Citation: Joyal, André, Ross Street, and Dominic Verity. "Traced monoidal categories." Mathematical proceedings of the cambridge philosophical society. Vol. 119. No. 3. Cambridge University Press, 1996.
Web: https://people.math.rochester.edu/faculty/doug/otherpapers/jsv.pdf
Tags: Mathematical, Monoidal-categories, Non-finite/semisimple
In this paper, the authors deal with traced monoidal categories. The setup is as follows. Suppose you have a braided monoidal category with a twist. To make it a ribbon fusion category, you need to add antiparticles in a way that is compatible with your monoidal structure, your braiding, and your twist. Upon introducing this rigid structure, there is a canonical trace operation that presents itself. The main result of this paper is that having a well-behaved trace map in fact allows you to define antiparticles. Hence, rigid structure on a category with monoidal+braided+twisted structure is equivalent to a well-behaved trace.
In this paper, the result is framed as the fact that the forgetful 2-functor from the 2-category of ribbon categories to the 2-category of traced categories has a left biadjoint. This can be seen as a case where the general theory of 2-categories helps clarify the abstract study of 1-categories.