Let $V$ and $W$ be classes of algebraic structures, and suppose we have some canonical way of constructing objects of $W$ from objects of $V$. Let's call this construction $C$, so that for all $A\in V$, $C(A)\in W$.

Then by a *representation theorem*, I mean a way of associating to each $B\in W$ an object $U(B)\in V$ such that $B$ embeds into $C(U(B))$.

Examples:

$V = \operatorname{Sets}$, $W = \operatorname{Groups}$. For $X$ a set, let $C(X) = S_X$, the symmetric group on $X$. Then for any group $G$, we can take $U(G)$ to be the underlying set of $G$, and $G$ embeds in $S_{U(G)}$.

$V = \operatorname{Abelian Groups}$, $W = \operatorname{Rings}$. For $V$ a set, let $C(V) = \text{End}(V)$, the endomorphism ring of $V$. Then for any ring $R$, we can take $U(R)$ to be the additive group of $R$, and $R$ embeds in $\text{End}(U(R))$.

$V = \operatorname{Sets}$, $W = \operatorname{Boolean Algebras}$. For $X$ a set, let $C(X) = \mathcal{P}(X)$, the power set of $X$. Then for any Boolean algebra $B$, we can take $U(B)$ to be the set of ultrafilters on $B$, and $B$ embeds in $\mathcal{P}(U(B))$.

This setup does not seem to fit nicely into a categorical framework: In the first two examples above, the construction $C$ is not a functor, at least from the usual categories of sets and abelian groups. $C$ is a functor if we only force it to respect isomorphisms (by considering $V$ as a category with invertible morphisms as arrows), but we care about a noninvertible morphism in $W$ (the embedding $B\rightarrow C(U(B))$, so we won't be able to make $U$ into a functor to $V$. Finally, even if we could make $C$ and $U$ functors, it doesn't seem that the embeddings $B\rightarrow C(U(B))$ would cohere into a natural transformation $\operatorname{id}_W\rightarrow C\circ U$.

Of course, the third example is much better behaved, categorically speaking. I am aware of the family of examples related to this one, as described, for example, in Johnstone's book Stone Spaces.

But do we know anything about the general algebraic setup? In particular are there general conditions under which representation theorems must exist? If we know that a representation theorem exists, is there any sense in which we can compute what it must be?

Localisation, spectra and sheaf representationseems related to what you want. $\endgroup$