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Toposes and quasi-toposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and many of the latter arise by adding "good " quotients of equivalence relations to a simple category with finite limits. This construction is called the exact completion of the original category. Exact completions are not always toposes and it was not known, not even in the realizability and presheaf cases, when or why toposes arise in this way. Exact completions can be obtained as the composition of two related constructions. The first one assigns to a category with finite limits, the "best " regular category (called its regular completion) that embeds it. The second assigns to

: This paper is an introduction to recent research on hidden algebra and its application to software engineering; it is intended to be informal and friendly, but still precise. We first review classical algebraic specification for traditional "Platonic" abstract data types like integers, vectors, matrices, and lists. Software engineering also needs changeable "abstract machines," recently called "objects," that can communicate concurrently with other objects through visible "attributes" and state-changing "methods." Hidden algebra is a new development in algebraic semantics designed to handle such systems. Equational theories are used in both cases, but the notion of satisfaction for hidden algebra is behavioral, in the sense that equations need only appear to be true under all possible experiments; this extra flexibility is needed to accommodate the clever implementations that software engineers often use to conserve space and/or time. The most important results in hidden algebra are ...

Abstract We propose a notion of interval object in a category with finite products, providing a universal property for closed and bounded real line segments. The universal property gives rise to an analogue of primitive recursion for defining computable functions on the interval. We use this to define basic arithmetic operations and to verify equations between them. We test the notion in categories of interest.

Abstract. Locally partial-ordered spaces (local po-spaces) have been used to model concurrent systems. We provide equivalences for these spaces by

It advertises the joining of two themes: groups and symmetry; and categorical methods and analogues of set theory. Groups are expected to be associated with symmetry. Klein’s famous Erlanger Programme asserted that the study of a geometry was the study of the group of automorphisms of that geometry. The structure of group alone may not give all the expression one needs of the intuitive idea of symmetry. One often needs structured groups (for example topological, Lie, algebraic, order,...). Here we consider groups with the additional structure of directed graph, which we abbreviate to graph. This type of structure appears in [18, 14]. We shall associate with a graph A a group AUT(A) which is also a graph. The vertices of AUT(A) are the automorphism of the graph A and the edges between automorphisms give an expression of “adjacency ” of automorphisms. The vertices of this graph form a group, and so also do the edges. The automorphisms of A adjacent to the identity will be called the inner automorphisms of the graph A. One aspect of the problem is to describe these inner automorphisms in terms of the internal structure of the graph A. The second theme is that of regarding the usual category of sets and mappings as but one environment for doing mathematics, and one which may be replaced by others. We use the word “environment ” here rather than “foundation”, because the former word implies a more relativistic approach.

Data Types and Algebraic Semantics The history of programming languages, and to a large extent of software engineering as a whole, can be seen as a succession of ever more powerful abstraction mechanisms. The first stored program computers were programmed in binary, which soon gave way to assembly languages that allowed symbolic codes for operations and addresses. fortran began the spread of "high level" programming languages, though at the time it was strongly opposed by many assembly programmers; important features that developed later include blocks, recursive procedures, flexible types, classes, inheritance, modules, and genericity. Without going into the philosophical problems raised by abstraction (which in view of the discussion of realism in Section 4 may be considerable), it seems clear that the mathematics used to describe programming concepts should in general get more abstract as the programming concepts get more abstract. Nevertheless, there has been great resistance to u...

into two principal paradigms: model-theoretic syntax (MTS), which

It is shown that every (small) topos is equivalent to the category of global sections of a sheaf of so-called hyperlocal topoi, improving on a result of Lambek & Moerdijk. It follows that every boolean topos is equivalent to the global sections of a sheaf of well-pointed topoi. Completeness theorems for higher-order logic result as corollaries. The main result of this paper is the following. Theorem (Sheaf representation for topoi). For any small topos E, there is a sheaf of categories e E on a topological space, such that: (i) E is equivalent to the category of global sections of e E, (ii) every stalk of e E is a hyperlocal topos. Moreover, E is boolean just if every stalk of e E is well-pointed. Before defining the term "hyperlocal," we indicate some of the background of the theorem. The original and most familiar sheaf representations are for commutative rings (see [12, ch. 5] for a survey); e.g. a well-known theorem due to Grothendieck [9] asserts that every commutative r...

Abstract. Information systems anticipate the real world. Classical databases store, organise and search collections of data of that real world but only as weak anticipatory information systems. This is because of the reductionism and normalisation needed to map the structuralism of natural data on to idealised machines with von Neumann architectures consisting of fixed instructions. Category theory developed as a formalism to explore the theoretical concept of naturality shows that methods like sketches arising from graph theory as only non-natural models of naturality cannot capture real-world structures for strong anticipatory information systems. Databases need a schema of the natural world. Natural computing databases need the schema itself to be also natural. Natural computing methods including neural computers, evolutionary automata, molecular and nanocomputing and quantum computation have the potential to be strong. At present they are mainly at the stage of weak anticipatory systems.