Injective iff Projective in Dual Category

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Let $\mathbf A$ be an abelian category.

Let $X$ be an object in $\mathbf A$.


$X$ is injective in $\mathbf A$

if and only if:

$X$ is projective in the dual category $\mathbf A^{\mathrm{op}}$ of $\mathbf A$.


Sufficient Condition

Let $X$ be injective in $\mathbf A$.

Let $f : B \to A$ be an epimorphism in $\mathbf A^{\mathrm {op} }$.

By Monomorphism iff Epimorphism in Dual Category:

$f : A \to B$ is a monomorphism in $\mathbf A$.

By definition of injective object, the mapping:

$\alpha: \map {\mathrm {Hom}_{\mathbf A} } {B, X} \to \map {\mathrm {Hom}_{\mathbf A} } {A, X}, \quad g \mapsto g \circ f$

is surjective.

In the above, $\map {\mathrm {Hom}_{\mathbf A} } {B, X}$ is the set of morphisms in $\mathbf A$ from $B$ to $X$, and $g$ is an arbitrary element of the domain of $\alpha$.

Further, by definition of dual category:

$\map {\mathrm {Hom}_{\mathbf A^{\mathrm {op} } } } {X, B} = \map {\mathrm {Hom}_{\mathbf A} } {B, X}$
$\map {\mathrm {Hom}_{\mathbf A^{\mathrm {op} } } } {X, A} = \map {\mathrm {Hom}_{\mathbf A} } {A, X}$


$\alpha : \map {\mathrm {Hom}_{\mathbf A^{\mathrm {op} } } } {X, B} \to \map {\mathrm {Hom}_{\mathbf A^{\mathrm {op} } } } {X, A}, \quad g \mapsto g \circ f = f \circ_{\mathbf A^{\mathrm{op}}} g$

is surjective.

In the above, $\circ_{\mathbf A^{\mathrm{op}}}$ denotes composition in $\mathbf A^{\mathrm {op} }$.

By definition of projective object, $X$ is projective in $\mathbf A^{\mathrm {op} }$.


Necessary Condition


Also see