Traces in monoidal categories

The main result of this paper is the construction of a trace and a trace pairing for endomorphisms satisfying suitable conditions in a monoidal category. This construction is a common generalization of the trace for endomorphisms of dualizable ob jects in a balanced monoidal category and the trace of nuclear operators on a locally convex topological vector space with the approximation property.

💡 Research Summary

The paper “Traces in Monoidal Categories” develops a unified framework for defining a trace and a trace pairing for endomorphisms that satisfy certain categorical conditions, thereby encompassing two classical notions of trace: (1) the trace of endomorphisms of dualizable objects in a balanced (or pivotal) monoidal category, and (2) the trace of nuclear operators on locally convex topological vector spaces (LCTVS) that enjoy the approximation property.

The authors begin by recalling the standard construction of a trace in a balanced monoidal category. If an object (X) is dualizable, there exist evaluation and coevaluation morphisms \