Affinization of monoidal categories

TL;DR

This paper introduces the concept of affinization for monoidal categories, providing a unifying framework that encompasses various constructions related to diagrams, algebras, and invariants in category theory.

Contribution

It formalizes the affinization process for monoidal categories, offering an alternative characterization and connecting it to the horizontal trace, with numerous examples from algebra and topology.

Findings

Affinization generalizes many existing constructions in monoidal categories.

When rigid, affinization is isomorphic to the horizontal trace.

The paper provides multiple examples from Hecke algebras, braids, and knot invariants.

Abstract

We define the affinization of an arbitrary monoidal category $\mathcal{C}$, corresponding to the category of $\mathcal{C}$-diagrams on the cylinder. We also give an alternative characterization in terms of adjoining dot generators to $\mathcal{C}$. The affinization formalizes and unifies many constructions appearing in the literature. In particular, we describe a large number of examples coming from Hecke-type algebras, braids, tangles, and knot invariants. When $\mathcal{C}$ is rigid, its affinization is isomorphic to its horizontal trace, although the two definitions look quite different. In general, the affinization and the horizontal trace are not isomorphic.