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