We demonstrate that an equivalence of categories using $\varepsilon$-interleavings as a fundamental component exists between the model of persistence modules as graded modules over a polynomial ring and the model of persistence modules as modules over the total order of the real numbers.
Higher interpolation and extension for persistence modules
Equivalence of models for equivariant $(\infty, 1)$-categories
Model structures for diagrammatic $(\infty, n)$-categories
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