Star-autonomous envelopes

Abstract: Symmetric monoidal categories with duals, a.k.a. compact monoidal categories, have a pleasing string diagram calculus. In particular, any compact monoidal category is closed with $[A,B] = (A^* \otimes B)$, and the transpose of $A \otimes B \to C$ to $A \to [B,C]$ is represented by simply bending a string. Unfortunately, a closed symmetric monoidal category cannot even be embedded fully-faithfully into a compact one unless it is traced; and while string diagram calculi for closed monoidal categories have been proposed, they are more complicated, e.g. with "clasps" and "bubbles". In this talk we obtain a string diagram calculus for closed symmetric monoidal categories that looks almost like the compact case, by fully embedding any such category in a star-autonomous one (via a functor that preserves the closed structure) and using the known string diagram calculus for star-autonomous categories. No knowledge of star-autonomous categories will be assumed.

category theory

Audience: researchers in the topic

ACT@UCR

Series comments: We will have discussions on the new Category Theory Zulip: categorytheory.zulipchat.com

See more information at this blog post.