This shows that V is unipotent in M . Suppose A is a coherent analytic category. Suppose P is an indecomposable object. We prove that P is simple. Since any proper strong mono is contained in a proper regular mono, it suffices to prove that P has exactly two regular subobjects.
P is a proper regular subobject of X .
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.