic. Springer Lecture Notes in Math. vol. 611 (1977).

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M. Makkai and G. E. Reyes. First Order Categorical Logic, Lecture Notes in Math. Vol. 611 (Springer-Verlag, Berlin, 1977).

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M. Makkai and G.E. Reyes (1977). First-Order Categorical Logic. Lec- ture Notes in Mathematics vol. 611, Springer.

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M. Makkai and G. Reyes, First Order Categorical Logic. Lecture Notes in Mathematics Vol.

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Michael Makkai and Gonzalo E. Reyes. 1977. First Order Categorical Logic. Lecture Notes in Mathematics 611. Springer, Berlin.

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M. Makkai and G. E. Reyes. First Order Categorical Logic, Lecture Notes in Math. Vol. 611 (Springer-Verlag, Berlin, 1977).

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