Let $\mathcal{P}$ be a (small) exact category. Without delving into any homotopy theory, we can provide characterisations of $K\_0(\mathcal{P})$ and $K\_1(\mathcal{P})$ as plain categorical constructions:


$K\_0(\mathcal{P})$ is the free abelian group on the objects of $\mathcal{P}$ under the relations $[P] = [P'] + [P'']$ if $0 \to P' \to P \to P'' \to 0$


$K\_1(\mathcal{P})$ is the free abelian group on the pairs $(P, \alpha)$, where $P$ is an object and $\alpha$ one of its automotphisms, under suitable relations.



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> Does $K\_2(\mathcal{P})$ admit a similar description?
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My question has been answered (and then some!). It has been pointed out however that I was slightly mistaken: the $K\_1(\mathcal{P})$ given above only coincides with the standard one ($\pi\_2(Q \mathcal{P})$) when short exact sequence of $\mathcal{P}$ splits. So I actually refer to the `Bass $K\_1$'.



Categorical description of the second K-group
Algebraic generators and relations for Quillen's K-group $K\_n(P)$ are given in this paper: "[Algebraic K-theory via binary complexes](http://dx.doi.org/10.1090/S0894-0347-2012-00743-7)". Those you mention don't give the Quillen K-group $K\_1(P)$ in general, but just for Quillen-exact categories where every short exact sequence splits; the group they give is known as Bass' $K\_1$.


