Classifying space BGL

Description

Let $R$ be a ring with identity. For $n \ge 1$, let $GL_n(R)$ be the general linear group of invertible $n \times n$ matrices over $R$. Embed $GL_n(R)$ in $GL_{n+1}(R)$ by $g \mapsto \begin{pmatrix} g & 0 \\ 0 & 1 \end{pmatrix}$, and let $GL(R) = \bigcup_{n \ge 1} GL_n(R)$ be the stable general linear group, viewed as a discrete group. The space $BGL(R)$ is a classifying space of $GL(R)$: a connected CW complex with $\pi_1(BGL(R)) = GL(R)$ and $\pi_k(BGL(R)) = 0$ for all $k \ge 2$.

Check: the geometric realization of the nerve of $GL(R)$, seen as a category with one object, has exactly these homotopy groups. Since $GL(R)$ is the union of the $GL_n(R)$, one can take $BGL(R) = \bigcup_n BGL_n(R)$.