From the work of Bauer and Lesnick, it is known that there is no functor from the category of pointwise finite-dimensional persistence modules to the category of barcodes and overlap matchings. In this work, we introduce sub-barcodes and show that there is a functor from the category of factorizations of persistence module homomorphisms to a poset of barcodes ordered by the sub-barcode relation. Sub-barcodes and factorizations provide a looser alternative to bottleneck matchings and interleavings that can give strong guarantees in a number of settings that arise naturally in topological data analysis. The main use of sub-barcodes is to make strong claims about an unknown barcode in the absence of an interleaving. For example, given only upper and lower bounds g ≥ f ≥ of an unknown real-valued function f , a sub-barcode associated with f can be constructed from and g alone. We propose a theory of sub-barcodes and observe that the subobjects in the functor category Fun(Int op , Mch) naturally correspond to sub-barcodes.