### Induced matchings and the algebraic stability of persistence barcodes

#### Abstract

$\DeclareMathOperator{\ker}{ker}\DeclareMathOperator{\coker}{coker}$We define a simple, explicit map sending a morphism $f\colon M\to N$ of pointwise finite dimensional persistence modules to a matching between the barcodes of $M$ and $N$. Our main result is that, in a precise sense, the quality of this matching is tightly controlled by the lengths of the longest intervals in the barcodes of $\ker f$ and $\coker f$.

As an immediate corollary, we obtain a new proof of the algebraic stability theorem for persistence barcodes, a fundamental result in the theory of persistent homology. In contrast to previous proofs, ours shows explicitly how a $\delta$-interleaving morphism between two persistence modules induces a $\delta$-matching between the barcodes of the two modules. Our main result also specializes to a structure theorem for submodules and quotients of persistence modules, and yields a novel “single-morphism” characterization of the interleaving relation on persistence modules.

As an immediate corollary, we obtain a new proof of the algebraic stability theorem for persistence barcodes, a fundamental result in the theory of persistent homology. In contrast to previous proofs, ours shows explicitly how a $\delta$-interleaving morphism between two persistence modules induces a $\delta$-matching between the barcodes of the two modules. Our main result also specializes to a structure theorem for submodules and quotients of persistence modules, and yields a novel “single-morphism” characterization of the interleaving relation on persistence modules.

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