Notes on Presentations of Strict Tensor Categories.

(October 2021)

First we’d like to prove the fact (well-known to planar algebra people) that when presenting a semisimple tensor category by generators and relations, we know we have “enough relations” as soon as we can evaluate any closed diagram. For simplicity we restrict attention to spherical categories. In this note, all categories are strict monoidal, strictly pivotal, and $k$ -linear.

Recall that a ( $k$ -linear) multitensor category is a category satisfying the axioms of a tensor category, except $\operatorname{\textrm{End}}({\bf 1})$ may not be 1-dimemnsional.

The following is just [BMPS2012], Prop. 3.5 translated to tensor category language.

Lemma 0.1.

Let $\operatorname{\mathcal{C}}$ be a spherical tensor category. Then the negligible morphisms form the unique maximal tensor ideal of $\operatorname{\mathcal{C}}$ .

Proof.

It suffices to show that any proper pivotal tensor ideal $\mathcal{I}$ consists only of negligible morphisms. Suppose $f\in\mathcal{I}$ is not negligible. Then there exists $g$ such that $\operatorname{\textrm{Tr}}(fg)\neq 0$ . Since $\operatorname{\mathcal{C}}$ is a tensor category, $\operatorname{\textrm{End}}_{\operatorname{\mathcal{C}}}({\bf 1})$ is one-dimensional, so $\operatorname{\textrm{id}}_{\bf{1}}\in\mathcal{I}$ (since $\operatorname{\textrm{Tr}}(fg)\in\mathcal{I}$ ). This implies $\mathcal{I}$ is not a proper tensor ideal. ∎

We say that a spherical tensor category $\operatorname{\mathcal{C}}$ is generated by a set of morphisms $F$ if the smallest tensor ideal containing $F$ is equal to $\operatorname{\mathcal{C}}$ .

Now also assume that $\operatorname{\mathcal{C}}$ is semisimple, tensor generated by a symmetrically self-dual object $X$ , and generated by a set of morphisms $F$ . Let $\operatorname{\mathcal{C}}(F)$ denote the free spherical tensor category generated by a single symmetrically self-dual object and the morphisms $F$ . Concretely, the objects of $\operatorname{\mathcal{C}}(F)$ are $0,1,2,\dots$ and the morphisms are unoriented planar tangles with coupons labeled by $F$ , modulo isotopy.

Suppose $R$ is a set of relations satisfied in $\operatorname{\mathcal{C}}$ and let $\operatorname{\mathcal{R}}$ be the spherical tensor ideal of $\operatorname{\mathcal{C}}(F)$ generated by $R$ . Then $\operatorname{\mathcal{C}}^{\prime}=\operatorname{\mathcal{C}}(F)/\operatorname{\mathcal{R}}$ is a pivotal multitensor category and we have a full pivotal tensor functor

\operatorname{\mathcal{C}}^{\prime}\xrightarrow{\Phi}\operatorname{\mathcal{C}}.

Since $\operatorname{\mathcal{C}}$ is semisimple, the functor factors through the semisimplification of $C^{\prime}$ so we get a pivotal tensor functor

\operatorname{\mathcal{C}}^{\prime}/\mathcal{N}(\operatorname{\mathcal{C}}^{\prime})\xrightarrow{\overline{\Phi}}\operatorname{\mathcal{C}}.

Proposition 0.2.

The functor $\overline{\Phi}$ is an equivalence if and only if the endomorphism algebra of the unit $\bf{1}$ in $\operatorname{\mathcal{C}}^{\prime}$ is one dimensional, i.e $\operatorname{\mathcal{C}}^{\prime}$ is a tensor category.

Proof.

The “only if” direction is obvious (if an endomorphsim of ${\bf 1}$ in $\operatorname{\mathcal{C}}^{\prime}$ is negligible, it is already $0$ ). For the other direction, suppose $\operatorname{\textrm{End}}_{\operatorname{\mathcal{C}}^{\prime}}({\bf 1})=k$ . Let $\mathcal{K}$ denote the kernel of $\Phi$ , which is a pivotal tensor ideal of $\operatorname{\mathcal{C}}^{\prime}$ . Clearly $\mathcal{K}$ contains $\mathcal{N}(\operatorname{\mathcal{C}}^{\prime})$ . On the other hand the lemma states $\mathcal{K}$ is contained in $\mathcal{N}(\operatorname{\mathcal{C}}^{\prime})$ , which completes the proof. ∎

Definition 0.3.

A semisimple presentation of a semisimple spherical tensor category $\operatorname{\mathcal{C}}$ is a collection of morphisms $F$ and relations $R$ such that

\operatorname{\mathcal{C}}\cong\big{(}\operatorname{\mathcal{C}}(F)/\mathcal{R}\big{)}/\textrm{negligibles}

where $\mathcal{R}$ is the pivotal tensor ideal of $\operatorname{\mathcal{C}}(F)$ generated by $R$ .

Remark 0.4.

A category of type $A$ has no generators and the single relation

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