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Abstract In our previous article (http://arxiv.org/abs/1607.06041), we established an equivalence between pointed pivotal module tensor categories and anchored planar algebras. This article introduces the notion of unitarity for both module tensor categories and anchored planar algebras, and establishes the unitary analog of the above equivalence. Our constructions use Baez’s 2-Hilbert spaces (i.e., semisimple$$\textrm{C}^*$$

C^{*}

-categories equipped with unitary traces), the unitary Yoneda embedding, and the notion of unitary adjunction for dagger functors between 2-Hilbert spaces.

Planar Algebras in Braided Tensor Categories

https://doi.org/10.1090/memo/1392

Henriques, André; Penneys, David; Tener, James (February 2023, Memoirs of the American Mathematical Society)

We generalize Jones’ planar algebras by internalising the notion to a pivotal braided tensor category

C

. To formulate the notion, the planar tangles are now equipped with additional ‘anchor lines’ which connect the inner circles to the outer circle. We call the resulting notion ananchored planar algebra. If we restrict to the case when

C

is the category of vector spaces, then we recover the usual notion of a planar algebra. Building on our previous work on categorified traces, we prove that there is an equivalence of categories between anchored planar algebras in

C

and pivotal module tensor categories over

C

equipped with a chosen self-dual generator. Even in the case of usual planar algebras, the precise formulation of this theorem, as an equivalence of categories, has not appeared in the literature. Using our theorem, we describe many examples of anchored planar algebras.

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