Bart Jacobs. Categorical Logic and Type Theory, volume 141 of Studies in Logic and the Foundations of Mathematics. Elsevier, 1999.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....substitute sets (interpreting types) by arbitrary objects in a given topos. Function symbols are interpreted as morphisms, cartesian products are categorial products, relation symbols are interpreted as subobjects, functional types are interpreted using exponentials, and so on (see, for instance, [7, 2, 10, 6, 11]) The fact that categorial semantics use topoi guarantees the minimum amount of categorial operations needed to interpret the logical symbols of higher order languages. Bell proposes in his book [2] a sequent calculus style axiomatization of hol called local set theory, which permits to describe ....

....i ) 2 ) i ) i ) type not occurring free in h 1 ; 2 i. Existential quanti cation: 9 x k ) for (8 x i (8 x k ( i ) i ) type not occurring free in . 2 Topos Semantics Higher order languages can be interpreted in any topos (see, for instance, [7, 2, 10, 6, 11]) In order to interpret terms in a given topos we need to introduce the notion of context. By a context we mean a nite sequence x = x 1 : x n of distinct variables. We denote by [ the empty context. Given a context x = x 1 : x n where the variables x 1 ; x n are of type ....

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B. Jacobs. Categorical Logic and Type Theory, volume 141 of Studies in Logic and the Foundations of Mathematics. Elsevier, Amsterdam, 1999.

....function symbols will be written in the usual arrow notation: we will write f : ## 1 # for f # O( ##1 , #1 ) ## n , #n ) # ) Notice that only types in U may appear in negative positions of these arities. When U = we get back the traditional notion of typed signature [11]; when T = U = 1, we get the notion of binding signature [4] Finally, notice that a symbol may be overloaded, that is it may have several arities. Example 1. The signature of untyped # calculus is ## = # , app . Let T # # be the smallest set containing a given set of atomic type ....

....op( #x1 )t1 , #x n )tn ) where op : ## 1 # and for i = 1, n: t i S# i . The usual conventions about free bound variables and # conversion apply. A more precise definition of the terms generated by a binding signature # is given in a type theoretic form, as e.g. in [11,18]. Let Var = v1 , v2 , be an enumeration of distinct elements, called variable symbols; a typing context (denoted by #) is a list v1 : #1 , vn : #n , where # i U . The term calculus for # is the system for deriving typing judgments of the usual form T , and whose rules ....

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B. Jacobs. Categorical Logic and Type Theory, volume 141 of Studies in Logic and the Foundations of Mathematics. Elsevier, 1999.

....by its value (h(1; i) on the empty list. Hence P can be identified with N (See [Lan86] Chapter 2) and the inclusion function N , f1g 0 f2; 3g N) N 1) is the map m 7 : h(1; i) The signature of the coalgebraic transition structure on P is N N (N N) N N 1) [Jac99] can be used as a reference for the concepts of fibred category. Conclusion and Future Work 25 and the corresponding map is n 7 hn; m:n m; m:case[n m 7 n m j n m 7 n]i Here we use the error handling techniques and change the function N N 1 to a total function N N. It is easy to ....

Bart Jacobs. Categorical Logic and Type Theory, volume 141 of Studies in Logic and The Foundations of Mathematics. Elsevier, 1999.

....C 1 such that the identities : C 0 C 1 belong to it: # # # P m C 1 Notice that this includes the empty category with empty distinguished subobject of morphisms. There is an evident forgetful functor U : Cat P Cat (throw away P ) which is quite evidently a bration. We refer to [Jac99, Bor94] for bred categorical matters. This bration inherits a good many properties from the related bration (considering only the object of morphisms of the categories) U : Set) P =Set, where =Set is the category of pointed sets and point preserving morphisms. The bres of this bration are ....

.... es the triangular identities with : X) X ) id (X ) X) id Just like in the case of categories, the collection of X ) organize themselves into a bivariant functor ) Bearing in mind the connection between cartesian closed categories and simply typed lambda calculus [LS86, Jac99], we see that bivariant functors arise naturally in the categorical semantics of such calculi. 5.6. Example. Consider a (small) cartesian closed category and the collection of bivariant functors T : from a given small category . Except for rather uninteresting , the category ....

B. Jacobs. Categorical logic and type theory, volume 141 of Studies in Logic and the Foundations of Mathematics. North Holland, 1999.

....and let cod : Sub(B ) B be the evident forgetful functor, taking the codomain of the subobject. The fact that B is a regular category is equivalent to the statement that B has nite limits and cod is a bration with sums (bi bration satisfying the Beck Chevalley condition) and quotients. See [Jac99] for a convenient account of bred categorical matters (for the logically minded reader) Since we are working with a distinguished class of monos M, we would require a similar fibration with sums structure for cod : M(B ) B , with M(B ) the evident full subcategory of Sub(B ) spanned by the ....

B. Jacobs. Categorical logic and type theory, volume 141 of Studies in Logic and the Foundations of Mathematics. North Holland, 1999.

....form ) are interpreted as subsets of the set of maps from (the carrier of) to (the carrier of) From the works of Lawvere (see for example [14, 15] it was proved that the usual axiomatizations of hol are sound and complete w.r.t. an extremely elegant topos semantics (see, for instance, [12, 3, 16, 9, 17]) In this research report we introduce two very simple Hilbert style axiomatizations of hol, which are sound and complete w.r.t. topos semantics. The rst one is obtained by adapting the sequent calculus style axiomatization of hol called local set theory, introduced by Bell in [3] de ned in a ....

....i is the rst variable of not occurring free in h 1 ; 2 i. Existential quanti cation: 9 x ) for i (8 x k ( i ) i ) i is the rst variable of not occurring free in . 2 Topos Semantics Higher order languages can be interpreted in any topos (see, for instance, [12, 3, 16, 9, 17]) In order to interpret terms in a given topos we need to introduce the notion of context. By a context we mean a nite sequence x = x 1 : x n of distinct variables. We denote by [ the empty context. Given a context x = x 1 : x n where the variables x 1 : x n are of type 1 ; ....

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B. Jacobs. Categorical Logic and Type Theory, volume 141 of Studies in Logic and the Foundations of Mathematics. Elsevier, Amsterdam, 1999.

....establish a bred equivalence, as in OpenSub j (E) # # # # # # # # # # , ClSub j (E) zz# # # # # # # # # # mm E : Proof. Easy using the already noted fact that X = X and X = X. Proposition 4.3. The bration OpenSub j (E) ## E of open subobjects is a higher order bration [5] with extensional entailment, in which the following hold (we label the connectives etc. in OpenSub j (E) ## E with a subscript ) 9 , Eq are as for ordinary subobjects. X Y = X Y ) X Y = X Y ) 8 ) f X = 8 f X) and thus : X) X ....

....except the generic object. Proof. The rst order structure is de ned categorically and thus preserved along equivalences. Therefore, the rst order structure is obtained from the wellknown description of the logical operations of the closed subobject bration (explicitly stated, e.g. in [5]) For example, for X;Y 2 OpenSub j (E) over I we have that X Y = X j Y , where j is the disjunction in the closed subobject bration, so X Y = X Y = X Y ) X Y = X Y (where we used that interior preserves as a left adjoint) It is easy to verify that true : 1 ....

B. Jacobs. Categorical Logic and Type Theory, volume 141 of Studies in Logic and the Foundations of Mathematics. Elsevier Science Publishers B.V., 1999.

....general notion of realizability is related to standard realizability over PCAs, and what the main results of the paper are, we briefly recall some basic facts about standard realizability over PCAs. To this end, let A be a PCA. Given A, one may construct the following well known categories (see [13] for a textbook treatment with references to the literature) the standard realizability tripos p : UFam(A) Set, a fibred (or indexed) category; the category of partitioned assemblies PartAsm(A) the category of modest sets Mod(A) the category of assemblies Asm(A) and the standard ....

.... applied to the tripos p, and it is equivalent to the exact completion of the left exact category PartAsm(A) and to the exact completion of the regular category Asm(A) The categories Mod(A) and Asm(A) are both locally cartesian closed and both model dependent predicate logic (see, e.g. [13]) We show how any WCPC category C gives rise to a pre1 tripos p : UFam(C ) Set (almost a tripos, but without a weak generic object) and a category F(C ) corresponding to the category of partitioned assemblies and how these may be used to construct categories of modest sets Mod(C ) and ....

[Article contains additional citation context not shown here]

B. Jacobs. Categorical Logic and Type Theory, volume 141 of Studies in Logic and the Foundations of Mathematics. Elsevier Science Publishers B.V., 1999.

....general notion of realizability is related to standard realizability over PCAs, and what the main results of the paper are, we briefly recall some basic facts about standard realizability over PCAs. To this end, let A be a PCA. Given A, one may construct the following well known categories (see [13] for a textbook treatment with references to the literature) the standard realizability tripos p : UFam(A) Set, a fibred (or indexed) category; the category of partitioned assemblies PartAsm(A) the category of modest sets Mod(A) the category of assemblies Asm(A) and the standard ....

.... applied to the tripos p, and it is equivalent to the exact completion of the left exact category PartAsm(A) and to the exact completion of the regular category Asm(A) The categories Mod(A) and Asm(A) are both locally cartesian closed 1 and both model dependent predicate logic (see, e.g. [13]) We show how any WCPC category C gives rise to a pretripos p : UFam(C ) Set (almost a tripos, but without a weak generic object) and a category F(C ) corresponding to the category of partitioned assemblies and how these may be used to construct categories of modest sets Mod(C ) and ....

[Article contains additional citation context not shown here]

B. Jacobs. Categorical Logic and Type Theory, volume 141 of Studies in Logic and the Foundations of Mathematics. Elsevier Science Publishers B.V., 1999.

....as in OpenSub j (E) K K K K K K K K K K Gamma , ClSub j (E) zzu u u u u u u u u u ffi mm E : Proof. Easy using the already noted fact that ffi X = ffi X and ffi X = X. Proposition 4.3. The fibration OpenSub j (E) fflffl E of open subobjects is a higher order fibration [5] with extensional entailment, in which the following hold (we label the connectives etc. in OpenSub j (E) fflffl E with a subscript ffi) ffl ffi , ffi , 9 ffi , Eq ffi are as for ordinary subobjects. ffl ffi = ffi , X ffi Y = ffi (X Y ) X oe ffi Y = ffi (X oe Y ) 8 ffi ) f X ....

....except the generic object. 12 Proof. The first order structure is defined categorically and thus preserved along equivalences. Therefore, the first order structure is obtained from the wellknown description of the logical operations of the closed subobject fibration (explicitly stated, e.g. in [5]) For example, for X;Y 2 OpenSub j (E) over I we have that X ffi Y = ffi X j Y , where j is the disjunction in the closed subobject fibration, so X ffi Y = ffi X Y = ffi (X Y ) ffi X ffi Y = X Y (where we used that interior preserves as a left adjoint) It is easy to verify ....

B. Jacobs. Categorical Logic and Type Theory, volume 141 of Studies in Logic and the Foundations of Mathematics. Elsevier Science Publishers B.V., 1999.

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Bart Jacobs. Categorical Logic and Type Theory, volume 141 of Studies in Logic and the Foundations of Mathematics. Elsevier, 1999.

No context found.

Bart Jacobs. Categorical Logic and Type Theory, volume 141 of Studies in Logic and The Foundations of Mathematics. Elsevier, 1999.

No context found.

Bart Jacobs. Categorical Logic and Type Theory, volume 141 of Studies in Logic and the Foundations of Mathematics. Elsevier, 1999.

No context found.

B. Jacobs. Categorical Logic and Type Theory, volume 141 of Studies in Logic and The Foundations of Mathematics. Elsevier, 1999.

No context found.

B. Jacobs. Categorical Logic and Type Theory, volume 141 of Studies in Logic and The Foundations of Mathematics. Elsevier, 1999.

No context found.

B. Jacobs. Categorical Logic and Type Theory, volume 141 of Studies in Logic and the Foundations of Mathematics. Elsevier, Amsterdam, 1999.

No context found.

Bart Jacobs. Categorical Logic and Type Theory, volume 141 of Studies in Logic and the Foundations of Mathematics. Elsevier, Amsterdam, 1999.

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