We present a formal description of softwate atchitectures for softwate reuse to support a view of systematic softwate reuse as the plugging of components into an architecture. The components ate object descriptions in the object calculus. Interconnection between the components is defined via synchronisation morphisms within a framework based on category theory. Component composition is defined via the pushout construction, giving the atchitecture as a "calculated" component, from which the atchitecture's properties may be derived. We show that the architectures described ate reusable in our Reuse Triplet that forms the motivation for our ongoing work on systematic softwate reuse. This work provides further support for the suggestion that category theory provides the appropriate level of mathematical abstraction to describe softwate atchitectures.