projective objectProjectiveObject2004-11-01 20:04:322009-01-23 23:50:44Definitioninjective object% this is the default PlanetMath preamble. as your knowledge
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% define commands hereLet $\mathcal{C}$ be a category. An object $P$ in $\mathcal{C}$ is said to be \emph{projective} if, given a diagram on the left, with $g$ a strong epimorphism, there is a morphism $h:P\to A$ making the diagram on the right commutative:
$$\xymatrix@+=4pc{
&{P}\ar[d]^{f}\\
{A}\ar[r]_{g}&{B}
}
\hspace{4cm}
\xymatrix@+=4pc{
&{P}\ar[d]^{f} \ar@{.>}[dl]_h \\
{A}\ar[r]_{g}&{B}
}
$$
An \emph{injective object} can be defined dually: an object $Q$ in $\mathcal{C}$ is \emph{injective} if, given a diagram on the left, with $g$ a strong monomorphism, there is a morphism $h:B\to Q$ making the digram on the right commutative:
$$\xymatrix@+=4pc{
{A}\ar[r]^{g} \ar[d] &{B} \\
Q &
}
\hspace{4cm}
\xymatrix@+=4pc{
{A}\ar[r]^{g} \ar[d] &{B} \ar@{.>}[dl]^h \\
Q &
}
$$
When $\mathcal{C}$ is an abelian category, we have the following: an object $P$ in $\mathcal{C}$ is projective iff
$$\operatorname{Hom}(P,-)\colon\mathcal{C}\to\mathbf{Ab}$$
is an exact functor, where $\mathbf{Ab}$ is the category of abelian groups. Dually, an object $Q$ is injective iff the $\operatorname{Hom}(-,Q)$ functor from $\mathcal{C}$ to $\mathbf{Ab}$ is exact.
\textbf{Example.} Let $R$ be a ring with 1. Consider the category of left $R$-modules $\mathcal{M}_R$. $\mathcal{M}_R$ is an abelian category. The projective objects in $\mathcal{M}_R$ are precisely the \PMlinkname{projective left $R$-modules}{ProjectiveModule}. So $R$ is itself a projective object in $\mathcal{M}_R$.
Dually, the injective objects in $\mathcal{M}_R$ are exactly the \PMlinkname{injective left $R$-modules}{InjectiveModule}.
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\bibitem{fb} F. Borceux \emph{Basic Category Theory, Handbook of Categorical Algebra I}, Cambridge University Press, Cambridge (1994)
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