  We study rank functions on a triangulated category $\mathcal{C}$ via its
abelianisation $\operatorname{mod}\mathcal{C}$. We prove that every rank
function on $\mathcal{C}$ can be interpreted as an additive function on
$\operatorname{mod}\mathcal{C}$. As a consequence, every integral rank function
has a unique decomposition into irreducible ones. Furthermore, we relate
integral rank functions to a number of important concepts in the functor
category $\operatorname{Mod}\mathcal{C}$. We study the connection between rank
functions and functors from $\mathcal{C}$ to locally finite triangulated
categories, generalising results by Chuang and Lazarev. In the special case
$\mathcal{C}=\mathcal{T}^c$ for a compactly generated triangulated category
$\mathcal{T}$, this connection becomes particularly nice, providing a link
between rank functions on $\mathcal{C}$ and smashing localisations of
$\mathcal{T}$. In this context, any integral rank function can be described
using the composition length with respect to certain endofinite objects in
$\mathcal{T}$. Finally, if $\mathcal{C}=\operatorname{per}(A)$ for a
differential graded algebra $A$, we classify homological epimorphisms $A\to B$
with $\operatorname{per}(B)$ locally finite via special rank functions which we
call idempotent.
