Can anybody give me an example of a "naturally-occurring" algebraic category $C$ in which:


1. $C$ has two non-isomorphic objects $A$ and $B$ which are bi-embeddable via monic maps; but
2. $C$ does NOT have any infinite collection $A\_{1}$, $A\_{2}$, ... of objects which are pairwise bi-embeddable (via monic maps) and pairwise nonisomorphic?


Alternatively: can anybody give a reason why, under some reasonable hypothesis about the category $C$, property 1 should imply that 2 fails ("there is an infinite collection of pairwise bi-embeddable, pairwise nonisomorphic objects")?



Bi-embeddability vs. isomorphism
I don't know if this counts as a "naturally-occuring" algebraic category but as I scanned through my internal list of categories of modules over a ring, I spotted that the following has (1) and (2).


Take $G$ to be the "$p$-adic dihedral group" that is a semidirect product of the p-adic integers and a cyclic group of order $2$ where the latter acts by $-1$ on the former. Then form the completed group algebra (Iwasawa algebra) with coefficients in the field with $p$ elements, $R=F\_{p}\left[\left[G\right]\right]$.


Now there are precisely two isomorphism classes of indecomposable projective modules [$P\_{1}$] and [$P\_{2}$] and $P\_{1}$ and $P\_{2}$ are bi-embeddable via monic maps (see section 9.6 of <http://www.dpmms.cam.ac.uk/~sjw47/char.pdf> for the proof of both these claims). 


It follows that if we take the category of all finitely generated projective modules over $R$ then it consists of objects that are direct sums of m copies of $P\_{1}$ and $n$ copies of $P\_{2}$, say. Two such objects will be bi-embeddable if and only if they have the same value of $m+n$ thus whilst there are arbitrarily large finite collections of pairwise bi-embeddable but pairwise nonisomorphic objects in this category there are no infinite ones.


I am sure there will be many more examples along these lines, I just happen to spend a lot of time thinking about Iwasawa algebras.


