Partial Traces on Additive Categories

In this paper, we study partial traces on additive categories. Haghverdi and Scott introduced partially traced symmetric monoidal categories generalizing traced symmetric monoidal categories given by Joyal, Street and Verity. The original example of a partial trace is given in terms of the execution formula on the category of vector spaces and linear functions. Malherbe, Scott and Selinger gave another example of a partial trace on the category of vector spaces, and they observed that we can define these two partial traces on arbitrary additive categories. A natural question is: what kind of partial traces does the category of vector spaces have? We give a (partial) answer to this question. Our main result is: every abelian category has a largest partial trace. Here, “largest” means that every partial trace on the abelian category is obtained by restricting the domain of the largest partial trace. As a corollary, we show that the partial trace given by Malherbe, Scott and Selinger is the largest partial trace on the category of vector spaces.