J. R. B. Cockett, Introduction to distributive categories, Mathematical Structures in Computer Science 3 (1993), no. 3, 277--307.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....theory is given in Figure 2. Semantics. A bicartesian category is a category with finite products (1, and finite coproducts (0, Bicartesian categories for which the canonical map (A B) A C) A (B C) is an isomorphism for all objects A; B; C are called distributive categories [8, 7]. A cartesian closed category (CCC) is a category with finite products and exponentials ( Bicartesian closed categories are referred to as BiCCCs; they are, of course, distributive. With respect to an interpretation I of base types in a BiCCC S, we write I[ for the interpretation of a ....

J. R. B. Cockett. Introduction to distributive categories. Mathematical Structures in Computer Science, 3:277--307, 1993.

....direct products and nal object 1, then a 1 = 1 a = a. Proof. The isomorphism between a and a 0 is given by I 1 a;0 and [Id a ; Init a ] and the isomorphism between a and a 1 is given by P 1 a;1 and hId a ; Final a i 2 De nition 2. 14 We say that a category is distributive [6, 7] if it satis es 1. has binary direct sums and products; 2. has an initial object 0, and 3. the canonical morphisms ldistr a;b;c : a b a c a (b c) rdistr a;b;c : a c b c a (b c) lzero a : 0 a 0, and rzero a : a 0 0 are isomorphisms. 2 5 De nition 2.15 Let C be a category. We call ....

J. R. B. Cockett. Introduction to distributive categories. Mathematical Structure in Computer Science, 3:277-307, 1993.

....applicable to elementary toposes. The main one says that, in the coproduct of spaces, the two components are embedded as complementary open subsets. The coproduct is therefore stable and disjoint, and the empty space is strict. 9.2. Theorem. C is extensive, i.e. it has stable disjoint coproducts [Coc93, CLW93] Tay99, x5.5] cf. JT84, Corollary to V 2 1] for locales. X Z Y 1 2 s 1 Proof. Given any commutative diagram in C as shown above, we must show that the top row is a coproduct of spaces (Z = X Y ) i the squares are pullbacks (inverse Theory and Applications ....

Robin Cockett. Introduction to distributive categories. Mathematical Structures in Computer Science, 3:277-307, 1993.

....following equations hold: F 1 ffi F A;X = 1 (7) F ff A;X;Y ffi F A;X ThetaY = F A ThetaX;X ffi ( F A;X Theta id Y ) ffi ff FA;X;Y (8) Polynomial functors turn out to be strong under the additional assumption that category C is distributive. A category C is said to be distributive [34, 8] if it possesses both finite products and coproducts and binary products distribute over coproducts. This means that, for any objects A, B and C, the canonical map [inl Theta id C ; inr Theta id C ] A Theta C B Theta C (A B) Theta C is an isomorphism whose inverse is the natural ....

R. Cockett. Introduction to Distributive Categories. Mathematical Structures in Computer Science, 3:277--307, 1993.

....of Categories, Vol. 4, No. 10 213 Recall that for a category A, we may construct the category of finite families in A op , denoted Fam(A op ) This category is the free category with finite sums on A op , and is extensive (see section 1 of [1] proposition 2.4 of [3] or section 5. 2 of [4]) We use the notation P(A) for Fam(A op ) and think of it as a category of polynomials with exponents drawn from A. We denote objects of P(A) as sums of monomials, the family (A i ) i2I being denoted X i2I X A i Since sums in P(A) are given by disjoint unions of families, the sum of ....

....the injections in A are Ms (as they are in the case A = Sets f and M is the injections) there is no reason for (m j n) to be an M even if m and n are Ms. Indeed, the product of two Stirling monomials is not usually a Stirling monomial. This situation is referred to as familial finite products in [4], however, as the category SP(A) is an object of primary interest here, we shall work with products in the category of families. Theory and Applications of Categories, Vol. 4, No. 10 214 2.6. Construction. Suppose that A admits finite sums. For A, B 2 A, consider a sum C = A B in A with ....

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J.R.B. Cockett. Introduction to distributive categories. Mathematical Structures in Computer Science, 3:277--307, 1993.

....3 after we recall the basic concepts of distributive categories in Section 2. 4 2 Distributive Categories Distributive categories are the appropriate formal setting for studying abstract data type specifications and control flow aspects of programs. See [LS91, Wal92a] for an introduction and [Coc91, CF92, CLW93] for more details. Here by a distributive category we mean one in the sense of Lawvere and Schanuel 1 : a category with with finite sums (coproducts) and finite limits, such that in the diagram Y g y X fflffl f Z oo z h A i A A B B oo i B (5) where the bottom row is a ....

J.R.B. Cockett. Introduction to distributive categories. Mathematical Structures in Computer Science, 1:1--20, 1991.

....1996 Abstract In a distributive category (a category in which the product distributes over the coproduct) coproducts can be used to model conditional expressions. We develop such a theory of conditionals. 1 Introduction The category Set of sets and total functions is a distributive category [1], which is to say that the categorical product in Set (namely cartesian product) distributes over the coproduct (disjoint sum) In such a category, coproducts can be used to model conditionals. In this paper we show how this is done, developing a theory of conditionals as we go. We write a 2 A ....

Robin Cockett, An Introduction to Distributive Categories. Mathematical Structures in Computer Science, 1 (1991) p1--20.

.... arithmetic (see Cockett [Coc90] A lextensive category has finite limits, finite coproducts, and the property that in the following diagram (2) 1) B A B A Y Z X b 1 b 0 y x (1) and (2) are pullbacks exactly when the top row is a coproduct (see Carboni, Lack and Walters [CLW92] or Cockett [Coc93]) It is important to realize that, foundationally, this is all the structure that is required: although we shall need some extra ingredients to carry through the subsequent construction of processes, the remaining ingredients are a matter of choice rather than axiomatization. Let C be a ....

Cockett, J.R.B., Introduction to distributive categories. Mathematical Structures in Computer Science, 3:277--307, 1993.

....to the limit cone in X gives the mediating morphism in F= G: GZ GK FZ FK k z Gu Fu with commutivity forced since GZ is the limit in Y . 2 Dually, F= G has pointwise D colimits whenever X has D colimits and F preserves them. 5 It is often desirable to have coproducts which are extensive [Coc93]. A category with finite coproducts is extensive if given any commuting diagram (2) 1) B A B A Y Z X b 1 b 0 (1) and (2) are pullbacks if and only if the top row is a coproduct. An extensive category with finite limits is called lextensive. Proposition 3 F= G is lextensive whenever X is ....

J.R.B. Cockett. Introduction to distributive categories. Mathematical Structures in Computer Science, 3:277--307, 1993.

....the model category Tran of transition systems used to construct SProc as well as the functorial structure of Tran used later to construct ASProc. 2. 1 Notation For generality, we describe the category of transition systems and its functorial structure in a lextensive category (see [CLW92] or [Coc93]) Such categories have finite limits, finite coproducts and the property that in the following diagram (2) 1) B A B A Y Z X b 1 b 0 y x (1) and (2) are pullbacks if and only if the top row is a coproduct. Although the path construction on transition systems is described in Set, we conjecture ....

Cockett, J.R.B., Introduction to distributive categories. Mathematical Structures in Computer Science, 3:277--307, 1993.

....study of cartesian weak distributive categories is to understand the categorical proof theory of the = fragment of intuitionistic logic. This proof theory should be a cartesian weakly distributive category. However, there is another possible candidate for this proof theory. A distributive category [Co93] has finite products and coproducts such that the comparison map from the coproduct hi Theta b 0 ji Theta b 1 i : A Theta B A Theta C Gamma A Theta (B C) is an isomorphism. A somewhat surprising observation is: Proposition 3.1 An elementary distributive category is a cartesian weakly ....

....map is essentially the identity. Similarly the bottom map is essentially the Boolean negation map. This implies that, in any distributive category which is also weakly distributive, the Boolean negation has a fixed point. This happens only when the distributive category is a preorder (see [Co93]) 2 Finally, we ought to note that dropping the offending condition from the definition (i.e. equation (13) so as to keep distributive categories as examples) is really not an option, for, as the reader can verify from Figure 1, this condition is necessary if the semantics is to satisfy ....

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Cockett J.R.B. Introduction to distributive categories. Mathematical Structures in Computer Science 3 (1993) 277 - 308.

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J. R. B. Cockett, Introduction to distributive categories, Mathematical Structures in Computer Science 3 (1993), no. 3, 277--307.

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J. R. B. Cockett. Introduction to distributive categories. Mathematical Structures in Computer Science, 3:277--307, 1993.

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