Do non-associative objects have a natural notion of representation?

Question

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!

Answer 1

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).

Answer 2

Let me concentrate on your first question (frankly speaking, the way you formulate your second question slightly lacks motivation).

The case where there is a reasonable suggestion, assumes that you work with some type of nonassociative algebras over a field, and all identities follow from the multilinear ones. In other words, the category of algebras you are studying is the category of algebras over some operad $O$. In this case, there is a nice way to describe a module over such an algebra. For an algebra $A$, a module structure on $V$ is given by a collection of operations defined by all possible operations from $O$, where you are allowed to plug an element from $V$ into one slot of operations, and plug elements of $A$ into other slots. To write down the module axioms, take the defining identities of $O$, and form new identities, marking one element there in all possible ways; now treat the unmarked elements as elements of $A$, and marked elements as belonging to $V$.

For example, for associative algebras the original identity is $(ab)c=a(bc)$, which leads to the following definition. A module structure is defined by two operations, $a,v\mapsto av$ and $a,v\mapsto va$ satisfying the identities $(ab)v=a(bv)$, $(av)b=a(vb)$, $(va)b=v(ab)$. This means that in the case of associative algebras we defined bimodules. Also, for Lie algebras we get the module structure which, as it is immediate to check, coincides with the usual module/representation structure. In general, this construction provides a reasonable "enveloping algebra" for your nonassociative algebra. Thus, one way to approach your question is to study representations of the enveloping algebra, and sometimes it's the best you can get.

Answer 3

Lets begin by observing that to define a "non-associative action", we actually don't need a magma; a mere set will do.

Definition 0. Whenever $S$ is a set, a representation of $S$, also known as an $S$-unary algebra, consists of a set, call it $X$, together with a function $S \times X \rightarrow X,$ denoted $a,x \mapsto ax$.

A convention: the notation $abx$ means $a(bx)$, the notation $abcx$ means $a(b(cx))$, etc.

For example:

• a $0$-unary algebra is basically just a set.
• a $1$-unary algebra is basically a set $X$ equipped with a function $f:X \rightarrow X$. In the literature, these are called monounary algebras.
• a $2$-unary algebra is basically a set $X$ equipped with a pair of functions $f,g:X \rightarrow X$. In the literature, these are called biunary algebras.
• etc.

It should be clear that $S$-unary algebras are ubiquitous; for example, any time we have a set $X$ together with a self-map $f:X \rightarrow X$, the pair $(X,f)$ is a $1$-unary algebra. A famous example: the Collatz Conjecture is a statement about the $1$-unary algebra $(\mathbb{N},\xi)$, where $\xi$ is the Collatz function.

Definition 1. Whenever $S$ is a set, write $S_*$ for the $S$-unary algebra freely generated by $\{1\}$. The elements of $S_*$ are called words in $S$.

We're all familiar with the "symmetrical" viewpoint on $S_*$ in which we view it as a monoid denoted $S^*$. In particular, its the free monoid on $S$. In fact, the "asymmetrical" viewpoint in which its viewed as an $S$-unary algebra is pretty important, too; this basically amount to taking a treelike viewpoint. For example, here's a depiction of $\{f,g\}_*$:

Lets now collect together the main facts about $S_*$:

Theorem 0. Generalized Peano Postulates. Let $S$ denote a set. Then:

1. For all $a \in S$ and all $x,y \in S_*$, we have $ax = ay \rightarrow x=y.$
2. For all $a \in S$ and all $x \in S_*$, we have $ax = 1 \rightarrow \bot$
3. For all $a,b \in S$ and all $x \in S_*$, we have $ax = by \rightarrow a=b.$
4. (Axiom of Induction.) The only $S$-unary subalgebra of $S_*$ that contains $1$ is $S_*$ itself.

Furthermore, the above facts characterize $S_*$ among all $S$-unary algebras.

Okay. Define that $\mathbb{N}$ is the $\{s\}$-unary algebra freely generated by $\{0\}$. So basically, $\mathbb{N}$ is just $\{s\}_*$ with a slight change in notation. $$\mathbb{N} = \{0,s0,ss0,sss0,\ldots\}$$

By noting that Condition 3 trivializes when $S =\{s\}$ has only one element, we essentially rediscover Peano's original axioms for $\mathbb{N}$.

Theorem 1. Peano Postulates for \mathbb{N}.

1. For all $x,y \in \mathbb{N}$, we have $sx = sy \rightarrow x=y.$
2. For all $x \in \mathbb{N}$, we have $sx = 0 \rightarrow \bot$
3. (Axiom of Induction.) The only monounary subalgebra of $\mathbb{N}$ that contains $0$ is $\mathbb{N}$ itself.

Furthermore, the above facts characterize $\mathbb{N}$ among all monounary algebras.

So I'd say non-associative actions are pretty important!

On the other hand, they're (in some sense) unnecessary:

Theorem 2. An $S$-unary algebra is the same thing as an $S^*$-set, i.e. a set $X$ equipped with an (associative) action of the monoid $S^*$.

One application is that these ideas give a really slick definition of the term "planar tree." First, note that if $M$ is a monoid and $X$ is an $M$-set, then the closed subsets of $X$ always form an Alexandroff topology (as opposed to a mere closure system). Ergo, since an $S$-unary algebra is just an $S^*$-set, hence its closed subsets automatically form an Alexandroff topology. This means, in particular, that $S_*$ is automatically an Alexandroff space.

Definition 3. An $S$-planar tree is an open subset of $S_*$.

Of course, Alexandroff spaces are the same thing as preorders; under this translation, "open subset" means the same thing as "lowerset". So we could equally well define that an $S$-planar tree is a lowerset of $S_*$.

Answer 4

Taking the cue from Lie algebras you could try considering something like the enveloping algebra. In the Lie case a representation of $g$ is just a usual representation of $U(g)$ so maybe here you can make the same construction.

Answer 5

There is an instant method to produce "modules" over about anything: given an object $C$ in any category $\mathcal C$ with pullbacks, one defines $$ C-\textrm{mod}:=\textrm{Ab}(\mathcal C/C) $$ (the category of internal abelian groups in the slice over $C$).

Of course as every quickie, it is not always what you want. But at least it is always there :)

Answer 6

There is a thing occurring in the overkill context of Lurie's Higher Algebra which seems related to this question.

Namely, in Ch 3.3, Lurie introduces a fairly general notion of "module for an algebra of an operad". More precisely, we fix the following data:

• A unital operad $O$ (here "unital" means that for every color $C$ of the operad, there is a unique nullary $C$-valued operation). For example, $O$ could be the operad for unital magmas. I don't know how essential the unitality hypothesis is.

• A "fibration of generalized operads" $C \to O$. I believe we're supposed to think of this as a category $C$ with "$O$-monoidal structure", or an $O$-algebra object in $Cat$. For example, if $O$ is the operad for unital magmas, then any monoidal category acquires such a structure by restriction along the map from $O$ to the associative operad.

• An $O$-algebra $A$ in $C$. I'm not 100% sure I'm unwinding the definitions correctly (Lurie has several subtly different concepts he denotes by "$Alg$" with various decorations, and I'm not straight on what's what), but I believe that when $O$ is the operad for unital magmas and $C$ is a monoidal category, then $A$ is just a unital magma object in $C$.

Given this data, Lurie defines a category of "$O$-module objects over $A$", denoted

$$Mod_A^O(C)$$

I don't have a great grasp on this -- part of the issue is that in fact, Lurie doesn't isolate this construction, instead jumping straight to the construction of this category plus a full $O$-monoidal structure (and even packaging it all up in a bigger construction which allows $A$ to vary), which requires his technical "coherence" hypothesis. (Caveat: Lurie doesn't prove that $Mod_A^O(C)$ is a category in general -- without the "coherence" hypothesis, it might just be some simplicial set. But I'll proceed under the assumption that even though coherence is needed to get an $O$-monoidal structure on $Mod_A^O(C)$, it's probably the case that $Mod_A^O(C)$ is at least a category in greater generality.) However, Lurie does say that

• When $O$ is the commutative operad, we recover the usual notion of module over a commutative algebra $A$;

• When $O$ is the associative operad, we recover the notion of an $(A,A)$-bimodule.

That second point really throws me, and bears repeating -- for Lurie, a "module over an associative algebra object" is neither a left nor a right module, but rather a bimodule (I hasten to add that he does develop the theory of left and right modules separately, without reference to this more general context). For example, Lurie's notion does not recover the notion of a group acting on a set. Lurie explains that the motivation for doing it this way (the idea of which I think he attributes to John Francis) is that he wants to have an $O$-monoidal structure on the category of $A$-modules -- and of course left $A$-modules don't generally have a monoidal structure, but $(A,A)$-bimodules do. I have no idea whether the operad for unital magmas satisfies the technical "coherence" condition guaranteeing that $Mod_A^O(C)$ does in fact have an $O$-monoidal structure; if it doesn't, then the main motivation for introducing this notion evaporates and perhaps it ends up not being useful. But it's still there.

If I have the chance to unwind what this construction yields in the case of the operad for magmas and ordinary 1-categories, I will edit to add more. Or perhaps somebody more familiar with this stuff can chime in.

Answer 7

For monoids (which are associative) the Krohn–Rhodes theorem gives a powerful decomposition result: every finite monoid is a quotient of a submonoid of an alternating wreath product of finite groups and monoids. Google suggests that Joel Wanderwerf may have generalized this theorem to arbitrary algebras in a 1996 article for the Semigroup Forum journal, but I don't have access to this Springer journal so I can't say for sure.