Categorical description of the second K-group 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.

Does $K_2(\mathcal{P})$ admit a similar description?


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 $K_1$'.
 A: Algebraic generators and relations for Quillen's K-group $K_n(P)$ are given in this paper: "Algebraic K-theory via binary complexes".  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' $K_1$.
A: The $K_1$ group you describe is the automorphism $K_1$, which is in general not isomorphic to Quillen's $K_1$ of an exact category. It coincides with Quillen's when exact sequences in $\mathcal P$ split. For a description by generators and relations of $K_1$ of any exact category see:
Nenashev, A.
K1 by generators and relations. (English summary) 
J. Pure Appl. Algebra 131 (1998), no. 2, 195–212. 
generators are pairs of short exact sequences on the same objects, and relations are given by $3\times 3$ diagrams. A generalization of this result to all $K_n$ is given in
Grayson, Daniel R.
Algebraic K-theory via binary complexes. 
J. Amer. Math. Soc. 25 (2012), no. 4, 1149–1167. 
