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In a recent investigation, I constructed a category $\mathcal{C}$ with the following property. For objects $X,Y \in \mathcal{C}$, the morphism set $\text{Mor}(X,Y)$ is a commutative monoid with addition denote by $+$. However, $\mathcal{C}$ is not CMon enriched, where CMon is the monoidal category of commutative monoids. The reason is that $+$ is not distributive w.r.t. the composition of morphisms.

My question is whether there is a name for such a category and is there any study for such a category?

I am interested in looking at functors into $\mathcal{C}$ such that the monoid structures of morphism sets are preserved. I really appreciate any answer.

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    $\begingroup$ I guess the specific example you came across is too specialised to describe here — but can you give some “toy example” of such a situation? $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented Dec 22, 2022 at 10:58
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    $\begingroup$ Roughly speaking (not very precise), an object $X$ is a measurable space. $\text{Mor}(X,Y)$ are like random variables which can be added. Compositions are defined in a more involved way but the key point is that they are not distributive w.r.t. the addition. To study relations with other categories with similar structures, I want to look at functors that preserve the monoid structure. I would like to know if there is already some theory on such categories. $\endgroup$
    – F J
    Commented Dec 22, 2022 at 11:04
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    $\begingroup$ Any category can be endowed with such a structure since any set can be made a commutative monoid. $\endgroup$
    – Fernando Muro
    Commented Dec 22, 2022 at 13:13
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    $\begingroup$ @FernandoMuro Or at least -- any nonempty set! $\endgroup$
    – Tim Campion
    Commented Dec 22, 2022 at 16:59
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    $\begingroup$ @FJ: Since you have a concrete category in mind, to provide a meaningful answer you really need to spell out the details of your category. It is not possible to gather what kind of structure you have just from the vague descriptions we have so far. $\endgroup$
    – Dmitri Pavlov
    Commented Dec 22, 2022 at 20:03

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