Formal Aspects of Computing

, Volume 13, Issue 1, pp 32–49 | Cite as

Presheaves as Configured Specifications

  • Steven Vickers
  • Gillian Hill

Abstract.

The paper addresses a notion of configuring systems, constructing them from specified component parts with specified sharing. This notion is independent of any underlying specification language and has been abstractly identified with the taking of colimits in category theory. Mathematically it is known that these can be expressed by presheaves and the present paper applies this idea to configuration.

We interpret the category theory informally as follows. Suppose ? is a category whose objects are interpreted as specifications, and for which each morphism u : XY is interpreted as contravariant ‘instance reduction’, reducing instances of specification Y to instances of X. Then a presheaf P: Set ?op represents a collection of instances that is closed under reduction. We develop an algebraic account of presheaves in which we present configurations by generators (for components) and relations (for shared reducts), and we outline a proposed configuration language based on the techniques. Oriat uses diagrams to express colimits of specifications, and we show that Oriat's category Diag(?) of finite diagrams is equivalent to the category of finitely presented presheaves over ?.

Keywords: Category; Colimit; Configuration; Diagram; Presheaf; Specification 

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Copyright information

© British Computer Society 2001

Authors and Affiliations

  • Steven Vickers
    • 1
  • Gillian Hill
    • 2
  1. 1.Department of Pure Mathematics, The Open University, Milton Keynes, UKGB
  2. 2.Department of Computer Science, City University, and Department of Computing, Imperial College, London, UKGB

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