Commutative-monoid enrichment

A commutative-monoid enrichment gives each hom-set a commutative monoid structure, with zero and addition, so that composition distributes over addition in both variables and annihilates zero. This is called a semi-additive structure in some conventions, including Cambridge Part III paper 119 of 2017. Other conventions reserve semi-additive category for categories with this enrichment and finite biproducts. A one-object category arising from a semiring has the enrichment without generally having those finite products.

For a multiplicative monoid automorphism $\varphi :\mathbb{N}\to \mathbb{N}$ fixing $0$, define $a\oplus b={\varphi }^{-1}(\varphi (a)+\varphi (b))$. This transports a semiring structure to the fixed underlying multiplication, hence a commutative-monoid enrichment on its one-object category. Swapping prime $2$ with an odd prime $p$ gives $1\oplus 1=p$. Infinitely many such primes therefore give infinitely many distinct additions, although the transported enriched categories are isomorphic.