Normal sections and direct product decompositions

Bourn, D;Gran, Marino
(2004) Communications in Algebra — Vol. 32, n° 10, p. 3825-3842 (2004)

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In any finitely complete category, there is an internal notion of normal monomorphism. We give elementary conditions guaranteeing that a normal section s : Y --> X of an arrow f : X --> Y produces a direct product decomposition of the form X similar or equal to Y x W. We then show how these conditions gradually vanish in various algebraic contexts, such as Maltsev, protomodular and additive categories.
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Bourn, D., & Gran, M. (2004). Normal sections and direct product decompositions. Communications in Algebra, 32(10), 3825-3842. https://doi.org/10.1081/AGB-200027750 (Original work published 2004)