Projective object

In category theory, the notion of a projective object generalizes the notion of a projective module. Projective objects in abelian categories are used in homological algebra. The dual notion of a projective object is that of an injective object.

An object $P$ in a category ${\mathcal {C}}$ is projective if for any epimorphism $e:E\twoheadrightarrow X$ and morphism $f:P\to X$ , there is a morphism ${\overline {f}}:P\to E$ such that $e\circ {\overline {f}}=f$ , i.e. the following diagram commutes:

That is, every morphism $P\to X$ factors through every epimorphism $E\twoheadrightarrow X$ .^[1]^[2]

In a locally small category ${\mathcal {C}}$ , the following statement is equivalent: $P$ is projective if the hom functor

\operatorname {Hom}(P,-)\colon {\mathcal {C}}\to {\mathbf {Set}}

preserves epimorphisms.^[3]

Let ${\mathcal {C}}$ be an abelian category. In this context, an object $P\in {\mathcal {C}}$ is called a projective object if

\operatorname {Hom}(P,-)\colon {\mathcal {C}}\to {\mathbf {Ab}}

is an exact functor, where ${\mathbf {Ab}}$ is the category of abelian groups.

Properties

The coproduct of two projective objects is projective.^[4]
The retract of a projective object is projective.^[5]

Enough projectives

Let ${\mathcal {A}}$ be an abelian category. ${\mathcal {A}}$ is said to have enough projectives if, for every object $A$ of ${\mathcal {A}}$ , there is a projective object $P$ of ${\mathcal {A}}$ and an exact sequence

P\longrightarrow A\longrightarrow 0.

In other words, the map $p\colon P\to A$ is "epic", or an epimorphism.

Examples

The statement that all sets are projective is equivalent to the axiom of choice.

The projective objects in the category of abelian groups are the free abelian groups.

Let $R$ be a ring with 1. Consider the (abelian) category of left $R$ -modules ${\mathcal {M}}_{R}$ . The projective objects in ${\mathcal {M}}_{R}$ are precisely the projective left R-modules. Consequently, $R$ is itself a projective object in ${\mathcal {M}}_{R}.$ Dually, the injective objects in ${\mathcal {M}}_{R}$ are exactly the injective left R-modules.

The category of left (right) $R$ -modules also has enough projectives. This is true since, for every left (right) $R$ -module $M$ , we can take $F$ to be the free (and hence projective) $R$ -module generated by a generating set $X$ for $M$ (we can in fact take $X$ to be $M$ ). Then the canonical projection $\pi \colon F\to M$ is the required surjection.

References

↑ "projective object in nLab". ncatlab.org. Retrieved 2017-10-17.
↑ Awodey, Steve (2010). Category theory (2nd ed.). Oxford: Oxford University Press. p. 33. ISBN 9780199237180. OCLC 740446073.
↑ Mac Lane, Saunders (1978). Categories for the Working Mathematician (Second ed.). New York, NY: Springer New York. p. 114. ISBN 1441931236. OCLC 851741862.
↑ Awodey, Steve (2010). Category theory (2nd ed.). Oxford: Oxford University Press. p. 72. ISBN 9780199237180. OCLC 740446073.
↑ Awodey, Steve (2010). Category theory (2nd ed.). Oxford: Oxford University Press. p. 33. ISBN 9780199237180. OCLC 740446073.

Mitchell, Barry (1965). Theory of categories. Pure and applied mathematics. 17. Academic Press. ISBN 978-0-124-99250-4. MR 0202787.

This article incorporates material from Projective object on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

This article incorporates material from Enough projectives on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] "projective object in nLab". ncatlab.org. Retrieved 2017-10-17.

[2] Awodey, Steve (2010). Category theory (2nd ed.). Oxford: Oxford University Press. p. 33. ISBN 9780199237180. OCLC 740446073.

[3] Mac Lane, Saunders (1978). Categories for the Working Mathematician (Second ed.). New York, NY: Springer New York. p. 114. ISBN 1441931236. OCLC 851741862.

[4] Awodey, Steve (2010). Category theory (2nd ed.). Oxford: Oxford University Press. p. 72. ISBN 9780199237180. OCLC 740446073.

[5] Awodey, Steve (2010). Category theory (2nd ed.). Oxford: Oxford University Press. p. 33. ISBN 9780199237180. OCLC 740446073.