  Persistent homology has become a ubiquitous tool in topological data
analysis. Throughout this paper, we consider persistent homology of a fixed
dataset which is assumed to be a point cloud in a Euclidean space. Further, we
use the Rips construction for computation of persistent homology which, in our
setup, takes as a parameter a step size $h$. For a fixed $h$, one important
data visualization for the persistent homology is the persistence barcode. From
another point of view, each persistence barcode corresponds to a specific
isomorphism class of quiver representations of an equioriented A-type Dynkin
quiver. Using algebro-geometric facts regarding type-A quiver representations,
we define the codimension of a persistence barcode as the codimension of the
corresponding isomorphism class in the space of all quiver representations.
Further, we prove a new formula for the codimension and, as an application we
prove that for a fixed dataset, the value of the codimension of the persistence
barcode stabilizes as the step size parameter tends to zero. As a consequence,
we define a new topological statistic for a dataset which we call the quiver
codimension of the dataset, and which is equal to the stabilized value of the
barcode codimensions.
