Higher colimits, derived functors and homology

Abstract

A theory of higher colimits over categories of free presentations is developed. It is shown that different homology functors such as Hoshcshild 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 it is shown 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. A Hopf-type formula for odd dimensional cyclic homology of an algebra over a field of characteristic zero is also proved.arXiv:1805.07754v2 [math.KT]

Dimension quotients, Fox subgroups and limits of functors

“…The theory of (derived) colimits in the categories of presentations is developed in [10]. It is shown in that paper that a lot of interesting functors like cyclic homology or certain K-functors can be described via derived colimits.…”