Freyd, P., Remarks on algebraically compact categories, in Applications of Categories in Computer Science, pp. 95-106, eds.: Fourman, Johnstone, Pitts, LMS Lecture Note Series 177, Cambridge University Press 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of R 62 14 External computational adequacy 68 15 Applications 73 15.1 Realizability models . 73 15.2 Models of axiomatic domain theory . 74 1 Introduction In his work on algebraic compactness, Freyd [9, 10] identi ed the categorical structure required to model recursive types. Many examples of algebraically compact categories are known. Domain theory provides the classical example of the category of cpos [3] More generally, axiomatic domain theory has successfully abstracted the particularities ....

....The research in the present paper constitutes a development of the techniques of synthetic domain theory. Nevertheless, our applications to axiomatically given classes of models demonstrate that our results should be viewed equally much as a contribution to the eld of axiomatic domain theory [9, 10, 39, 4, 3, 5, 6]. It is the author s view that embedding categories of predomains within models of intuitionistic set theory is the correct approach to obtaining an axiomatic account of domain theoretic constructions that applies uniformly across the di erent types of model. At present, it is the only known ....

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P.J. Freyd. Remarks on algebraically compact categories. In Applications of Categories in Computer Science, pages 95-106. LMS Lecture Notes 177, CUP, 1992.

....EPSRC Research Grant no. K06109 and an EPSRC Advanced Research Fellowship. tial (relative to L) exponentials, to interpret function types; and finally, to interpret recursive types, the derived category, pP, of partial maps, induced by L on P, must be algebraically compact in the sense of Freyd [7, 8], at least with repect to functors defined by type expressions. The above identifies the structure required by a model of FPC, but does not indicate where to find examples of models. Nevertheless, several sources of such models are known. Domain theory provides the classical example of the ....

....functors: pP # pP , 5) where (3) and (4) require Axiom 1, and (5) requires Axiom 2. N.b. although extends product on P, it is not a cartesian product on pP, whereas is a binary coproduct functor on pP. Our goal is to prove the algebraic compactness, in the sense of Freyd [7, 8], of the internal category pP. We recall this notion for ordinary categories. Given an endofunctor F on an arbitrary category K, a bifree algebra is an initial F algebra a : FA A for which a 1 is also a final F coalgebra (by Lambek s Lemma, an initial algebra is always an isomorphism) A ....

P.J. Freyd. Remarks on algebraically compact categories. In Applications of Categories in Computer Science, pages 95--106. LMS Lecture Notes 177, CUP, 1992.

....and continuous with respect to E, we can find a game D solving the required equation by setting D = I) 2.8.2 Canonicity of solutions Having demonstrated that solutions of certain equations exist, it is reasonable to ask whether they are in some sense canonical. In a series of papers [14 16], Freyd has recently proposed and investigated a suitable notion of canonicity called the minimal invariant condition which captures abstractly the key properties guaranteed by the well known limit colimit coincidence theorem for recursively defined objects the category of Scott domains [34] ....

P. J. Freyd. Remarks on algebraically compact categories. In M. P. Fourman, P. T. Johnstone, and A. M. Pitts, editors, Applications of Categories in Computer Science: Proceedings of the LMS Symposium, Durham, 1991.

....Domain Theory] 1 to outline the first abstract setting for specifying both algebraic (in the ADJ jargon) and recursive types, but these ideas were not pursued further. In [Fre90] aiming at an axiomatic treatment of recursive types, Freyd revisited the previous approaches. And, in [Fre91, Fre92] he proposed a universal approach (in the category theoretic sense) for solving recursive type equations. There he introduced algebraically compact categories and established their fundamental property: that bifunctors on them have canonical and minimal fixed points. This has been a first ....

P.J. Freyd. Remarks on algebraically compact categories. In [FJP92], pages 95--106, 1992.

....of their arguments and contravariant in others. We may either do like Smyth and Plotkin ( Smyth Plotkin 82] and move to a category where the morphisms are retracts, that is, pairs ( 0) as above, fulfilling c = id, whereby all functors become covariant, or we may do like Freyd ( Freyd 91] and [Freyd 92] and split the positive and negative occurrences of a variable and obtain covariancy that way. For a further development in this direction see ( Pitts 93] Paul Taylor remarks in his thesis ( Taylor 86] 2.2.12, p. 43) that in order to obtain the limit colimit coincidence, it is really enough ....

Freyd, P., Remarks on algebraically compact categories, in Applications of Categories in Computer Science, pp. 95-106, eds.: Fourman, Johnstone, Pitts, LMS Lecture Note Series 177, Cambridge University Press 1992.

....Rel (the category of sets and relations, with hom sets ordered by inclusion) pCpo (the category of cpos and partial continuous functions, with hom sets ordered pointwise) Prof , and Prof M . From Theorem 2.1, Corollary 2.2, and Lemma 3. 1, we can deduce pseudoalgebraic compactness (see [8, 6]) Corollary 3.2 (Pseudo algebraic compactness) Kcats are pseudo algebraically compact with respect to pseudo Cat functors. Thus, every pseudo Cat functor T : K Theta K K on a Kcat K has a free pseudo dialgebra T (D; D) D characterised by the following universal property: for every ....

P. Freyd. Remarks on Algebraically Compact Categories. LMS vol. 177, 21

....the coherence isomorphisms [47] 68 CHAPTER 4. PROFUNCTORS In terms of Kan extensions, the theorem above is saying that Lan y P (F ) H and yQ (D) H # . As we shall see later, just like Rel, also Prof (or better Cocont as far as our definitions will be concerned) is compact closed [35, 36], though in a bicategorical 4.2.2 A domain theoretic analogy We discuss now the intuition that the presheaf construction is analogue to a powerdomain [101] one. As remarked in [138] and [25] in fact, Prof can be described as the bicategory of free algebras for a pseudo monad over the categorical ....

....by solving appropriate recursive domain equations. In this chapter we give a generalisation of classical results [125, 120] about the solution of recursive domain equations that, following the axiomatic approach of [30, 33, 105, 34] will justify our intuitive understanding. In fact, after Freyd [35, 36], we shall consider the notion of algebraic compactness and use a pseudo version of the Basic Lemma (see [125] to deduce pseudo algebraic compactness for a class of 2 categories that include Cocont which is the 2 categorical equivalent of Prof . We develop a domain theoretical approach to open ....

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Peter J. Freyd. Remarks on algebraically compact categories. In M.P. Fourman, P.T. Johnstone, and A.M. Pitts, editors, Applications of Categories in Computer Science, volume 177 of London Mathematical Society Lecture Note Series, pages 95--106. Cambridge University Press, 1992.

....opposite category S op . There always is a canonical morphism I T, but in all the obvious examples that come to mind, this morphism is not an isomorphism. According to Freyd s Versality Principle, domains for the denotational semantics are to be found in categories where it is (see [4] 5] [6]; in fact, Freyd shows that such categories occur naturally also in other situations, very far from computer science) Now the SDT approach to construct such categories is as follows: one considers the Sigma lift endofunctor of a topos S and tries to choose Sigma in such a way that the morphism ....

....See [15] V.2.1 for details. Let us begin with a theorem stating existence of an initial E algebra under some conditions on the endofunctor E. This seems to be a typical folklore theorem: I ve heard versions of it from Alex Simpson, Paul Taylor, Pino Rosolini; see also the first proposition in [6]. To the author s knowledge, the earliest (and, it seems, the most general) version is Theorem V(2.2.2) of [15] We shall extract from it the particular case we need. First let us recall the notion of unique existentiation (u. e. pullback from [3] proposition 2.21) Given any f : X Y , there ....

P. Freyd, Remarks on algebraically compact categories, Applications of Categories in Computer Science (M. P. Fourman, P. T. Johnstone, A. M. Pitts, eds.), London Math. Soc. Lect. Note Ser. 177, 1992, 95 -- 106.

....of initial and terminal L algebras in Set. Theorem 5.1 In Set there exists an initial L algebra OE : L Gamma and a terminal L coalgebra AE : Gamma L where is a subobject of S(N) and is the least sub L algebra of AE Gamma1 . As OE and AE are known to be isomorphisms, c.f. [3], the following definitions make sense. Definition 5.4 The maps oe , OE ffi j : Gamma oe , AE Gamma1 ffi j : Gamma are called successor maps on and , respectively. The unique L algebra morphism from OE to AE Gamma1 is called : Gamma satisfying AE ffi ffi OE = ....

....1997) one may establish the existence of solutions of domain equations provided the universe Set is impredicative. First one shows that C is algebraically compact, i.e. for any functor F : C Gamma C there exists an an initial F algebra ff whose inverse is a terminal F coalgebra. By Freyd s [3] the category C op Theta C is algebraically compact, too. Thus, any internal mixed variant functor F : C op Theta C Gamma C admits a canonical solution D = F (D; D) by taking the initial terminal algebra for the functor F x : C op Theta C Gamma C op Theta C where F x ( 1 ; 2 ....

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P. Freyd. Remarks on algebraically compact categories. In Applications of Categories in Computer Science, volume 177 in Notes of the London Mathematical Society, 1992.

....support recursive datatypes; see, for example, 25, 15, 19, 3, 16] 1 The goal of axiomatic domain theory is to axiomatize the structure common to such models. In his notion of algebraic compactness, Peter Freyd isolated the crucial universal property of (possibly contravariant) recursive types [6, 7]. In one modern formulation, a model supporting the definition of recursive datatypes should provide a cartesian closed category of predomains together with a lifting monad whose associated partial category is algebraically compact. This formulation leaves two questions unanswered. First, which ....

P.J. Freyd. Remarks on algebraically compact categories. In Applications of Categories in Computer Science, pages 95--106. LMS Lecture Notes 177, Cambridge University Press, 1992.

....trees, where the extensionality principle is not as neat as the one for streams. Further applications of coinduction in computer science can be found in relational semantics [MT88] To model recursive data types we adopt a non standard approach recently revived in the work of Peter Freyd [Fre91, Fre92]. The meaning of a recursive data type will be a final coalgebra. We recall the basic definitions. Let C be a category (which we think of as the universe of denotations for the data types) and let F be an endofunctor on it (which we think of as the denotation of a data type constructor with a free ....

.... been customary to interpret recursive types as initial algebras (the notion dual to that of final coalgebra) We remark that in certain categories of domains this traditional view and the viewpoint adopted here coincide for initial algebras and final coalgebras are canonically isomorphic (see e.g. [Fre90, Smy91, Fre92, FP92, Fio94]) The mysterious definition of bisimulation in our example is an instance of an abstract notion of bisimulation on a coalgebra (taken from [AM89] motivated by concurrency theory. This has two important methodological consequences. 1. We are able to provide bisimulations for recursive data ....

[Article contains additional citation context not shown here]

P.J. Freyd. Remarks on algebraically compact categories. In M.P. Fourman, P.T. Johnstone, and A.M. Pitts, editors, Applications of Categories in Computer Science, volume 177 of London Mathematical Society Lecture Note Series, pages 95--106. Cambridge University Press, 1992.

....arguments and contravariant in others. We may either do like Smyth and Plotkin ( Smyth Plotkin 82] and move to a category where the morphisms are retracts, that is, pairs (OE; as above, fulfilling ffi OE = id, whereby all functors become covariant, or we may do like Freyd ( Freyd 91] and [Freyd 92] and split the positive and negative occurrences of a variable and obtain covariancy that way. For a further development in this direction see ( Pitts 93] Paul Taylor remarks in his thesis ( Taylor 86] 2.2.12, p. 43) that in order to obtain the limit colimit coincidence, it is really enough ....

Freyd, P., Remarks on algebraically compact categories, in Applications of Categories in Computer Science, pp. 95-106, eds.: Fourman, Johnstone, Pitts, LMS Lecture Note Series 177, Cambridge University Press 1992.

....one has that the category D of domains is obtained as the co Kleisli category of a comonad on the category of strict maps S. For example, Cppo is the co Kleisli category of the lifting comonad on Cppo . In axiomatic domain theory, S satisfies a curious property, first identified by Freyd [13, 14]: a wide class of endofunctors on S have initial algebras whose inverses are final coalgebras (in Freyd s terminology, S is algebraically compact) Following [7] we call such initial final algebras coalgebras bifree algebras. In the example of Cppo , every Cppo enriched endofunctor has a ....

.... Theta ( Gamma) on D have bifree algebras. In spite of the above reformulation, we believe that it is usually more appropriate to consider the bifree algebras as living in S. A common application of the results in this section will involve using a category S that is algebraically compact [13, 14], in which case the existence of sufficiently many bifree algebras in S is guaranteed. The canonical example of this situation is when S is Cppo , which is algebraically compact with respect to Cppo enriched endofunctors [10] The results in this section thus apply to the co Kleisli category of ....

P. Freyd. Remarks on algebraically compact categories. In Applications of Categories in Computer Science, pages 95-- 106. LMS Lecture Notes 177, Cambridge University Press, 1992.

....for a specific model of the language, it is not clear a priori to what extent the specific model affects the result. The latter question was recently addressed by Pitts [17] who showed that Freyd s universal characterization of the solution of a domain equation by the minimal invariant property [6, 5, 7] is sufficient to validate the construction of a wide class of relational interpretations of recursive types. The starting point for the present work is the observation that for a sufficiently rich language with recursive functions and recursive types the minimal invariance property of the model ....

Peter Freyd. Remarks on algebraically compact categories. In M. P. Fourman, P.T. Johnstone, and A. M. Pitts, editors, Applications of Categories in Computer Science. Proceedings of the LMS Symposium, Durham 1991, volume 177 of London Mathematical Society Lecture Note Series, pages 95--106. Cambridge University Press, 1991.

....closure under sums, products, and exponentials, the admission of fixed point operators, and the solution of recursive domain equations. Further impulse to the abstract theory derived from the characterization of the categorical properties of the solutions of domain equations due to Peter Freyd [5,3,4]. In particular, after his insight of considering canonical maps from initial algebras to final coalgebras, the main infinitary axiom of SDT, see [8] reads as follows: L1) c is an iso, where 1 is a terminal object, L is (the underlying endofunctor of) a lifting monad [8,2] internalizing ....

P. Freyd. Remarks on algebraically compact categories. In M.P. Fourman, P.T. Johnstone, and A.M. Pitts, editors, Proc. Symposium in Applications of Categories to Computer Science. Cambridge University Press, 1992.

....Such a theorem holds for pCpo [Plo85] the category of small cpos (posets, possibly without bottom, closed under lubs of chains) and partial continuous functions. The aim of this paper is to generalise to a wide class of order enriched categories (Section 2) one can compare this endeavour to [SP82, Fre90, Fre92] where a similar programme was carried out for the solution of recursive domain equations. Research partially supported by Fundaci on Antorchas and The British Council grant ARG 2281 14 6 and SERC grant RR30735. y Research supported by an SERC Senior Fellowship. In order to provide a direct ....

....) a ; F i. This property allows us to turn mixed variance functors on a category C into covariant symmetric functors on C. Note that InvCAT is cartesian with terminal object (1; Id) and products (A; a ) Theta (B; b ) A Theta B; a Theta ( b ) 5 Recursive types In [Fre91, Fre92], Peter Freyd defined an algebraically complete category as one such that each of its endofunctors has an initial algebra and remarked that this should be understood in a 2 categorical setting; that is, a setting in which the phrase every endofunctor refers to an understood class. For us, this ....

[Article contains additional citation context not shown here]

P.J. Freyd. Remarks on algebraically compact categories. In M.P. Fourman, P.T. Johnstone, and A.M. Pitts, editors, Applications of Categories in Computer Science, volume 177 of London Mathematical Society Lecture Note Series, pages 95--106. Cambridge University Press, 1992.

....equations f p ffi f e = id D and f e ffi f p v id E . The requirement of strictness in the case of ep pairs is redundant. The generalization to pre ep pairs makes the technical details work more smoothly, and also corresponds closely to Freyd s constructions on mixed variance functors [15, 16]. This category has enough structure for interpreting recursive and polymorphic types. Recursive types are interpreted using the inverse limit construction. A chain hD i ; f i j i 0i is a collection such that f i : T 0 s;h 2 T 0 s;h 1 T 1 s;h 2 T 1 s;h 1 T 2 s;h 2 T 2 s;h 1 ....

P. J. Freyd. Remarks on algebraically compact categories. In M. P. Fourman, P. T. Johnstone, and A. M. Pitts, editors, Applications of Categories in Computer Science, volume 177 of London Mathematical Society Lecture Note Series, pages 95--106. Cambridge University Press, 1992.

....Freyd in (Freyd, 1991) is that data types should be defined in algebraically compact categories , that is, in categories where endofunctors have both initial algebras and final coalgebras which, moreover, do coincide in the sense that they are canonically isomorphic . See also (Freyd, 1990; Freyd, 1992). This gives a useful mixed induction coinduction principle. Cf (Pitts, 1994a; Pitts, 1994b) One of the main examples of algebraically compact categories is the category Cppo of complete pointed partial orders and strict continuous functions: regarded as an orderenriched category, it ....

....canonical morphism is itself an isomorphism, then the initial algebra and the final coalgebra are canonically isomorphic. Categories in which all endofunctors have both an initial algebra and a final coalgebra and, moreover, they are canonically isomorphic are called algebraically compact in (Freyd, 1992). An example of such a category is Cppo , when regarded as a Cpocategory. This can be proved by means of the limit colimit coincidence of categories of embedding projection pairs (Smyth and Plotkin, 1982, Theorem 2) In particular, the one element set f g (with trivial order) is a null ....

Freyd, P. (1992). Remarks on algebraically compact categories. In Fourman, M., Johnstone, P., and Pitts, A., editors, Applications of Category Theory in computer science, volume 177 of London Mathematical Society Lecture Notes Series, pages 95--106. Cambridge University Press.

....ordered by inclusion) pCpo (the category of cpos and partial continuous functions, with hom sets ordered pointwise) Prof , and Prof M . Moreover, Kcats are closed under duals and products. From Theorem 2.1, Corollary 2.2, and Lemma 3. 1, we can deduce pseudo algebraic compactness (see [11, 6]) Corollary 3.2 (Pseudo algebraic compactness) Kcats are pseudo algebraically compact with respect to pseudo Cat functors. Thus, every pseudo Cat functor T : K op Theta K K on a Kcat K has a free pseudo dialgebra T (A; A) A characterised by the following universal property: for ....

P. Freyd. Remarks on algebraically compact categories. In M. Fourman, P. Johnstone, and A. Pitts, editors, Applications of Categories in Computer Science, volume 177 of London Mathematical Society Lecture Note Series, pages 95--106. Cambridge University Press, 1992.

....to capture the co inductive properties of recursively defined domains. Indeed we will see that our results go through for a very abstract notion of relation which includes, for example, continuous endofunctions of a domain. A second crucial idea, present in Freyd s recent work on recursive types [6, 7, 8], is to treat separately the positive and negative occurrences of a type in the body of the declaration that defines it. To explain this further, consider the ML declaration datatype ty = C 1 of oe 1 j Delta Delta Delta j Cn of oe n (1) where the types oe 1 ; oe n are built up from ....

.... fixed point property (provided the relations R in the positive place are suitably admissible ) Of course we will deduce this property from the mixed initiality finality property of recursively defined domains which Freyd has focused upon in his work on algebraically compact categories [7, 8]. The above discussion does not make precise exactly what is a relation on a domain. In fact we need remarkably few properties of relations in order to establish our main result. These properties can be conveniently axiomatized using the framework employed by O Hearn and Tennent in their recent ....

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P. J. Freyd, Remarks on algebraically compact categories. In [5], pp 95--106.

.... of Domain Theory, a full clarification of just what canonicity of solutions means, and how it can be translated into proof principles for reasoning about these canonical solutions, has only emerged over the past two or three years, through the work of Peter Freyd and Andrew Pitts [Freyd, 1991, Freyd, 1992, Pitts, 1993a] We make extensive use of their insights in our presentation. Equational theories We present a general theory of the construction of free algebras for inequational theories over continuous domains. These results, and the underlying constructions in terms of bases, appear to be ....

....made on this programme. One step that can be made relatively cheaply is to generalize from concrete categories of domains to categories enriched over some suitable subcategory of DCPO. Much of the force of Domain Theory carries over directly to this more general setting [Smyth and Plotkin, 1982, Freyd, 1992] Moreover, this additional generality is not spurious. A recent development in the semantics of computation has been towards a refinement of the traditional denotational paradigm, to reflect more intensional aspects of computational behaviour. This has led to considering as semantic universes ....

[Article contains additional citation context not shown here]

P. J. Freyd. Remarks on algebraically compact categories. In M. P. Fourman, P. T. Johnstone, and A. M. Pitts, editors, Applications of Categories in Computer Science, volume 177 of L.M.S. Lecture Notes, pages 95--106. Cambridge University Press, 1992.

.... of Domain Theory, a full clarification of just what canonicity of solutions means, and how it can be translated into proof principles for reasoning about these canonical solutions, has only emerged over the past two or three years, through the work of Peter Freyd and Andrew Pitts [Freyd, 1991, Freyd, 1992, Pitts, 1993a] We make extensive use of their insights in our presentation. Equational theories We present a general theory of the construction of free algebras for inequational theories over continuous domains. These results, and the underlying constructions in terms of bases, appear to be ....

....can be made relatively cheaply is to generalize from concrete categories of domains to categories enriched over some suitable subcategory of DCPO. Much of the force of Domain Theory carries over directly to this more general setting [Smyth and Plotkin, 1982, 156 Samson Abramsky and Achim Jung Freyd, 1992] Moreover, this additional generality is not spurious. A recent development in the semantics of computation has been towards a refinement of the traditional denotational paradigm, to reflect more intensional aspects of computational behaviour. This has led to considering as semantic universes ....

[Article contains additional citation context not shown here]

P. J. Freyd. Remarks on algebraically compact categories. In M. P. Fourman, P. T. Johnstone, and A. M. Pitts, editors, Applications of Categories in Computer Science, volume 177 of L.M.S. Lecture Notes, pages 95--106. Cambridge University Press, 1992.

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Freyd, P., Remarks on algebraically compact categories, in Applications of Categories in Computer Science, pp. 95-106, eds.: Fourman, Johnstone, Pitts, LMS Lecture Note Series 177, Cambridge University Press 1992.

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P. Freyd. Remarks on algebraically compact categories. In Applications of Categories in Computer Science, pages 95-- 106. LMS Lecture Notes 177, Cambridge University Press, 1992.

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P. J. Freyd, Remarks on algebraically compact categories, in: M. P. Fourman, P. T. Johnstone, A. M. Pitts (Eds.), Applications of Categories in Computer Science, Vol. 177 of LMS Lecture Note Series, Cambridge Univ. Press, 1992, pp. 95--106.

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