The connecting homomorphism for K-theory of generalized free products

We interpret the connecting homomorphism of the long exact sequence of algebraic K-groups associated to a category with cofibrations and two notions of weak equivalence. One notion of weak equivalence is finer than the other and certain technical conditions involving mapping cylinders must also be satisfied. Such situations arise when one considers certain free product diagrams in the category of rings, such as those that arise in applications of the Seifert-van Kampen theorem where all fundamental group homomorphisms are injective. The techniques follow Waldhausen’s approach to algebraic K-theory of categories with cofibrations and weak equivalences.