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.

Definition

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

That is, every morphism [math]\displaystyle{ P\to X }[/math] factors through every epimorphism [math]\displaystyle{ E\twoheadrightarrow X }[/math].

If C is locally small, i.e., in particular [math]\displaystyle{ \operatorname{Hom}_C(P, X) }[/math] is a set for any object X in C, this definition is equivalent to the condition that the hom functor (also known as corepresentable functor)

[math]\displaystyle{ \operatorname{Hom}(P,-)\colon\mathcal{C}\to\mathbf{Set} }[/math]

preserves epimorphisms.

Projective objects in abelian categories

If the category C is an abelian category such as, for example, the category of abelian groups, then P is projective if and only if

[math]\displaystyle{ \operatorname{Hom}(P,-)\colon\mathcal{C}\to\mathbf{Ab} }[/math]

is an exact functor, where Ab is the category of abelian groups.

An abelian category [math]\displaystyle{ \mathcal{A} }[/math] is said to have enough projectives if, for every object [math]\displaystyle{ A }[/math] of [math]\displaystyle{ \mathcal{A} }[/math], there is a projective object [math]\displaystyle{ P }[/math] of [math]\displaystyle{ \mathcal{A} }[/math] and an epimorphism from P to A or, equivalently, a short exact sequence

[math]\displaystyle{ 0 \to K \to P \longrightarrow A \longrightarrow 0. }[/math]

The purpose of this definition is to ensure that any object A admits a projective resolution, i.e., a (long) exact sequence

[math]\displaystyle{ \dots P_2 \to P_1 \to P_0 \to A \to 0 }[/math]

where the objects [math]\displaystyle{ P_0, P_1, \dots }[/math] are projective.

Projectivity with respect to restricted classes

(Semadeni 1963) discusses the notion of projective (and dually injective) objects relative to a so-called bicategory, which consists of a pair of subcategories of "injections" and "surjections" in the given category C. These subcategories are subject to certain formal properties including the requirement that any surjection is an epimorphism. A projective object (relative to the fixed class of surjections) is then an object P so that Hom(P, −) turns the fixed class of surjections (as opposed to all epimorphisms) into surjections of sets (in the usual sense).

Properties

• The coproduct of two projective objects is projective.
• The retract of a projective object is projective.

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 [math]\displaystyle{ R }[/math] be a ring with identity. Consider the (abelian) category [math]\displaystyle{ R }[/math]-Mod of left [math]\displaystyle{ R }[/math]-modules. The projective objects in [math]\displaystyle{ R }[/math]-Mod are precisely the projective left R-modules. Consequently, [math]\displaystyle{ R }[/math] is itself a projective object in [math]\displaystyle{ R }[/math]-Mod. Dually, the injective objects in [math]\displaystyle{ R }[/math]-Mod are exactly the injective left R-modules.

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

The projective objects in the category of compact Hausdorff spaces are precisely the extremally disconnected spaces. This result is due to (Gleason 1958), with a simplified proof given by (Rainwater 1959).

In the category of Banach spaces and contractions (i.e., functionals whose norm is at most 1), the epimorphisms are precisely the maps with dense image. (Wiweger 1969) shows that the zero space is the only projective object in this category. There are non-trivial spaces, though, which are projective with respect to the class of surjective contractions. In the category of normed vector spaces with contractions (and surjective maps as "surjections"), the projective objects are precisely the [math]\displaystyle{ l^1 }[/math]-spaces.

[math]\displaystyle{ l^1(S) = \{(x_s)_{s \in S}, \sum_{s \in S} ||x_s|| \lt \infty \}. }[/math]

References

• Awodey, Steve (2010), Category theory (2nd ed.), Oxford: Oxford University Press, ISBN 9780199237180, OCLC 740446073 
• Gleason, Andrew M. (1958), "Projective topological spaces", Illinois Journal of Mathematics 2 (4A): 482–489, doi:10.1215/ijm/1255454110 
• Mac Lane, Saunders (1978), Categories for the Working Mathematician (Second ed.), New York, NY: Springer New York, pp. 114, ISBN 1441931236, OCLC 851741862 
• Mitchell, Barry (1965). Theory of categories. Pure and applied mathematics. 17. Academic Press. ISBN 978-0-124-99250-4. 
• Pothoven, Kenneth (1969), "Projective and Injective Objects in the Category of Banach Spaces", Proceedings of the American Mathematical Society 22 (2): 437–438, doi:10.2307/2037073 
• Rainwater, John (1959), "A Note on Projective Resolutions", Proceedings of the American Mathematical Society 10 (5): 734–735, doi:10.2307/2033466 
• Semadeni, Z. (1963), "Projectivity, injectivity and duality", Rozprawy Mat. 35, http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.desklight-2c5c2dc5-5782-45c0-b156-c7853b8977b3 

External links

'"projective object in nLab". https://ncatlab.org/nlab/show/projective+object.