F. W. Lawvere. Equality in hyperdoctrines and comprehension schema as an adjoint functor. In A. Heller, editor, Applications of Categorical Algebra, volume 17 223 of Proceedings of Symposia in Pure Mathematics, pages 1--14, New York, 1968.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....subsets of the set of maps from (the interpretation of) 1 to (the interpretation of) 2 . The trick consists of enlarging adequately the class of models, reducing therefore the set of validities, which can be now captured by proof theoretic methods. From the works of Lawvere (see for example [8, 9]) it was proved that the usual proof methods for hol are sound and complete w.r.t. an extremely elegant topos semantics. The discover of Lawvere that category theory is able to interpret logical languages in a natural way, opens the possibilities to consider topoi as a large class of new ....

F. Lawvere. Equality in hyperdoctrines and comprehension schema as an adjoint functor. In Proceedings of the American Mathematical Society Symposium on Pure Mathematics XVII, pages 1-14, 1970.

....of sets, and the distributivity of cartesian product over disjoint union. For the general axiomatics of the situation, see [AHS02] 6 The Model We shall use the hyper doctrine formulation of model of System F, as originally proposed by Seeley [See87] based on Lawvere s notion of hyperdoctrines [Law70], and simpli ed by Pitts [Pit88] a good textbook presentation can be found in [Cro93] We begin with a key de nition: G U (k) Sub(U) G(k) where U is the universe of System F types constructed in Section 6. 6.1 The Base Category We rstly de ne a base category B . The objects are ....

F. W. Lawvere. Equality in hyperdoctrines and the comprehension schema as an adjoint functor, Proc. Symp. on Applications of Categorical Logic, 1970.

....from it [RR88] Peter Freyd and Andre Scedrov started instead from binary relations (composing an allegory) FS90] Their theory also accounts for the way in which general sets may be obtained from iterated powersets, traditionally known as the von Neumann hierarchy. Bill Lawvere s treatment [Law70] describes and generalises the behaviour of comprehension in toposes and other categories in which it already exists, but does not explain how it creates new sets [Tay99, Exercises 9.45#] Other ways of expanding the class of types include the regular and exact completions of categories. The ....

Bill Lawvere. Equality in hyperdoctrines and the comprehension schema as an adjoint functor. In Alex Heller, editor, Applications of Categorical Algebra, number 17 in Proceedings of Symposia in Pure Mathematics, pages 1--14. American Mathematical Society, 1970.

....Also, truth has a right adjoint fg = dom: Sub(C ) C describing comprehension: it is given by (P X) 7 P . Thus the adjoint situation on the right, above, generalises the one one the left. This description of comprehension as right adjoint to truth is due to [3] and goes ultimately back to [16]. See also [18] Next assume we have an endofunctor F : C C . Its category of coalgebras, comes with a forgetful functor U : CoAlg(F ) C . Thus we can form the pullback of functors: CoAlgSub(F ) ## U (2) This means that the objects of the category CoAlgSub(F ) are given by ....

F.W. Lawvere. Equality in hyperdoctrines and comprehension scheme as an adjoint functor. In A. Heller, editor, Applications of Categorical Algebra, pages 1--14, Providence, 1970. AMS.

....and fibrations; the former represent the class of fibrations for which a definite (coherent) choice of a cleavage has been made. We introduce in this section the basic terminology of fibred category theory together with pointers to the related notion of elementary existential doctrine [20] of which the presheaf models of Section 3 will be an example. A more detailed introduction to fibrations can be found in, e.g. 17] 13 Definition 1.23 (Cartesian arrows) Let # : be a functor. An arrow in f : e # e is cartesian (with respect to #) if for every other arrow g : e ....

....# : Groth(F ) that projects any pair onto its the second component is a fibration. A cartesian lifting for b# with respect to # : b # b is given by the pair F (#)c , ##. Our main example of a bifibration will be given by an elementary existential doctrine in the sense of Lawvere [20] whose categories of attributes will be presheaf categories. We will consider the following two conditions on fibrations, which usually arise in the context of categorical logic [22, 20] Definition 1.31 Let be a pseudo functor. If # : b # we write # # for P(#) Suppose that for any ....

[Article contains additional citation context not shown here]

F. W. Lawvere. Equality in hyperdoctrines and comprehension schema as an adjoint functor. In Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVII, New York,

....adding a new strict initial object (typically denoted ) to it. Recursive type constructor: We use the results of Section 3, and take parameterised free pseudo algebras (see [6] 5 Relational structures Following [20, 22] we consider relational structures in the spirit of categorical logic [16] (c.f. 11] A relational structure R on an Cat 0 category K induces a Cat 0 category of relations fK j Rg with objects fC j Rg given 9 by an object C of K together with a relation R on it, maps are required to be parametric (i.e. relation preserving) Our main result here is that the ....

F.W. Lawvere. Equality in hyperdoctrines and the comprehension schema as an adjoint functor. In Applications of Categorical Algebra, pages 1--14, Amer. Math. Soc., 1970.

....that it is possible to give an axiomatization of hol sound and complete w.r.t. a wider class of models, called general models, in which types of the 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 ....

F. Lawvere. Equality in hyperdoctrines and comprehension schema as an adjoint functor. In Proceedings of the American Mathematical Society Symposium on Pure Mathematics XVII, pages 1-14, 1970.

....and fibrations, the former represent the class of fibrations for which a definite (coherent) choice of a cleavage has been made. We introduce therefore in this section the basic terminology of fibred category theory together with pointers to the related notion of elementary existential doctrine [71] of which the presheaf models of Chapter 3 will be an example. 22 CHAPTER 1. CATEGORICAL BACKGROUND Definition 1.4.1 (Cartesian arrows) Let # : be a functor. An arrow in f : e # e is cartesian (with respect to #) if for every other arrow g : e ## e such that #(g) ## with # = #(f) ....

....to # : b # b is given by the pair F (#)c , ##. The category of elements construction of Definition 1.2.16 is an example of application of the Grothendieck construction (cf. Example 1.4. 5) Our main example of a bifibration will be given by a Lawvere s elementary existential doctrine [71] whose categories of attributes will be presheaf categories. We will naturally consider the following two conditions on top of our fibrations. CAT be a pseudo functor. If # : b # we write # # for P(#) Suppose that for any #, # # has a left adjoint # . Beck Chevalley Condition: ....

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F. William Lawvere. Equality in hyperdoctrines and comprehension schema as an adjoint functor. In Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVII, New York,

....substitution of a term for a variable in a logical formula. We make a slightly non standard de nition: De nition 6. A bration p : E E 0 is called cartesian closed if E and F are cartesian closed and p is a cartesian closed functor. In order to obtain logical relations, following Lawvere [7], one typically uses a logic interpreted as a bration with structure. If the logic admits , and 8, then, provided the base is itself cartesian closed, it yields a cartesian closed bration. The observation that these connectives are all that is required to de ne product and exponential of ....

F. Lawvere. Equality in hyperdoctrines and the comprehension schema as an adjoint functor. In Proc New York Symposium on Applications of Categorical Algebra, pages 1-14, 1970.

....all u 2 e ( S ) and h 2 H. Proof. For u 2 e ( S ) and h; k 2 H we have jjujj h k i jjujj (h k) i T (u) 1 H = h k) i T (u) h k) i (u h) k. The proof of the second claim is analogous. 3. PARTITIONS AND RELATIVE COMPLEMENTS We begin by discussing Lawvere equality [17] in our context. Consider the hyperdoctrine on S given by H, i.e. E(e ( H) Definition 3.1. For I 2 S let eq I : e (I) e (I) H be de ned as eq I (i; j) jji = I jjj for i; j 2 e (I) where i = I j stands for e (e I ) i; j) with e I : I I S the classi er of the ....

....from the fact that e (I I) e ( S ) e (e I ) e (I) e (1 S ) e ( I ) e (true) is a pullback and that e ( I ) e (I) as e preserves nite limits. Theorem 3.1. The hyperdoctrine E(e ( H) over S has Lawvere equality in the sense of [17], i.e. for I 2 S and r : e (I) e (I) H, the following two conditions are equivalent (1)8i; j 2 e (I) eq I (i; j) r(i; j) 2)8i 2 e (I) 1 H r(i; i) Proof. By Lemma 3.1, instantiating h by r(i; j) we get that condition (1) is equivalent to 8i; j 2 e (I) i = e (I) ....

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F. W. Lawvere, Equality in hyperdoctrines and comprehension schema as an adjoint functor, Proc. of Symposia in Pure Mathematics of the American Math. Society 17 1-14.

....scheme as follows: B =J ## Im E J ## f g Dept. of Mathematics and Statistics, McGill University, 805 Sherbrooke St. W. Montreal, QC, Canada H3A 2K6. e mail:hermida math.mcgill.ca. where Im : B=J E J takes f : I J to f ( I ) its image) This is in fact the original formulation in [Law70]. The above adjunction gives the desired factorization: I ## f # # # # # # # # # # ## f fIm(f)g ## J where f is the unit of the adjunction. We shall see in the two speci c situations below that the construction of the fIm(f)g will guarantee automatically its monicness (for the chosen ....

F.W. Lawvere. Equality in hyperdoctrines and comprehension scheme as an adjoint functor. In A. Heller, editor, Applications of Categorical Algebra. AMS Providence, 1970.

....and Pitts 1980, p 222) Thus the following question naturally arose: Question. Is there a common generalisation, with useful properties, of the constructions of H valued sets and of the effective topos Drawing upon Lawvere s treatment of logic in terms of hyperdoctrines (Lawvere 1969; Lawvere 1970), I came up with an answer to this question based on a structure of indexed collections of posets with certain properties. Peter Johnstone (my PhD supervisor) suggested naming these structures with the acronym tripos standing for Topos Representing Indexed Partially Ordered Andrew M. Pitts 2 ....

....of this note is to point out that there is a slightly more general class of hyperdoctrines than triposes answering the above Question. The generalisation hinges upon a careful analysis of the comprehension properties that a hyperdoctrine may possess (different from the ones in the classic paper by Lawvere (1970) to do with reflecting predicates into subobjects) Thus there are hyperdoctrines that generate toposes in just the same way that triposes do, yet whose powerobject structure is weaker than that required of triposes. This is explained, and examples given, in Section 4. The scene is set by ....

Lawvere, F. W. (1970). Equality in hyperdoctrines and the comprehension schema as an adjoint functor. In A. Heller (Ed.), Applications of Categorical Algebra, pp. 1--14. American Mathematical Society, Providence RI.

....doing so we deviate from standard methods of presenting logics and the formulation here is closer to a programming language in that it includes environments as well as contexts. The other source for this theory of classes is the notion of a comprehension schema introduced into category theory in [Lawvere 1970]. Lawvere formulates a comprehension schema as an adjoint in an indexed category and shows how such schemata capture the extent of a predicate. We describe comprehension schemata later in the paper and use this link to categories to provide models for the formal systems of classes. There is a ....

....by induction on their derivation. These results rely on the decision procedure for equality in C Der . As an application, we consider a model theory for the systems. For the first order system C Der , models in strict indexed categories with comprehension schemata are formalised (based upon [Lawvere 1970]) a semantics of judgements in such models is given, and a soundness theorem is proved (completeness holds as well) This provides a basis for a semantics of C Env , given straightforwardly using the coherence result. Finally, we show how this relates to type classes in programming languages. 2 ....

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F.W. Lawvere, Equality in Hyperdoctrines and the Comprehension Schema as an Adjoint Functor. In Proc. Symp. in Pure Math., XVII: Applications of Categorical Algebra, Am. Math. Soc. pp. 1--14, 1970.

....M be a Omega b in N ] Table 2. The equations of ILT 4 3 Categorical semantics of ILT The basis for our categorical model of ILT is Ehrhard s notion of a D category for modelling dependent types [Ehr88] which goes back to Lawvere s idea of hyperdoctrines satisfying the comprehension axiom [Law70]. Hyperdoctrines model many sorted predicative logic, where predicates are indexed over sorts or sets. A suitable adjunction allows the interpretation of the comprehension axiom, that is the creation of a subset defined by a predicate indexed over a set. Ehrhard generalized this idea in terms of ....

F.W. Lawvere. Equality in hyperdoctrines and comprehension schema as an adjoint functor. Proc. Sympos. Pure Math., XVII:1--14, 1970.

....but take the total viewpoint as the more fundamental one: the first one will be a derived notion. In order to obtain a categorical model for Hoare triples, we turn to a conceptually very clear categorical model for first order logic. It is based on the notion of hyperdoctrines as introduced by Lawvere (1969, 1970). A detailed description can be found in Pitts (1989b) see also Hyland, Johnstone and Pitts 1980) We give an informal description. The model is an indexed category. It consists of a category C (the base category) that is used for giving meaning to the sorts and terms: to every sort one assigns ....

....one can model equality if the model possesses an equality predicate ffi (which denotes [ x = x 0 ] Since [ x=x 0 ] Delta [ this predicate ffi should be a value of the left adjoint to Delta at . These fundamental insights in the nature of first order logic are due to Lawvere (1969, 1970). 4.2. Soundness In this section we show that the total proof system for the logic is sound with respect to the total interpretation. We need a substitution lemma. To fix notation: Suppose OE(x) is a formula in context x = x 1 ; xn where x i : A i say. Assume that for all 1 i n there ....

Lawvere, F.W. (1970), Equality in hyperdoctrines and the comprehension schema as an adjoint functor. In A. Heller, editor, Proc. New York Symp. on Applications of Categorical Algebra, pages 1--14. Amer. Math. Soc.

....to guarantee full completeness for ML types. This axiomatization is put to use in [AL99,AL00] in order to provide a concrete denotational model fully complete for the whole class of ML types. The axioms presented in this paper are given on the models of system F originated from Lawvere ([Law70]) which are called hyperdoctrines (see also [Pit88] As in [Abr97] our axiomatization works in the context of adjoint models and, although the full completeness result applies to intuitionistic types, it makes essential use of the linear decomposition of these types. Our axiomatization consists ....

....hyperdoctrine. Adjoint hyperdoctrines arise as co Kleisli indexed categories of linear indexed categories. In what follows, we assume that all indexed categories which we consider are strict (see e.g. AL91,Cro93] for more details on indexed categories) Definition 2 (2 Theta hyperdoctrine, [Law70,Pit88]) A 2 Theta hyperdoctrine is a triple (C; G;8) where: C is the base category, it has finite products, and it consists of a distinguished object U which generates all other objects using the product operation Theta. We will denote by U m , for m 0, the objects of C. G : C op CCCat ....

F.Lawvere. Equality in hyperdoctrines and the comprehension schema as an adjoint functor, Proc. Symp. on Applications of Categorical Logic, 1970.

.... C has nite products and exponentials of the form A , for some distinguished object P is closed under implication and universal quanti cation along rst projections there is some distinguished generic predicate t 2 P[ The interpretation of equality is equivalent to that proposed in [Law70]: the predicate =A is necessarily unique, and 9 satis es the Beck Chevalley and Frobenious Reciprocity conditions. The de nition of tripos is a minor simpli cation of the original one (see [Pit81, HJP80] Lemma 3.5 If t 2 P[ and t 0 2 P[ 0 ] are skeletal generic predicates, then the unique ....

F.W. Lawvere. Equality in hyperdoctrines and comprehension schema as an adjoint functor. In A. Heller, editor, New York Symp. on Applications of Categorical Algebra. AMS, 1970.

....in a general type theory with dependent types, unifying for instance the notion of proofs by induction and functions defined by induction. Ideas closely related to the Curry Howard correspondence had also appeared earlier in Lawvere s work on categorical logic from the 60s. His hyperdoctrines [18], which are categorical models of first order predicate logic, model propositions as objects and proofs as morphisms, and has a quite similar structure to type theory. The notions of constructor and concrete data type turned out to be similar to the notions of introduction rule and logical ....

F. W. Lawvere. Equality in hyperdoctrines and comprehension schema as an adjoint functor. In A. Heller, editor, Applications of Categorical Algebra, Proceedings of Symposia in Pure Mathematics. AMS, 1970.

....in a general type theory with dependent types, unifying for instance the notion of proofs by induction and functions defined by induction. Ideas closely related to the Curry Howard correspondence had also appeared earlier in Lawvere s work on categorical logic from the 60s. His hyperdoctrines [14], which are categorical models of first order predicate logic, model propositions as objects and proofs as morphisms, and has a quite similar structure to type theory. The notions of constructor and concrete data type turned out to be similar to the notions of introduction rule and logical ....

F. W. Lawvere. Equality in hyperdoctrines and comprehension schema as an adjoint functor. In A. Heller, editor, Applications of Categorical Algebra, Proceedings of Symposia in Pure Mathematics. AMS, 1970.

....functions. The type theoretic properties of ECC allow us to define deliverables using the simple type structure of the programs, and the logic of quantifiers. The abstract categorical structure which is sufficient to describe these constructions was elaborated more than twenty years ago by Lawvere [54,55]. 5.1.1 Hyperdoctrines Definition 5.1.1 (Lawvere [54,55] hyperdoctrine A hyperdoctrine is an indexed category H over a cartesian closed base category C, with cartesian closed fibres, called the attributes of C, together with the following additional structure : Quantification for each C f D ....

....to define deliverables using the simple type structure of the programs, and the logic of quantifiers. The abstract categorical structure which is sufficient to describe these constructions was elaborated more than twenty years ago by Lawvere [54,55] 5.1.1 Hyperdoctrines Definition 5.1. 1 (Lawvere [54,55]) hyperdoctrine A hyperdoctrine is an indexed category H over a cartesian closed base category C, with cartesian closed fibres, called the attributes of C, together with the following additional structure : Quantification for each C f D in C, the pullback functor f :H[D] H[C] has ....

F.W.Lawvere, Equality in hyperdoctrines and the comprehension schema as an adjoint functor, in: Proceedings of the AMS symposium on Applications of Category Theory, AMS, Providence R.I., 1970.

....Johnstone, and Pitts 1980, p 222) Thus the following question naturally arose: Question. Is there a common generalisation, with useful properties, of the constructions of H valued sets and of the effective topos Drawing upon Lawvere s treatment of logic in terms of hyperdoctrines (Lawvere 1969; Lawvere 1970), I came up with an answer to this question based on a structure of indexed collections of posets with certain properties. Peter Johnstone (my PhD supervisor) suggested naming these structures with the acronym tripos standing for Topos Representing Indexed Partially Ordered Set 1 and the ....

....of this note is to point out that there is a slightly more general class of hyperdoctrines than triposes answering the above Question. The generalisation hinges upon a careful analysis of the comprehension properties that a hyperdoctrine may possess (different from the ones in the classic paper by Lawvere (1970) to do with reflecting predicates into subobjects) Thus there are hyperdoctrines that generate toposes in just the same way that triposes do, yet whose powerobject structure is weaker than that required of triposes. This is explained, and examples given, in Section 4. The scene is set by ....

Lawvere, F. W. (1970). Equality in hyperdoctrines and the comprehension schema as an adjoint functor. In A. Heller (Ed.), Applications of Categorical Algebra, pp. 1--14. American Mathematical Society, Providence RI.

....op CAT, assigning to each set A 2 C a category of predicates P(A) over it, usually cartesian closed; and to each 1 It is hard to see a reason for this, other than subjective. The main obstacles to categorical proof theory, e.g. as analyzed by Dana Scott in [46] had been removed by 1970 [31, 32]. function f : A B in C the substitution functor P(f) P(B) P(A) Q(y) 7 Q(f(x) 2 The quantifiers are represented as the functors adjoint to the substitution along a projection p : A Theta B A: 9y a Pp a 8y : P(A Theta B) P(A) R(x; y) 7 8y:R(x; y) This representation ....

....be formalized in section 2. The uniformity is, of course, not always welcome and this idea should not be used as a Procrustean bed. Linear logic, for instance, does not arise entirely in terms of adjunctions. But for the basic constructs, the adjunctions seem to be working remarkably well. In [32], Lawvere has described the comprehension scheme, assigning to each predicate P (x) a set fxjP (x)g, as an adjoint functor. In [33] he has explained how maps, as total and singlevalued relations, can be presented as self adjoint bimodules. This idea is central in the present paper. In the next ....

[Article contains additional citation context not shown here]

F.W. Lawvere, Equality in hyperdoctrines and comprehension schema as an adjoint functor, Proc. Sympos. Pure Math XVII(1970), 1--14

....X to Kan extensions. The new edition [Mac97] also contains important material on topos theory, 2categories, bicategories, and presheaves. Beyond that, I would recommend categorical logic and fibrations [Jac99, Pho92] enriched category theory [Law73a] and any further writing by Lawvere such as [Law70, Law69, Law91]. Links . http: www.mta.ca #cat dist categories.html . http: www.acsu.buffalo.edu #wlawvere ....

F. William Lawvere. Equality in hyperdoctrines and comprehension schema as an adjoint functor. In A. Heller, editor, Proceedings 17th AMS Symp. in Pure Maths. on Applications of Categorical Algebra, New York, NY, USA, 10--11 Apr 1968, volume 17 of Proceedings Symposia in Pure Maths., pages 1--14. American Mathematical Society, Providence, RI, 1970.

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F. W. Lawvere. Equality in hyperdoctrines and comprehension schema as an adjoint functor. In A. Heller, editor, Applications of Categorical Algebra, volume 17 223 of Proceedings of Symposia in Pure Mathematics, pages 1--14, New York, 1968.

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F. W. Lawvere. Equality in hyperdoctrines and the comprehension schema as an adjoint functor, Proc. Symp. on Applications of Categorical Logic, 1970.

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