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## "Traced monoidal categories", Andre Joyal, Ross Street, Dominic Verity, 1996

*Reviewed February 4, 2024*

*Citation:* Joyal, AndrĂ©, 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.