A **magma** is a set $M$ equipped with a binary operation $* : M \times M \to M$. In abstract algebra we typically begin by studying a special type of magma: groups. Groups satisfy certain additional axioms that "symmetries of things" should satisfy. This is made precise in the sense that for any object $A$ in a category $C$, the invertible morphisms $A \to A$ have a group structure again given by composition. An alternate definition of "group," then, is "one-object category with invertible morphisms," and then the additional axioms satisfied by groups follow from the axioms of a category (which, for now, we will trust as meaningful). Groups therefore come equipped with a natural notion of representation: a representation of a group $G$ (in the loose sense) is just a functor out of $G$. Typical choices of target category include $\text{Set}$ and $\text{Hilb}$.

It seems to me, however, that magmas (and their cousins, such as non-associative algebras) don't naturally admit the same interpretation; when you throw away associativity, you lose the connection to composition of functions. One can think about the above examples as follows: there is a category of groups, and to study the group $G$ we like to study the functor $\text{Hom}(G, -)$, and to study this functor we like to plug in either the groups $S_n$ or the groups $GL_n(\mathbb{C})$, etc. on the right, as these are "natural" to look at. But in the category of magmas I don't have a clue what the "natural" examples are.

**Question 1:** Do magmas and related objects like non-associative algebras have a "natural" notion of "representation"?

It's not entirely clear to me what "natural" should mean. One property I might like such a notion to have is an analogue of Cayley's theorem.

For certain special classes of non-associative object there is sometimes a notion of "natural": for example, among not-necessarily-associative algebras we may single out Lie algebras, and those have a "natural" notion of representation because we want the map from Lie groups to Lie algebras to be functorial. But this is a very special consideration; I don't know what it is possible to say in general.

(If you can think of better tags, feel free to retag.)

**Edit:** Here is maybe a more focused version of the question.

**Question 2:** Does there exist a "nice" sequence $M_n$ of finite magmas such that any finite magma $M$ is determined by the sequence $\text{Hom}(M, M_n)$? (In particular, $M_n$ shouldn't be an enumeration of all finite magmas!) One definition of "nice" might be that there exist compatible morphisms $M_n \times M_m \to M_{n+m}$, but it's not clear to me that this is necessarily desirable.

**Edit:** Here is maybe another more focused version of the question.

**Question 3:** Can the category of magmas be realized as a category of small categories in a way which generalizes the usual realization of the category of groups as a category of small categories?

**Edit:** Tom Church brings up a good point in the comments that I didn't address directly. The motivations I gave above for the "natural" notion of representation of a group or a Lie algebra are in some sense external to their equational description and really come from what we would like groups and Lie algebras to do for us. So I guess part of what I'm asking for is whether there is a sensible external motivation for studying arbitrary magmas, and whether that motivation leads us to a good definition of representation.

**Edit:** I guess I should also make this explicit. There are two completely opposite types of answers that I'd accept as a good answer to this question:

One that gives an "external" motivation to the study of arbitrary magmas (similar to how dynamical systems motivate the study of arbitrary unary operations $M \to M$) which suggests a natural notion of representation, as above. This notion might not look anything like the usual notion of either a group action or a linear representation, and it might not answer Question 3.

One that is "self-contained" in some sense. Ideally this would consist of an answer to Question 3. I am imagining some variant of the following construction: to each magma $M$ we associate a category whose objects are the non-negative integers where $\text{Hom}(m, n)$ consists of binary trees with $n$ roots (distinguished left-right order) and $m$ "empty" leaves (same), with the remaining leaves of the tree labeled by elements of $M$. Composition is given by sticking roots into empty leaves. I think this is actually a 2-category with 2-morphisms given by collapsing pairs of elements of $M$ with the same parent into their product. An ideal answer would explain why this construction, or some variant of it, or some other construction entirely, is natural from some higher-categorical perspective and then someone would write about it on the nLab!

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