I wonder whether there is any categorical property that picks out exactly the finite algebras of some algebraic theory without referring to the underlying set.
What I mean is: I don't want to use the the structure of a functor $\mathcal{A} \to \mathsf{Set}$ from my algebraic category. This is (obviously) not so interesting, but more importantly I want the property to make sense in the absence of any such structure too (obviously I could have a different interpretation then, I'm open to anything interesting).
No cheating allowed, of course! You can't refer to the free object on a singleton, because that doesn't make any sense in a general category (e.g. you can't use $\mathcal{A}(\mathbb Z, G)$ to count the elements in a group $G$ as $\mathbb{Z}$ is determined by some property involving the functor $\mathcal{A} \to \mathsf{Set}$; even though you can characterise $\mathbb{Z}$ by different means, this probably doesn't make much sense for other kinds of algebras besides groups).
Here are some necessary conditions for an algebra $A$ to be finite (but I don't see how any of them are sufficient):
- the poset of subobjects $\operatorname{Sub} A$ is finite
- $A$ is finitely generated ($\mathcal{A}(X,\_)$ preserves filtered colimits of monos)
- the monoid of endomorphisms $\operatorname{End} A$ is finite
- $A$ is Dedekind-finite, i.e.: every mono (injection) $A\to A$ is an isomorphism
I'd also be interested in a property for some (not to small) subclass of all theories or a statement, that what I'm trying to do is infeasible.
As alluded to in the question, it is possible to characterize the finitely generated algebras $F$ (e.g., as those for which $\mathcal{A}(F,-)$ preserves filtered colimits of monomorphisms).
If $F$ is finitely generated, then a homomorphism $F\to A$ is determined by the images of a finite set of generators, so if $A$ is finite then $\mathcal{A}(F,A)$ is finite.
Conversely, if $F$ is the free algebra on one generator, then $\mathcal{A}(F,A)$ has the same cardinality as $A$.
So the finite algebras $A$ are characterized as those for which $\mathcal{A}(F,A)$ is finite for every finitely generated algebra $F$.