  We generalize the concept of a norm on a vector space to one of a norm on a
category. This provides a unified perspective on many specific matters in many
different areas of mathematics like set theory, functional analysis, measure
theory, topology, and metric space theory. We will especially address the two
last areas in which the monotone-light factorization and, respectively, the
Gromov-Hausdorff distance will naturally appear. In our formalization a
Schr\"oder-Bernstein property becomes an axiom of a norm which constitutes
interesting properties of the categories in question. The proposed concept
provides a convenient framework for metrizations.
