  Rank functors on a quiver $Q$ are certain additive functors from the category
of representations of $Q$ to the category of finite-dimensional vector spaces.
Composing with the dimension function on vector spaces gives a rank function on
$Q$. These induce functions on $\rep(Q, \alpha)$, the variety of
representations of $Q$ of dimension vector $\alpha$, and thus can be used to
define "rank loci" in $\rep(Q, \alpha)$ as collections of points satisfying
finite lists of linear inequalities of rank functions. Although quiver rank
functions are not generally semicontinuous like the rank of a linear map, we
show here that they do have the geometric property that these rank loci are
constructible subvarieties. The same is true for loci defined by rank functions
in Schofield's subrepresentation bundles on $\rep(Q, \alpha)$, and in quiver
Grassmannians.
