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# projective object

Let $\mathcal{C}$ be a category. An object $P$ in $\mathcal{C}$ is said to be *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:

$\xymatrix@+=4pc{&{P}\ar[d]^{{f}}\\ {A}\ar[r]_{{g}}&{B}}\xymatrix@+=4pc{&{P}\ar[d]^{{f}}\ar@{.>}[dl]_{h}\\ {A}\ar[r]_{{g}}&{B}}$ |

An *injective object* can be defined dually: an object $Q$ in $\mathcal{C}$ is *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:

$\xymatrix@+=4pc{{A}\ar[r]^{{g}}\ar[d]&{B}\\ Q&}\xymatrix@+=4pc{{A}\ar[r]^{{g}}\ar[d]&{B}\ar@{.>}[dl]^{h}\\ Q&}$ |

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.

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 projective left $R$-modules. So $R$ is itself a projective object in $\mathcal{M}_{R}$.

Dually, the injective objects in $\mathcal{M}_{R}$ are exactly the injective left $R$-modules.

# References

- 1
F. Borceux
*Basic Category Theory, Handbook of Categorical Algebra I*, Cambridge University Press, Cambridge (1994)

## Mathematics Subject Classification

18E10*no label found*

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