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 $g\geq f\geq \ell$ of an unknown real-valued function $f$, a sub-barcode associated with $f$ can be constructed from $\ell$ and $g$ alone. We propose a theory of sub-barcodes and observe that the subobjects in the category of functors from intervals to matchings naturally correspond to sub-barcodes.

We revisit the Contextual Pattern Matching Problem, defined as follows: preprocess a text T[1, n], so that given a query consisting of a string P and a length P, the occurrences of all distinct strings XPY where |X|=|Y|=P can be reported. This problem was introduced by Navarro, who presented an O($\overline{r}\log(n/\overline{r}))$ space data structure, where $\overline{r}$ is the maximum of the number of runs in the BWT of the text $\mathrm{T}[1,n]$ and its reverse. His solution reports all c contextual occurrences in $O(|P|+c\log n)$ time. However, the only known bounds on $\overline{r}$ are $\overline{r}=O(r\log^{2}n)$ where r is the number of runs in the BWT of T, making it desirable to avoid using structures with space dependent on $\overline{r}$. We demonstrate that this is possible without a significant sacrifice in query time by providing an $O(r\log(n/r))$ space solution that answers queries in $O(|P|+c\log P\cdot\log(n/r))$ time.