Rig categories concept which is equivalent to monoid concept in monoidal categories

Question

In monoidal categories, there is a notion of monoid. Is there an "equivalent" concept in rig categories (i.e., categories with two monoidal structures which are related like + and * in a rig)?

Answer 1

The question is not really precise. And you certainly don't look for an equivalent concept. Perhaps you are looking for the notion of a rig object internal to a rig category? This is an object $x$ equipped with morphisms $z : 0 \to x$ (zero) , $a : x \oplus x \to x$ (addition), $u : 1 \to x$ (unit) and $m : x \otimes x \to x$ (multiplication) such that the evident diagrams commute.

In more detail, we require that $(x,a,z)$ is a commutative monoid object, $(x,m,u)$ is a monoid object, and that $m$ distributes over $a$ in the following sense: $$x \otimes (x \oplus x) \xrightarrow{x \otimes a} x \otimes x \xrightarrow{m} x$$ agrees under the identification $x \otimes (x \oplus x) \cong (x \otimes x) \oplus (x \otimes x)$ with $$(x \otimes x) \oplus (x \otimes x) \xrightarrow{m \otimes m} x \oplus x \xrightarrow{a} x.$$ Likewise, we demand that two certain morphisms $(x \oplus x) \otimes x \rightrightarrows x$ are equal.

Finally, we require that $z$ is absorbing for $m$, i.e. that $x \otimes 0 \xrightarrow{x \otimes z} x \otimes x \xrightarrow{m} x$ equals $x \otimes 0 \cong 0 \xrightarrow{z} x$. Likewise for $0 \otimes x \to x$.