Axiomatic Sheaf Theory : Some Constructions and Applications

I see model theory as becoming increasingly detached from set theory, and the Tarskian notion of set-theoretic model being no longer central to model theory. In much of modern mathematics, the set-theoretic component is of minor interest, and basic notions are geometric or category-theoretic. In algebraic geometry, schemes or algebraic spaces are the basic notions, with the older “sets of points in affine or projective space” no more than restrictive special cases. The basic notions may be given sheaf-theoretically, or functorially. To understand in depth the historically important affine cases, one does best to work with more general schemes. The resulting relativization and “transfer of structure” is incomparably more flexible and powerful than anything yet known in “set-theoretic model theory”.It seems to me now uncontroversial to see the fine structure of definitions as becoming the central concern of model theory, to the extent that one can easily imagine the subject being called “Definability Theory” in the near future.Tarski's set-theoretic foundational formulations are still favoured by the majority of model-theorists, and evolution towards a more suggestive language has been perplexingly slow. None of the main texts uses in any nontrivial way the language of category theory, far less sheaf theory or topos theory. Given that the most notable interactions of model theory with geometry are in areas of geometry where the language of sheaves is almost indispensable (to the geometers), this is a curious situation, and I find it hard to imagine that it will not change soon, and rapidly.

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Sheaf semantics for concurrent interacting objects

Mathematical Structures in Computer Science ◽

10.1017/s0960129500001420 ◽

1992 ◽

Vol 2 (2) ◽

pp. 159-191 ◽

Cited By ~ 25

Author(s):

Joseph A. Goguen

Keyword(s):

Computer Security ◽

Object Oriented ◽

Continuous Linear ◽

Sheaf Theory ◽

Electrical Circuits ◽

Object Based ◽

Time Concepts ◽

Hardware Description ◽

Description Languages ◽

Object Oriented Systems

This paper uses concepts from sheaf theory to explain phenomena in concurrent systems, including object, inheritance, deadlock, and non-interference, as used in computer security. The approach is very; general, and applies not only to concurrent object oriented systems, but also to systems of differential equations, electrical circuits, hardware description languages, and much more. Time can be discrete or continuous, linear or branching, and distribution is allowed over space as well as time. Concepts from categpru theory help to achieve this generality: objects are modelled by sheaves; inheritance by sheaf morphisms; systems by diagrams; and interconnection by diagrams of diagrams. In addition, behaviour is given by limit, and the result of interconnection by colimit. The approach is illustrated with many examples, including a semantics for a simple concurrent object-based programming language.

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Homotopy Theory of Diagrams and CW-Complexes Over a Category

Canadian Journal of Mathematics ◽

10.4153/cjm-1991-046-3 ◽

1991 ◽

Vol 43 (4) ◽

pp. 814-824 ◽

Cited By ~ 8

Author(s):

Robert J. Piacenza

Keyword(s):

Classical Theory ◽

Homotopy Theory ◽

Homotopy Groups ◽

Weak Equivalence ◽

Equivariant Homotopy ◽

Topological Category ◽

Sheaf Theory ◽

Closed Model ◽

Base Category ◽

Closed Model Category

The purpose of this paper is to introduce the notion of a CW complex over a topological category. The main theorem of this paper gives an equivalence between the homotopy theory of diagrams of spaces based on a topological category and the homotopy theory of CW complexes over the same base category.A brief description of the paper goes as follows: in Section 1 we introduce the homotopy category of diagrams of spaces based on a fixed topological category. In Section 2 homotopy groups for diagrams are defined. These are used to define the concept of weak equivalence and J-n equivalence that generalize the classical definition. In Section 3 we adapt the classical theory of CW complexes to develop a cellular theory for diagrams. In Section 4 we use sheaf theory to define a reasonable cohomology theory of diagrams and compare it to previously defined theories. In Section 5 we define a closed model category structure for the homotopy theory of diagrams. We show this Quillen type homotopy theory is equivalent to the homotopy theory of J-CW complexes. In Section 6 we apply our constructions and results to prove a useful result in equivariant homotopy theory originally proved by Elmendorf by a different method.

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On q-th Derivative of Vector Bundles

Nagoya Mathematical Journal ◽

10.1017/s0027763000024211 ◽

1967 ◽

Vol 29 ◽

pp. 121-126

Author(s):

Akikuni Kato

Keyword(s):

Vector Bundle ◽

Vector Bundles ◽

Present Note ◽

Higher Order ◽

Enumerative Geometry ◽

Sheaf Theory ◽

Definition Of

In the present note we shall be concerned with the improvement of fundamental definitions in higher order enumerative geometry which has been recently given by W. F. Pohl. Pohl’s definition of q-th derivative of vector bundle is very complicated. We shall give a simpler and more reasonable definition of the q-th derivative of vector bundle in terms of sheaf theory and simplify the proofs in [P]. We shall also give a definition of higher order singularity of map.

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