M has two or one direct factors. If M has only one direct factor it must be the initial object 0 . This shows that the only regular subobjects are M or 0 . Suppose A is a coherent analytic category. Any finitely cogenerated regular mono is a direct mono.
V is locally direct.
Then any finitely cogenerated regular mono u is an intersection of direct monos. P is a proper regular subobject. Thus m is locally direct, which is contained in a proper direct fact of P . This shows that P is simple. The opposite of any Stone geometry is a regular category.