Distributivity in linear monoidal categories

Question

Let $\mathcal{C}$ be a linear monoidal category, that is a monoidal category (with tensor product $\otimes$) enriched over $\mathbf{Vect}$.

Now as far as I can tell the axioms for a linear category and monoidal category respectively do not intersect, that is to say in particular that nowhere do we require the tensor product of morphisms to be bilinear with respect to addition in the hom-spaces.

But we could try to impose such a condition: suppose $\{f_i: a \rightarrow a^\prime\}_{i=1}^n, \{g_j: b \rightarrow b^\prime\}_{j=1}^m$ are finite families of morphisms, then quotient by the appropriate ideal to get the identification $$ \sum_if_i \otimes \sum_j g_j = \sum_i \sum_j (f_i \otimes g_j) \tag{$\ast$}$$ as morphisms from $a\otimes a^\prime$ to $b\otimes b^\prime$ for all objects $a,a^\prime,b,b^\prime$ in $\mathcal{C}$. (Just to be clear, here the summation is vector addition within the hom-space.)

Making such an identification plays well with composition of morphisms: we can check that for any morphisms $\sum_i f_i$, $\sum_k f^\prime_k$ from $a \rightarrow a^\prime$ and $\sum_j g_j$, $\sum_l g^\prime_l$ from $b \rightarrow b^\prime$, the relation $$ \left( \sum f_i \otimes \sum g_j \right) \circ \left( \sum f^\prime_k \otimes \sum g^\prime_l \right) = \left( \sum \sum f_i\otimes g_j \right) \circ \left( \sum \sum f^\prime_k\otimes g^\prime_l \right) $$ is coherent (by bilinearity of $\otimes$ with respect to $\circ$ and $\circ$ with respect to $+$).

I also expect (I haven't done the work) that other relations like associativity etc are also coherent.

Is this structure something that category theorists regularly work with? It seems to me that we could just as well have worked in $\mathcal{C}$ keeping the abstract tensor product of linear combinations separate from the linear combination of the tensor products, i.e. formally distinguish between the two sides of $(\ast)$. So I'm wondering: if we don't force said relation, does it arise naturally from the axioms? I already mentioned I don't believe it does, but I'm not entirely sure.

Also a search around the literature and nLab for terms like "linear/distributive monoidal/tensor category" doesn't turn up anything to do with this particular notion, I'd be grateful if anyone could tell me the right keywords to look up.

In fact I know that generic Temperley-Lieb is linear monoidal, but don't know if typical formulations impose $(\ast)$ or not. Any word on that would be appreciated.

Lastly, if anyone has any examples of linear monoidal categories that do or do not satisfy $(\ast)$ I'd be interested to hear about them. Thanks a lot!

Answer 1

It's standard to assume that a "linear monoidal category" means that the linear and monoidal structures are compatible. For example, see Def 0.1.2 of Catègories Tensorielles, Def 1.12.3 of Tensor Categories or Def 3.2.4 of Dualizable Tensor Categories. If I ran across a paper that didn't make this compatibility explicit, I'd assume that the authors were just being sloppy and meant to include it unless they were really explicit that they didn't want compatibility.

Answer 2

The term "linear monoidal category" doesn't mean "linear and monoidal category", but rather "monoidal (linear category)", i.e. a (weak) monoid in the monoidal bicategory of linear categories (over some fixed base ring $R$). The monoidal product of two linear categories $C,D$ has as objects pairs of objects of $C,D$ with hom-modules $\hom((a,b),(c,d)) = \hom(a,c) \otimes_R \hom(b,d)$. It follows that a linear monoidal category is a category which has a linear structure and a monoidal structure which are compatible in the sense you have described (i.e. both $a \otimes -$ and $- \otimes b$ are linear endofunctors). The point of my answer is: All this follows from the general definitions of enriched category theory, you don't have to reinvent this notion.

Since you have asked for an example where the compatibility is not satisfied: Take any $R$-algebra whose underlying set also carries another structure of a commutative monoid. Then this becomes a "linear and monoidal category" with one object and it is a linear monoidal category iff the two multiplications coincide (Eckmann-Hilton).