Open Maps, Behavioural Equivalences, and Congruences

Abstract

<p>Spans of open maps have been proposed by Joyal, Nielsen,<br />and Winskel as a way of adjoining an abstract equivalence, P-bisimilarity, to a category of models of computation M, where P is an arbitrary subcategory of observations. Part of the motivation was to recast and generalise Milner's well-known strong bisimulation in this categorical setting.<br />An issue left open was the congruence properties of P-bisimilarity. We address the following fundamental question: given a category of models of computation M and a category of observations P, are there any conditions under which algebraic constructs viewed as functors preserve P-bisimilarity? We define the notion of functors being P-factorisable,<br />show how this ensures that P-bisimilarity is a congruence with respect to such functors. Guided by the definition of P-factorisability we show how it is possible to parameterise proofs of functors being P-factorisable<br />with respect to the category of observations P, i.e., with respect to a behavioural equivalence.</p><p>Keywords: Open maps, P-bisimilarity, P-factorisability, congruences, process algebra, category theory.</p>