in the spring of 1972, quillen found another approach to the construction of higher "k"-theory which was to prove enormously successful. this new definition began with an exact category, a category satisfying certain formal properties similar to, but slightly weaker than, the properties satisfied by a category of modules or vector bundles. from this he constructed an auxiliary category using a new device called his ""q"-construction." like segal's γ-objects, the "q"-construction has its roots in grothendieck's definition of "k". unlike grothendieck's definition, however, the "q"-construction builds a category, not an abelian group, and unlike segal's γ-objects, the "q"-construction works directly with short exact sequences. if "c" is an abelian category, then "qc" is a category with the same objects as "c" but whose morphisms are defined in terms of short exact sequences in "c". the "k"-groups of the exact category are the homotopy groups of ω"bqc", the loop space of the geometric realization (taking the loop space corrects the indexing). quillen additionally proved his " theorem" that his two definitions of "k"-theory agreed with each other. this yielded the correct "k" and led to simpler proofs, but still did not yield any negative "k"-groups.