Ring and tensor categories

We now combine the linear, abelian, monoidal, and rigid structures. Following Etingof et al. (2015), Definition 4.1.1, a multitensor category over an algebraically closed field $k$ is a locally finite $k$-linear abelian rigid monoidal category whose tensor product is bilinear on morphisms. It is a tensor category when

\[\label{eq:tensor-category-unit} \operatorname{End}_{\mathcal C}(\mathbb 1)\cong k.\]

Because tensoring with an object in a rigid category has both adjoints, it is exact. Dropping rigidity gives the more general terminology of Etingof et al. (2015), Definition 4.2.3: a multiring category is a locally finite $k$-linear abelian monoidal category whose tensor product is bilinear and exact in each variable, and it is a ring category when equation \eqref{eq:tensor-category-unit} holds. Thus every multitensor category is a multiring category, and every tensor category is a ring category.

A ring category is a category with specified linear and monoidal properties; it should not be confused with the Grothendieck ring constructed from such a category later in this chapter.

Over a general coefficient field, TensorCategories.jl also supports the non-split convention of Mäurer and Thiel (2024), §2.1, in which a simple tensor unit need not have endomorphism algebra $k$. Algorithms which require the split condition must impose it separately.

The interface

The corresponding package predicates are is_multiring, is_ring, is_multitensor, and is_tensor. Stronger structural declarations imply the weaker ones through generic fallbacks. In particular, every multiring category reports is_locally_finite(C) == true. The predicates report what a backend has declared or established; they do not reconstruct the axioms from the available methods.

In the non-split convention, a category with a simple but non-scalar tensor unit can be reported as is_tensor(C) == true and is_ring(C) == true, even though equation \eqref{eq:tensor-category-unit} fails. Algorithms that require the split EGNO condition must additionally test is_split_semisimple(C). This broader behavior is part of the current package interface.

Example: Group representations

Both $\operatorname{Vec}_k$ and $\operatorname{Rep}_k(G)$ are tensor categories in the sense above, whether or not $\operatorname{Rep}_k(G)$ is semisimple. The representation category therefore illustrates why tensor category and fusion category are not synonyms.

using TensorCategories, Oscar
G = cyclic_group(3)
C = representation_category(GF(3), G)
@assert is_tensor(C) && is_ring(C)
@assert !is_semisimple(C)
(is_tensor(C), is_semisimple(C))

(true, false)

Continue with fusion and multifusion categories.