Spans of open maps have been proposed by Joyal, Nielsen, and Winskel as a way of adjoining an abstract equivalence, ℘-bisimilarity, to a category of models of computation , where ℘ 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. An issue left open was the congruence properties of ℘-bisimilarity. We address the following fundamental question: given a category of models of computation and a category of observations ℘, are there any conditions under which algebraic constructs viewed as functors preserve ℘-bisimilarity? We define the notion of functors being ℘ factorisable, show how this ensures that ℘-bisimilarity is a congruence with respect to such functors. Guided by the definition of ℘-factorisability we show how it is possible to parametrise proofs of functors being ℘-factorisable with respect to the category of observations ℘, i.e., with respect to a behavioural equivalence.
