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In this article we generalize the theory of $k$-minimal models in rationalhomotopy theory to the context of persistence. Our main motivation is thecomputation of barcode decompositions of rational homotopy groups of a space.We give an explicit construction of minimal models of maps between commutativedifferential-graded algebras, which we then use to construct minimal models oftame persistent CDGAs. We then build a model structure on persistent CDGAs forwhich these minimal models become cell complexes. Applications of our resultsinclude an explicit construction of Postnikov towers for copersistent spacesand a decomposition of copersistent Postnikov invariants.

Persistent $k$-minimal models and the interval sphere model structure