Magnitude Homology Is the Associated Graded of the Length Filtration

Abstract

Magnitude homology is graded by length and knows nothing of persistence. Its persistent refinement knows nothing of where its bars begin and end. We show that the two are one construction: filtering the length nerve by sublevel sets of the length yields the persistence module, and the associated graded of that filtration is the magnitude complex. A long exact sequence exchanges them, and each side gains what it lacked. Magnitude homology locates the critical values of the barcode, so a graded computation lists the lengths at which an endpoint can occur, and the barcode acquires a stability estimate of $(n+1)\delta$ in degree $n$ under a perturbation of size $\delta$, while a computed perturbation moves a barcode by more than $\delta$, so the factor cannot be dropped. We apply this to quantitative equational theories, whose free algebras are metric spaces built from syntax: an inclusion of theories induces a morphism of the presenting monads and a comparison of barcodes with an explicit bound, so the invariant measures axiomatic strength. Four examples are computed, one in every degree.