Of interest to topos theorists ... because it shows that supports need not split
(“(SS) fails”) even in a topos where
(SG) Subobjects of 1 Generate, and
(IC) locally, epimorphisms split (“Internal” axiom of Choice).
In particular, epimorphisms need not actually split,
i.e., (AC) may fail, i.e., the implication
(AC) => (IC) & (SG) is not reversible.
[NB: in contrast, for Grothendieck topoi, each of the implications
(AC) => (IC) & (SG) => (B) & (SG) is reversible, as
are (AC) => (B) & (SS) => (B) & (SG) , so
any counterexample is necessarily non-Grothendieck.]
The topos: locally finite sheaves over Heath’s V-space.
(Local finiteness => (IC); 0-dimensionality of the
base => (SG).)
Same topos can then be used as counterexample to the reversibility of
another couple of handfuls of implications among other conjunctions of
similar topos-theoretic conditions.
[For details, consult that
30-year-old reference mentioned at the outset.]
[ Ref. ]