We define Hochschild and cyclic homology groups for an exact category which generalize the usual definitions when one considers finitely generated projective modules. They satisfy additivity as well as many of the usual properties one expects from the homology groups of an algebra. The Dennis trace and its lift to negative homology are also (multiplicatively) generalized to this setting. We use the S construction of Waldhausen and a formal generalization of the usual cyclic complex from Hochschild homology for our definition.
