Robert Goldblatt. Topoi, the Categorial Analysis of Logic. North-Holland, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....with some basics of term rewriting, including con uence, termination and narrowing; these are explained, for example, in [7] and [16] We sometimes use basic notions from category theory, including category, functor, and initial object. For an introduction to category theory, see [3] or [27]. We use the bbold font to denote categories, e.g. C. Given morphisms f : A B and g : B C, we let f ; g denote their composition, a morphism A C; also, we let 1 A denote the identity morphism at an object A. Sections 4.1 and 7 use colimits, and Section 7 also uses universal constructions. ....

Robert Goldblatt. Topoi, the Categorial Analysis of Logic. North-Holland, 1979.

..... reduce (3 ( 3 i) 2) i) 2 i) 2) reduce (2 ( 3 i) j) 5 i) 7 j) reduce (1 ( 1 i) j) 2 j) C.8 Categories and Coproducts This subsection specifies categories and coproducts. Some familiarity with category theory may be needed (e.g. sections 2.3 and 3. 9 of [88]) on the other hand, the code may also provide a more concrete understanding of the categorical concepts; see also [136] Note how universal morphisms are defined as a subsort, and also the use of sort constraints. Recall that the semantics of op as is not yet implemented. The use of memo has ....

Robert Goldblatt. Topoi, the Categorial Analysis of Logic. North-Holland, 1979.

....on world w. Note that these definitions are applicable with any category W of possible worlds. The above treatment of procedures is not in general extensional in the usual sense; however, for any category W, the functor category S (where S is the category of sets and functions) is a topos [4, 7, 1], and so there is an interpretation of logical connectives such as implication ( universal quantification (8) and identity (j) that validates the laws of intuitionistic logic, and also the following formal law of extensionality: 8 : P ( j Q( P j Q where is not free in P; Q: ....

R. Goldblatt. Topoi, The Categorial Analysis of Logic. North-Holland, Amsterdam, 1979.

....from ours, we do not survey it here; see [6] and its references. Space precludes many details, including proofs, some de nitions, and further examples; see the full technical report [19] Parts of the paper assume familiarity with basics of category theory and algebraic speci cation (e.g. [17, 21]) 2 Examples This section introduces parameterized programming through examples, beginning with the rst order case, and then higher order examples. Example 1 The simplest non trivial module has only a single sort: th TRIV is sort Elt . end The keyword th introduces theories, which ....

Robert Goldblatt. Topoi, the Categorial Analysis of Logic. North-Holland, 1979.

....this paper calls them theories . We believe that this kind of type is the most relevant to formal methods for program development, and to programming in the large. This paper assumes familiarity with category, functor, natural transformation and colimit, for which see texts and papers such as [29], 38] and [4] Lawvere theories and indexed categories are also mentioned but not assumed. Section 2.5 assumes adjoints, and some other advanced concepts are defined as they arise. The identity morphism at A is denoted 1 A , and composition is indicated by semicolon, so that for example, f ; ....

Robert Goldblatt. Topoi, the Categorial Analysis of Logic. North-Holland, 1979.

....Classifier for vague regular subgraphs. T F M M F M F F T F F M F T F M F F M F M The classifying graph for crisp subgraphs is already well known, and is described in [LS97] Classifying graphs are a special case of the concept of a subobject classifier in category theory [Gol84] However, the classifiers for vague and for granular graphs in the present paper do not appear to have been discussed before. 5 Kinds of Granulation for Graphs Four kinds of granulation for graphs are considered in this section. For a given graph, G, there will in general be many different ....

R. Goldblatt. Topoi. The Categorial Analysis of Logic. North-Holland, 1984.

....kinds of fuzzy subset are obtained by considering functions from X to a suitable set of fuzzy truth values. The general idea of identifying parts of a structure, X , with functions from X to another structure is wellknown and leads to the concept of a sub object classi er in category theory (Goldblatt 1984). The details of this technique applied to crisp subgraphs can be found in (Lawvere Schanuel 1997) but the details of the approach to coarse graphs given below have not appeared before. There is some material on classi ers for vague graphs and for a simpler kind of coarse graph in (Stell ....

Goldblatt, R. (1984), Topoi. The Categorial Analysis of Logic, North-Holland.

....for axiomatic semantics. An alternative is higher order intuitionistic logic [8] or in nitary coherent logic [12] The existence proof for predicate transformers in these logics is analogous to the classical case. The main advantage is the close connection of these logics to the theory of topoi [2, 3, 8, 10, 11, 14]. Here, we consider the higher order logic of Fourman and Scott [8, 22] Each theory T of such a logic de nes a topos IE(T ) of de nable types. Conversely, each topos E de nes a higher order language L(E) and a canonical de nitionally complete theory T (E) with E = IE(T (E) In particular, ....

.... theory [19] to the case of dynamics in the data, in which case the polymorphic calculus is chosen as the underlying type system [20] Throughout the text, we assume some familiarity with basic notions of category theory [1, 13] Furthermore, topos theory [2, 11, 14] and its connection to logic [3, 8, 10] must also be presupposed. 2 The Classical Calculus This section gives a brief review of Dijkstra s classical calculus [6, 16] Assume that S is a program speci cation and that X is the nite set of variables occurring 1 It is also possible to use rst order logic for arithmetic, but this ....

R. Goldblatt: Topoi { The Categorial Analysis of Logic, North-Holland, Studies in Logic, vol. 98, 1984

....or phrases at different worlds need to be determined. The established formalism for applying the possible worlds ideas to programming language semantics is the formalism of functor categories. Excellent introductions CHAPTER 2. ALGOL LIKE LANGUAGES 13 to Category Theory can be found in [HS79, Gol79, BW90] a quite detailed exposition of functor categories for programming language semantics is given in [Ten91] Only a brief introduction will be given here, to establish notation and to outline the fundamental concepts. 2.3 Functor Category Semantics Let W be some category to be used as the ....

R. Goldblatt. Topoi, The Categorial Analysis of Logic. North-Holland, Amsterdam, 1979.

....with some basics of term rewriting, including confluence, termination and narrowing; these are explained, for example, in [7] and [15] We sometimes use basic notions from category theory, including category, functor, and initial object. For an introduction to category theory, see [3] or [26]. We use the bbold font to denote categories, e.g. C. Given morphisms f : A B and g : B C, we let f ; g denote their composition, a morphism A C; also, we let 1A denote the identity morphism at an object A. Sections 4.1 and 7 use colimits, and Section 7 also uses universal constructions. ....

Robert Goldblatt. Topoi, the Categorial Analysis of Logic. North-Holland, 1979.

....topos. To wit, a topos is a cartesian closed category with certain additional properties which are inspired by typical properties of Set; most notable among these is the existence of representations in a certain sense for partial morphisms, i.e. morphisms de ned only on a subobject of the domain [1, 18, 25]. Typical topoi, besides Set, are categories of sheaves; in fact, the early development of topos theory was largely motivated by the needs of algebraic geometry [19] Topos theory has aquired particular importance due to its close relation to constructive logic. In topology, the concept of topos ....

R. Goldblatt: Topoi. The Categorial Analysis of Logic, rev. ed., NorthHolland, Amsterdam, 1984.

....otherwise, they have short proofs, and the reader is urged to use them as exercises to test and improve his her understanding. Some may find this a relatively painless way to learn some basic category theory. The standard advanced introduction to category theory is Mac Lane [29] while Goldblatt [26] provides a gentler approach that includes topoi; neither book discusses Lawvere theories, but (for example) Schubert [42] does. The only prerequisites for this paper are some elementary set theory and an interest in its subject matter. I thank Timothy Fernando and Peter Rathmann for their ....

....category is higher order unification. Topoi, which were introduced by Lawvere and Tierney, are Cartesian closed categories with additional structure that captures set like structures that arise in geometry, algebra and logic, thus providing a surprising unification of many areas of mathematics; [26] provides an introduction to these topics, and of course, we now know what it would mean to solve equations in topoi. 8 Summary This paper develops a very general approach to the solution of equations, starting from the simple example of polynomials with integer coefficients, and gradually ....

Robert Goldblatt. Topoi, the Categorial Analysis of Logic. North-Holland, 1979.

....their approach does not seem to be closely related to the present paper. Prerequisites and Notation Basic category theory and some intuition for concurrency are needed to read this paper. The former can be found many places, including [32] which requires some mathematical sophistication, and [26], which may be especially recommended because it discusses sheaves, though in a different formulation from ours, and because it begins rather gently. An introduction to category theory for computing scientists is [3] Some underlying intuitions for basic categorical concepts are given in [16] and ....

....replacing u : U A 1 Theta : Theta A n by u : U U(A 1 Theta : Theta A n ) Proposition 19: If C is a structure category, then its forgetful functor U preserves products, and more generally, all limits. Proof: It is well known that right adjoints preserve limits; e.g. see [32] or [26]. 2 This means that the underlying set of a product object in C can be taken to be the product of the underlying sets of the component objects; for example, we can get a product of two vector spaces by giving a vector space structure to the product of their underlying sets. In the following ....

[Article contains additional citation context not shown here]

Robert Goldblatt. Topoi, the Categorial Analysis of Logic. North-Holland, 1979.

....our purpose is to provide intuition, and the definitions can be found in any textbook. Thus, if you are a newcomer to category theory, you will need to use some text in connection with this paper. Unfortunately, no existing text is ideal for computing scientists, but perhaps that by Goldblatt [36] comes closest. The classic text by Mac Lane [47] is warmly recommended for those with sufficient mathematics background, and Herrlich and Strecker s book [39] is admirably thorough; see also [2] and [45] The paper [22] gives a relatively concrete and self contained account of some basic category ....

....can be composed to yield a third. The category Set of sets embodies a contrary point of view, that each function has a domain in which its arguments are meaningful, a codomain in which its results are meaningful, and composition of functions is only allowed when meaningful in this sense. See [36] for related discussions. 1.2 Relations. Just as with functions, it seems desirable to take the view that the composition of relations is only meaningful when the domains match. Thus, we may define a relation from a set A 0 to a set A 1 to be a triple (A 0 ; R; A 1 ) with R A 0 Theta A 1 , and ....

[Article contains additional citation context not shown here]

Robert Goldblatt. Topoi, the Categorial Analysis of Logic. North-Holland, 1979.

....of clarifying relationships between algebra and topology. As time passed they became increasingly recognized as a powerful tool for exploiting analogies throughout mathematics [21] In the early 1960s they led to revolutionary and still controversial developments in mathematical logic [17]. It gradually became clear that category theory was a part of a deeper subject, higher dimensional algebra , in which the concept of a category is generalized to that of an n category . But only by the 1990s did the real importance of categories for physics become evident, with the discovery ....

R. Goldblatt, Topoi, the Categorial Analysis of Logic, North-Holland, New York, 1979.

....are again conservative for initiality in a reasonable semiexact institution. 1.4 Prerequisites This paper assumes familiarity with basic category theory, including categories, functors, limits, colimits, and adjoints. The necessary material may be found in [43] 42] or in a more gentle style, [39]. By way of notation, we use ; for composition, we let 1 A denote the identity morphism at an object A, and we let jCj denote the class of objects of a category C. We have found that certain readers may have doubts about set theoretic foundations when category theory is used. In fact, this ....

Robert Goldblatt. Topoi, the Categorial Analysis of Logic. North-Holland, 1979.

....spectrum specification design languages and calculi. We make moderate use of the theory of categories, as well as of the theory of sequential processes. The reader may find all the necessary category theoretic notions in the first What is an object, after all 4 chapters of Goldblat s book [22]. And those from the theory of processes in Hennessy s book [24] although we use a simple model of processes first proposed in [4] 2. WHAT IS AN OBJECT A computer system, when observed as a whole, is a symbolic machine that is able to perceive, manipulate, store, produce and transmit ....

Goldblatt, R.: Topoi, the Categorial Analysis of Logic. North-Holland, 1979.

....relations and D. Puppe [15] established a notion of I categories that was a start point of categorical theory of relations. Peter Freyd [3] investigated theory of allegories as a basis for theory of relations and Max Kelly [10] studied relations relative to factorization systems. Topos theory [4, 5] is well known as a categorical model of higher order intuitionistic set theory and has been extensively studied by categorists and logicians. This paper presents relational set theory as categorical set theory slightly different from topoi or allegories to give another categorical perspective of ....

R. Goldblatt, Topoi, The categorial analysis of logic, North-Holland, Amsterdam, 1979.

.... ( 2 i) 2) reduce (2 ( 3 i) j) 5 i) 7 j) reduce (1 ( 1 i) j) 2 j) C.8 Categories and Coproducts This subsection presents the theories of categories and coproducts. Some familiarity with category theory may be needed to follow this example (e.g. sections 2.3 and 3. 9 of [69]) on the other hand, the code may also provide a more concrete understanding of the categorical concepts. Note how universal morphisms are defined as a subsort, and also the use of sort constraints. Recall that the semantics of op as is not yet implemented. The use of memo has quite a ....

Robert Goldblatt. Topoi, the Categorial Analysis of Logic. North-Holland, 1979.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC