We study the inverse problem for persistent homology: For a fixed simplicial complex K, we analyse the fiber of the continuous map PH on the space of filters that assigns to a filter f : K Ñ R the total barcode of its associated sublevel set filtration of K. We find that PH is best understood as a map of stratified spaces. Over each stratum of the barcode space the map PH restricts to a (trivial) fiber bundle with fiber a polyhedral complex. Amongst other we derive a bound for the dimension of the fiber depending on the number of distinct endpoints in the barcode. Furthermore, taking the inverse image PH ́1 can be extended to a monodromy functor on the (entrance path) category of barcodes. We demonstrate our theory on the example of the simplicial triangle giving a complete description of all fibers and monodromy maps. This example is rich enough to have a Möbius band as one of its fibers.