Tangled Circuits

The contribution of the paper is the introduction and application of several new such categories, and appropriate functors between them. The authors and collaborators have previously studied similar systems using symmetric monoidal categories ( [8,9,10,11,16,17,18,5]), with separable algebras instead of Frobenius algebras. These earlier works did not take into consideration any tangling of the wires. Further we will see in section ?? the importance of considering Frobenius algebras rather than the more special separable algebras even in the symmetric monoidal case (no tangling).

We propose a definition for a category of tangled circuit diagrams, in which it is possible to distinguish, for example, the first and second of the following circuit diagrams, while the second and third are equal.

The notion of tangled circuit diagram is parametrized by a multigraph (or tensor scheme) of components (such as the component R in the example above). Given such a multigraph M, a tangled circuit diagram (or more briefly, a circuit diagram) is an arrow in the free braided strict monoidal category on M in which objects of the multigraph M are equipped with symmetric Frobenius algebra structures; we denote this category by TCircD M . The objects of the multigraph M may be thought of as types of wires. Given any object A of M it is straightforward to see that there is an appropriate functor from Freyd, Yetter’s category Tangle ( [6]) to TCircD M since a symmetric Frobenius structure on A induces a tangle algebra structure on A. As a result any invariants of tangled circuit diagrams provide also invariants for tangles and knots. We conjecture that such functors Tangle / / TCircD M are faithful. We also conjecture that there is a topological description of TCircD M related to Freyd, Yetter’s description of Tangle and to cobordisms.

The category Rel whose objects are sets, and whose arrows are relations is symmetric monoidal with the tensor of sets being the cartesian product, and each object has a symmetric Frobenius (even separable) algebra structure provided by the diagonal functions and their reverse relations. In fact this was the motivating example for the introduction in [3] of the Frobenius equations (equivalent axioms had been given earlier by Lawvere in [14]). We describe here a modification of Rel which we call TRel G , which depends on a group G, and which is braided rather than symmetric. We further describe a commutative Frobenius algebra in TRel G which hence yields a representation of TCircD M , and this representation enables us, for example, to distinguish the two different circuits above. We discuss distinguishing closed circuits, a problem analogous to classifying knots, using TRel G .

The principal category we have using in the earlier work on circuits and communicating-parallel algebras of processes has been the category Span(Graph) of spans of graphs (and for sequential systems Cospan(Graph)). Already in the original paper [8] the separable algebra structure on each object played a crucial role. The relation between another model of circuits, namely Mealy automata and Span(Graph) was discussed in [9]. One of the motivations of the present work is to produce an semantic algebra in which the twisting of wires is also (at least partially) expressible.

To this end we introduce first a simple braided modification TSpan G of Span(Set), depending on a group G, with a commutative Frobenius algebra. It is clear that a similar construction TSpan G (C) could be made for a group object G in a category C with limits in the place of Set.

Again, there is a representation of Tangle (via a representation of TCircD) which takes a tangle to the span of colourings of the tangle (introduced by John Armstrong in [2]). Applied to knots the set of colourings is one of the simplest invariants for distinguishing knots (as a first example it allows one to show that a trefoil is not an unknot). The extended notion of colourings of tangled circuit diagrams gives further aid in distinguishing circuit diagrams.

The category Group op , the dual of the category of groups has finite limits. Further F , the free group on one generator is a group object in Group op . The category TSpan F (Group op ) is braided monoidal with F equipped with a commutative Frobenius structure. The induced representation

associates the cospan of groups introduced by John Armstrong in [1] to a tangle, and the knot group to a knot.

This example comes from the paper [9] where it is discussed in detail. However the Frobenius algebra structure was not noticed in that paper. The category is analogous to TSpan G (Graph) where the group