Flat covers: The direct method
 

We elaborate on the second existence proof of flat covers by Bican and El Bashir.
Flatness is based on a concept of purity. Purity, even in its most general form,
depends on Stenstr¨om’s concept of purity in a module category Mod(C) over a
small additive category C. To study pure monomorphisms in Mod(C), we enlarge C
to an abelian categorymos(C) of “small objects” such thatMod(C) can be identified
with the category of sheaves of abelian groups over mos(C) in its natural topology.
We show that purity in Mod(C) then boils down to a concept of “pure sections”
on open sets related to the objects of the site mos(C). Using this concept, we
associate a cardinal invariant pM to any C-module M which measures its “purity”.
The existence of flat covers in (not necessarily abelian) categories follows by an
estimation of the number of sections on an open set which might be of interest in
its own right.

Wolfgang Rump