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We define a simple, explicit map sending a morphism $f:M \rightarrow N$ ofpointwise finite dimensional persistence modules to a matching between thebarcodes of $M$ and $N$. Our main result is that, in a precise sense, thequality of this matching is tightly controlled by the lengths of the longestintervals in the barcodes of $\ker f$ and $\mathop{\mathrm{coker}} f$. As animmediate corollary, we obtain a new proof of the algebraic stability ofpersistence, a fundamental result in the theory of persistent homology. Incontrast to previous proofs, ours shows explicitly how a $\delta$-interleavingmorphism between two persistence modules induces a $\delta$-matching betweenthe barcodes of the two modules. Our main result also specializes to astructure theorem for submodules and quotients of persistence modules, andyields a novel "single-morphism" characterization of the interleaving relationon persistence modules.

Induced Matchings and the Algebraic Stability of Persistence Barcodes