It is known that for any ring $R$, $$K_{1}(R)=H_{1}(GL_{\infty}(R), \mathbb{Z})$$ $$ K_{2}(R)= H_{2}(E_{\infty}(R),\mathbb{Z})$$ $$ K_{3}(R)= H_{3}(St_{\infty}(R),\mathbb{Z})$$ where $GL_{\infty}= lim GL_{n}$, $E_{\infty}=lim E_{n}$ (elementary matrices) , $St_{\infty}=lim St_{n}$ (Steinberg groups).

Question:

Is there a same description of higher groups of algebraic K-theory. i.e. for each $n>0$ there exist a group $M^{n}=lim M_{i}$ such that $$ K_{n}(R)= H_{n}(M^{n}(R),\mathbb{Z})$$ where every thing is functorial. in our case $M^{1}= GL_{\infty}$, $M^{2}=E_{\infty}$ and $M^{3}=St_{\infty}$.

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