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We study the mod-$\ell$ homotopy type of classifying spaces forcommutativity, $B(\mathbb{Z}, G)$, at a prime $\ell$. We show that themod-$\ell$ homology of $B(\mathbb{Z}, G)$ depends on the mod-$\ell$ homotopytype of $BG$ when $G$ is a compact connected Lie group, in the sense that amod-$\ell$ homology isomorphism $BG \to BH$ for such groups induces amod-$\ell$ homology isomorphism $B(\mathbb{Z}, G) \to B(\mathbb{Z}, H)$. Inorder to prove this result, we study a presentation of $B(\mathbb{Z}, G)$ as ahomotopy colimit over a topological poset of closed abelian subgroups,expanding on an idea of Adem and G\'omez. We also study the relationshipbetween the mod-$\ell$ type of a Lie group $G(\mathbb{C})$ and the locallyfinite group $G(\bar{\mathbb{F}}_p)$ where $G$ is a Chevalley group. We seethat the na\"ive analogue for $B(\mathbb{Z}, G)$ of the celebratedFriedlander--Mislin result cannot hold, but we show that it does hold aftertaking the homotopy quotient of a $G$ action on $B(\mathbb{Z}, G)$.

On the mod-$\ell$ homology of the classifying space for commutativity