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Algebra  2013

# A Morphism Double Category and Monoidal Structure

Abstract:

We provide a recipe for “fattening” a category that leads to the construction of a double category. Motivated by an example where the underlying category has vector spaces as objects, we show how a monoidal category leads to a law of composition, satisfying certain coherence properties, on the object set of the fattened category. 1. Introduction and Geometric Background The interaction of point particles through a gauge field can be encoded by means of Feynman diagrams, with nodes representing particles and directed edges carrying an element of the gauge group representing parallel transport along that edge. If the point particles are replaced by extended one-dimensional string-like objects, then the interaction between such objects can be encoded through diagrams of the form (1) where the labels and describe classical parallel transport and , which may take values in a different gauge group, describes parallel transport over a space of paths. We will now give a rapid account of some of the geometric background. We refer to our previous work [1] for further details. This material is not logically necessary for reading the rest of this paper but is presented to indicate the context and motivation for some of the ideas of this paper. Consider a principal -bundle , where is a smooth finite dimensional manifold and a Lie group, and a connection on this bundle. In the physical context, may be spacetime, and describes a gauge field. Now consider the set of piecewise smooth paths on , equipped with a suitable smooth structure. Then, the space of -horizontal paths in forms a principal -bundle over . We also use a second gauge group (that governs parallel transport over path space), which is a Lie group along with a fixed smooth homomorphism and a smooth map such that each is an automorphism of , such that for all and . We denote the derivative by , viewed as a map , and denote by , to avoid notational complexity. Given also a second connection form on and a smooth -equivariant vertical -valued -form on , it is possible to construct a connection form on the bundle where is the -valued -form on specified by which is a Chen integral. Consider a path of paths in specified through a smooth map where each is -horizontal and the path is -horizontal. Let . The bi-holonomy is specified as follows: parallel translate along by , then up the path by , back along -reversed by and then down by , then the resulting point is The following result is proved in [1]. Theorem 1. Suppose that is smooth, with each being -horizontal and the path being -horizontal. Then, the parallel

References

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