Definition:Image (Category Theory)
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Definition
Let $\CC$ be a locally small category.
Let $f : X \to Y$ be a morphism in $\CC$.
An image of $f$ consists of an object $I$ and a monomorphism $m: I \to Y$ such that:
- $(1): \quad$ There exists a morphism $e : X \to I$ such that $f = m \circ e$.
- $(2): \quad$ For any object $I'$ with a morphism $e' : X \to I'$ and a monomorphism $m' : I' \to Y$ such that $f = m' \circ e'$, there exists a unique morphism $v : I \to I'$ such that $m = m' \circ v$.
Sources
- 1965: Barry Mitchell: Theory of Categories: $\S \text I.10$: Images.