  We study the mod-$\ell$ homotopy type of classifying spaces for
commutativity, $B(\mathbb{Z}, G)$, at a prime $\ell$. We show that the
mod-$\ell$ homology of $B(\mathbb{Z}, G)$ depends on the mod-$\ell$ homotopy
type of $BG$ when $G$ is a compact connected Lie group, in the sense that a
mod-$\ell$ homology isomorphism $BG \to BH$ for such groups induces a
mod-$\ell$ homology isomorphism $B(\mathbb{Z}, G) \to B(\mathbb{Z}, H)$. In
order to prove this result, we study a presentation of $B(\mathbb{Z}, G)$ as a
homotopy colimit over a topological poset of closed abelian subgroups,
expanding on an idea of Adem and G\'omez. We also study the relationship
between the mod-$\ell$ type of a Lie group $G(\mathbb{C})$ and the locally
finite group $G(\bar{\mathbb{F}}_p)$ where $G$ is a Chevalley group. We see
that the na\"ive analogue for $B(\mathbb{Z}, G)$ of the celebrated
Friedlander--Mislin result cannot hold, but we show that it does hold after
taking the homotopy quotient of a $G$ action on $B(\mathbb{Z}, G)$.
