What is the definition of the injective objects in the category of algebras?
2026-04-08 06:10:48.1775628648
The injective objects in the category of algebras
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An injective object in a locally small category $\mathcal{C}$ is an object $I$ such that the functor $\mathcal{C} (-, I) : \mathcal{C}^\mathrm{op} \to \mathbf{Set}$ sends monomorphisms in $\mathcal{C}$ to surjections. To put it in simpler terms, $I$ is injective if, for every monomorphism $f : A \to B$ and every morphism $a : A \to I$, there is a morphism $b : B \to I$ such that $a = b \circ f$.