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:
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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.
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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!
Best Answer
Since magmas in general don't have much structure, we can't reasonably expect a representation to preserve much structure. We can therefore define a left representation of a magma $M$ to be a set $V$ equipped with a map $M \times V \to V$. We do the analogous thing for general nonassociative algebras. Serge Lang liked to describe a notion of left regular representation of an algebra $A$, which is just the linear map $A \to \operatorname{End} (A)$ that takes an element to the linear transformation it induces by left multiplication. As expected, this map is a homomorphism if and only if the algebra is associative.
There are special cases of nonassociative algebras that admit good notions of representation, and in the cases I know, these arise from operads that have "good relationships" with the associative operad. The standard example is the natural map from the Lie operad to the Associative operad that yields the forgetful functor from associative algebras to Lie algebras. This functor admits the universal enveloping algebra functor as left adjoint. There is a formalism of enveloping operads, which generalizes this case. The upshot is that these special cases have a lot more structure than a simple composition law, so we can demand more from a representation (namely, that it respect the operad structure as manifested through the universal enveloping algebra).