Algebraic K-theory via binary complexes

In [11], Quillen defines the notion of exact category. Each such category N is an additive category together with a suitable collection of sequences called short exact sequences. Quillen constructs a space by gluing simplices to each other, using the exact sequences of N to determine the simplices and the gluing instructions. Then for each n ∈ N he defines KnN as a homotopy group of the space. The construction is engineered so that K0N is isomorphic to the Grothendieck group of N . Because they are defined as homotopy groups, these K-groups are difficult to compute, with the case where N is the category of finite-dimensional vector spaces over a finite field being one of the few cases where they are all known. Until now, there has been no description of the K-groups not involving homotopy theory. This paper1 resulted from contemplation of a result of Nenashev [10], which depends on his earlier work [8, 9] and on work of Sherman [13, 14]. Nenashev provides generators and relations for K1N , where N is an exact category; the generators are double exact sequences in N :