  Magnitude homology is an invariant of enriched categories which generalizes
ordinary categorical homology -- the homology of the classifying space of a
small category. The classifying space can also be generalized in a different
direction: it extends from categories to bicategories as the geometric
realization of the geometric nerve. This paper introduces a hybrid of the two
ideas: an iterated magnitude homology theory for categories with a second- or
higher-order enrichment. This encompasses, for example, groups equipped with
extra structure such as a partial ordering or a bi-invariant metric. In the
case of a strict 2-category, iterated magnitude homology recovers the homology
of the classifying space; we investigate its content and behaviour when
interpreted for partially ordered groups, normed groups, and strict
$n$-categories for $n > 2$.

Algebraic Topology;Category Theory;Group Theory;Metric Geometry