  Let $G$ be a compact connected Lie group and let $P$ be a principal
$G$-bundle over $K$. The gauge group of $P$ is the topological group of
automorphisms of $P$. For fixed $G$ and $K$, consider all principal $G$-bundles
$P$ over $K$. It is proved by Crabb--Sutherland and the second author that the
number of $A_n$-types of the gauge groups of $P$ is finite if $n<\infty$ and
$K$ is a finite complex. We show that the number of $A_\infty$-types of the
gauge groups of $P$ is infinite if $K$ is a sphere and there are infinitely
many $P$.
