  The Euler characteristic is an invariant of a topological space that in a
precise sense captures its canonical notion of size, akin to the cardinality of
a set. The Euler characteristic is closely related to the homology of a space,
as it can be expressed as the alternating sum of its Betti numbers, whenever
the sum is well-defined. Thus, one says that homology categorifies the Euler
characteristic. In his work on the generalisation of cardinality-like
invariants, Leinster introduced the magnitude of a metric space, a real number
that counts the "effective number of points" of the space and has been shown to
encode many invariants of metric spaces from integral geometry and geometric
measure theory. In 2015, Hepworth and Willerton introduced a homology theory
for metric graphs, called magnitude homology, which categorifies the magnitude
of a finite metric graph. This work was subsequently generalised to enriched
categories by Leinster and Shulman, and the homology theory that they
introduced categorifies magnitude for arbitrary finite metric spaces. When
studying a metric space, one is often only interested in the metric space up to
a rescaling of the distance of the points by a non-negative real number. The
magnitude function describes how the effective number of points changes as one
scales the distance, and is completely encoded by magnitude homology. When
studying a finite metric space in topological data analysis using persistent
homology, one approximates the space through a nested sequence of simplicial
complexes so as to recover topological information about the space by studying
the homology of this sequence. Here we relate magnitude homology and persistent
homology as two different ways of computing homology of filtered simplicial
sets.
