Magnitude homology is a derived functor

Abstract

We prove that the magnitude (co)homology of an enriched category can, under some technical assumptions, be described in terms of derived functors between certain abelian categories. We show how this statement is specified for the cases of quasimetric spaces, finite quasimetric spaces, and finite digraphs. For quasimetric spaces, we define the notion of a distance module over a quasimetric space, define the functor of (co)invariants of a distance module and show that the magnitude (co)homology can be presented via its derived functors. As a corollary we obtain that the magnitude cohomology of a quasimetric space can be presented in terms of Ext functors in the category of distance modules. For finite quasimetric spaces, we show that magnitude (co)homology can be presented in terms of Tor and Ext functors over a certain graded algebra, called the distance algebra of the quasimetric space. For finite digraphs, the distance algebra is a bound quiver algebra. In addition, we show that the magnitude cohomology algebra of a finite quasimetric space can be described as a Yoneda algebra.