Boolean and classical restriction categories

  • Authors:

  • Robin Cockett;Ernie Manes

  • Affiliations:

  • Department of computer science, university of calgary, calgary, t2n 1n4, alberta, canada;Department of mathematics and statistics, university of massachusetts, amherst, massachusetts 01003, u.s.a.

  • Venue:
  • Mathematical Structures in Computer Science
  • Year:
  • 2009

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Abstract

A restriction category is an abstract category of partial maps. A Boolean restriction category is a restriction category that supports classical (Boolean) reasoning. Such categories are models of loop-free dynamic logic that is deterministic in the sense that Q ⊂ [α]Q. Classical restriction categories are restriction categories with a locally Boolean structure: it is shown that they are precisely full subcategories of Boolean restriction categories. In particular, a Boolean restriction category may be characterised as a classical restriction category with finite coproducts in which all restriction idempotents split. Every restriction category admits a restriction embedding into a Boolean restriction category. Thus, every abstract category of partial maps admits a conservative extension that supports classical reasoning. An explicit construction of the classical completion of a restriction category is given.