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projective object
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(Definition)
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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:
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:
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.
- 1
- F. Borceux Basic Category Theory, Handbook of Categorical Algebra I, Cambridge University Press, Cambridge (1994)
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"projective object" is owned by CWoo.
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Cross-references: ring, functor, abelian groups, exact functor, iff, abelian category, strong monomorphism, injective, commutative, right, morphism, strong epimorphism, diagram, object, category
There are 8 references to this entry.
This is version 7 of projective object, born on 2004-11-01, modified 2009-01-23.
Object id is 6437, canonical name is ProjectiveObject.
Accessed 2955 times total.
Classification:
| AMS MSC: | 18E10 (Category theory; homological algebra :: Abelian categories :: Exact categories, abelian categories) |
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Pending Errata and Addenda
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![$\displaystyle \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} } $](http://images.planetmath.org:8080/cache/objects/6437/js/img1.png "$\displaystyle \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} } $")
![$\displaystyle \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 & } $](http://images.planetmath.org:8080/cache/objects/6437/js/img2.png "$\displaystyle \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 & } $")