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We generalize the concept of a norm on a vector space to one of a norm on acategory. This provides a unified perspective on many specific matters in manydifferent areas of mathematics like set theory, functional analysis, measuretheory, topology, and metric space theory. We will especially address the twolast areas in which the monotone-light factorization and, respectively, theGromov-Hausdorff distance will naturally appear. In our formalization aSchr\"oder-Bernstein property becomes an axiom of a norm which constitutesinteresting properties of the categories in question. The proposed conceptprovides a convenient framework for metrizations.

Norms on Categories and Analogs of the Schr\"oder-Bernstein Theorem