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The classifying space for commutativity, denoted by $B_\text{com} G$, of a Lie group $G$ is assembled from commuting tuples in $G$ as a subspace of the usual classifying space $BG$. The resulting space classifies principal $G$-bundles whose transition functions generate an abelian subgroup of $G$ whenever they are simultaneously defined. The relationship between the homotopy type of $G$ and the space $B_\text{com} G$ is much more interesting, and non-trivial compared to the case of $BG$. In this talk, I will present a work, joint with Ben Williams, where we study the mod-$\ell$ homotopy type of $B_\text{com} G$ at a prime $\ell$. The techniques involve a homotopy colimit decomposition over a topological category generalizing the construction of Adem-Gomez and application of results on mapping spaces between classifying spaces of compact Lie groups due to Dwyer-Wilkerson. We show that for a connected compact Lie group the mod-$\ell$ homotopy type of $B_\text{com}G$ depends on the mod-$\ell$ homotopy type of $BG$. More precisely, a mod-$\ell$ homology isomorphism between the classifying spaces of two such groups induces a mod-$\ell$ homology isomorphism between the corresponding classifying spaces for commutativity. Moreover, a Mislin-Friedlander type equivalence, replacing the Lie group by the discrete group obtained from the associated Chevalley group, fails in general, but can be restored after passing to the homotopy orbits by an action of $G$.