  In this paper, we describe the total space $E_{com} U(3)$ of the principal
$U(3)$-bundle associated with the classifying space for commutativity $B_{com}
U(3)$ as a homotopy colimit of a diagram of spaces and offer a computation of
the mod $2$ and mod $3$ cohomologies of $E_{com} U(3)$ by utilizing the
spectral sequence associated with a homotopy colimit. We investigate the
cohomology of different spaces in the homotopy colimit diagram. These spaces
are intriguing in their own right and contribute to the overall fascination of
the analysis. We also present the ring structure of the rational cohomology of
$E_{com} U(3)$.
