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. Thus the pushout of any direct mono is a direct mono, i.e. direct mono is co-universal. A locally finitely copresentable category is a Stone geometry iff its subcategory of finitely copresentable objects is a decidable lextensive category.