$\mathbb{CP}^1$ can be formed from $\mathbb{C}^\times = \text{GL}_1$ by gluing $\mathbb{C}$ by itself along $\mathbb{C}^\times$, a pushout of $1/z,z : \mathbb{C}^\times \rightarrow \mathbb{C}$.
I am hoping that someone can help me to figure out a simpler form for the coequilizer $\text{Id}, \text{Inv} : \text{GL}_n(\mathbb{C}) \rightarrow \text{Mat}_n(\mathbb{C})$. Call this $\mathbb{CP}^{1,n}$.
Just as $\mathbb{C}^\times$-principal bundles correspond to both $\mathbb{C}$ and $\mathbb{CP}^1$ bundles, it would also be nice to know whether $\mathbb{CP}^{1,n}$-bundles correspond to principal $\text{GL}_n(\mathbb{C})$-bundles.