epimorphism
|ep-i-mor-phism|
🇺🇸
/ˌɛpɪˈmɔrfɪzəm/
🇬🇧
/ˌɛpɪˈmɔːfɪzəm/
right-cancellative mapping (often surjective)
Etymology
'epimorphism' originates from Greek elements: the prefix 'epi-' meaning 'upon' or 'over' and 'morphism' from Greek 'morphē' meaning 'form' or 'shape', combined in modern mathematical usage to denote a type of mapping.
'morphism' was adopted in 20th-century mathematics (notably category theory) from Greek 'morphē'; 'epimorphism' was formed by prefixing 'epi-' to 'morphism' to describe a mapping with a specific cancellative property and became standard terminology in category theory and algebra.
Initially constructed as a technical compound meaning 'an 'epi-' type of morphism (a mapping with a certain property)', it has come to denote specifically a right-cancellative morphism in category theory and, in many algebraic contexts, a surjective homomorphism.
Meanings by Part of Speech
Noun 1
in category theory, a morphism f: A → B that is right-cancellative: for any object C and any pair of morphisms g1,g2: B → C, if g1 ∘ f = g2 ∘ f then g1 = g2.
In the category of sets, every epimorphism is a surjective function, but in other categories epimorphisms need not be surjective.
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Noun 2
in algebraic contexts (e.g., groups, rings), an epimorphism often means a surjective homomorphism, though this equivalence depends on the category.
A homomorphism of rings that is surjective is an epimorphism in the category of rings.
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Last updated: 2025/10/13 15:36