Definition:Epimorphism (Category Theory)
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This page is about Epimorphism in the context of Category Theory. For other uses, see Epimorphism.
Let $\mathbf C$ be a metacategory.
An epimorphism is a morphism $f \in \mathbf C_1$ such that:
- $g \circ f = h \circ f \implies g = h$
for all morphisms $g, h \in \mathbf C_1$ for which these compositions are defined.
That is, an epimorphism is a morphism which is right cancellable.
One writes $f: C \twoheadrightarrow D$ to denote that $f$ is an epimorphism.
Also known as
Often, epimorphism is abbreviated to epi.
Alternatively, one can speak about an epic morphism to denote an epimorphism.
- Monomorphism, the dual notion
The word epimorphism comes from the Greek morphe (μορφή) meaning form or structure, with the prefix epi- meaning onto.
Thus epimorphism means onto (similar) structure.
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 2.1$: Definition $2.1$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): epimorhism