Residually small variety

Remarks

Equivalently, the subdirectly irreducible members of $\mathcal{V}$ form a set up to isomorphism. The variety is residually finite when $\kappa$ can be chosen with every subdirectly irreducible member finite, and it has finite residual bound when $\kappa$ can be chosen finite.

Example: the variety of abelian groups is residually small. Its subdirectly irreducible members are the groups $\Z/p^n\Z$ and $\Z(p^\infty)$ for primes $p$ and $n \ge 1$, which are all countable, so $\kappa = \aleph_1$ works.

Non-example: the variety of all groups is residually large. For an infinite set $X$, the group of even permutations of $X$ that move only finitely many points is simple and has $\abs{X}$ elements. A nontrivial simple algebra is subdirectly irreducible, so no cardinal $\kappa$ bounds the sizes.