We develop a theory of higher colimits over categories of free presentations. We show that different homology functors such as Hochschild and cyclic homology of algebras over a field of characteristic zero, simplicial derived functors, and group homology can be obtained as higher colimits of simply defined functors. Connes’ exact sequence linking Hochschild and cyclic homology was obtained using this approach as a corollary of a simple short exact sequence. As an application of the developed theory, we show that the third reduced K-functor can be defined as the colimit of the second reduced K-functor applied to the fibre square of a free presentation of an algebra. We also prove a Hopf-type formula for odd-dimensional cyclic homology of an algebra over a field of characteristic zero.