Normal Sections and Direct Product Decompositions

Abstract

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 ≃ Y × W. We then show how these conditions gradually vanish in various algebraic contexts, such as Maltsev, protomodular and additive categories.

Key Words:

• Normal subobjects
• Direct product decompositions
• Maltsev categories
• Protomodular categories
• Modular varieties

2000 Mathematics Subject Classification:

• 08A30
• 08B10
• 18A20
• 18E10

Notes

^aThese two conditions are equivalent to the fact that τ is an isomorphism.

^#Communicated by A. Facchini.