Examples of semiring categories whose additive monoidal structure is not given by the coproduct

Question

One of the most common examples of semiring categories is given by distributive monoidal categories. Indeed, examples of the latter include the following:

• $(\mathsf{CMon},\oplus,\otimes_{\mathbb{N}},0,\mathbb{N})$;
• $(\mathsf{Ab},\oplus,\otimes_{\mathbb{Z}},0,\mathbb{Z})$;
• $(\mathsf{Mod}_R,\oplus,\otimes_{R},0,R)$;
• $(\mathsf{Sets},\coprod,\times,\emptyset,*)$;
• $(\mathsf{sSets},\coprod,\times,\emptyset_{\bullet},\Delta^0)$;
• $(\mathsf{Sets}_*,\vee,\wedge,*,2)$;
• $(\mathsf{Top}_*,\vee,\wedge,*,S^0)$.

On the other hand, examples of semiring categories which are not distributive monoidal categories (i.e. whose additive monoidal structure is not given by the coproduct) are harder to come by. The few ones I'm aware of are the following:

• $R_\mathsf{disc}$ for $R$ a semi/ring;
• The groupoid of finite sets and permutations $\mathbb{F}$;
• The $1$-truncation of the sphere spectrum $\tau_{\leq1}\mathbb{S}$, viewed as a semiring category.

Note that $\tau_{\leq1}\mathbb{S}$ is even an example of a ring category, meaning a semiring category $\mathcal{C}$ such that $\pi_0(\mathcal{C})$ is an abelian group. (That is, whose underlying additive monoidal structure is an "abelian $2$-group", also called a "gr-category", a "groupoidal category", or a "Picard groupoid".)

Question. What are some examples of semiring and ring categories which are not distributive monoidal categories?

In particular, are there any "large" such examples? For instance, do the categories $\mathsf{Sets}$, $\mathsf{CMon}$, $\mathsf{Ab}$, or $\mathsf{sSets}$ admit such 'exotic' semiring category structures?

Answer 1

A general recipe is to take a distributive monoidal category and a subcategory closed under direct sums and tensor products and containing all isomorphisms. Here the direct sum is not necessarily a coproduct anymore, it is just another monoidal structure. For example, this happens when there are morphisms $f : X \to Y$, $g : X' \to Y$ in the subcategory such that $(f;g) : X \oplus X' \to Y$ is not a morphism in the subcategory anymore, or simply when the unique morphism $0 \to X$ does not belong to the subcategory.

A typical example for this is the category of $R$-modules with the usual direct sum and tensor product with injective homomorphisms; the same for surjective homomorphisms. In the most extreme case, you could just take isomorphisms.