In the present paper, we develop the notion of a trace operator on a linearly distributive category, which amounts to essentially working within a subcategory (the core) which has the same sort of "type degeneracy" as a compact closed category. We also explore the possibility that an object may have several trace structures, introducing a notion of compatibility in this case. We show that if we restrict to compatible classes of trace operators, an object may have at most one trace structure (for a given tensor structure). We give a linearly distributive version of the "geometry of interaction" construction, and verify that we obtain a linearly distributive category in which traces become canonical. We explore the relationship between our notions of trace and fixpoint operators, and show that an object admits a fixpoint combinator precisely when it admits a trace and is a cocommutative comonoid. This generalises an observation of Hyland and Hasegawa.

Traced monoidal categories of circuits are introduced in order to give a compositional account of circuits. Behaviour functors of circuits are dened, and compositionality results are obtained in terms of the structure-preserving properties of these functors. The theory includes models of the asynchronous binary circuits constructed from instantaneous functional circuits and upbounded inertial binary delays, as well as asynchronous pulse circuits such as pipelines. Via the notion of a circuit network, an explicit connection is made with the General Multiple-Winner circuit model.