The normal decomposition of a morphism in categories without zeros

For a morphism $f:A\to B$ in a category $C$ with sufficiently many finite limits and colimits, we discuss an elementary construction of a functorial decomposition which, if $C$ happens to have a zero object, amounts to the standard decomposition In this way we obtain natural notions of normal monomorphism and normal epimorphism also in non-pointed categories, as special types of regular mono- and epimorphisms. We examine the factorization behaviour of these classes of morphisms in general, compare the generalized normal decompositions with other types of threefold factorizations, and illustrate them in some every-day categories. The concrete construction of normal decompositions in the slices or coslices of some of these categories can be challenging. Amongst many others, in this regard we consider particularly the categories of T₁-spaces and of groups.