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Concept# Projective object

Summary

In , the notion of a projective object generalizes the notion of a projective module. Projective objects in are used in homological algebra. The dual notion of a projective object is that of an injective object.
An in a category is projective if for any epimorphism and morphism , there is a morphism such that , i.e. the following diagram commutes:
That is, every morphism factors through every epimorphism .
If C is , i.e., in particular 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)
preserves epimorphisms.
If the category C is an abelian category such as, for example, the , then P is projective if and only if
is an exact functor, where Ab is the category of abelian groups.
An abelian category is said to have enough projectives if, for every object of , there is a projective object of and an epimorphism from P to A or, equivalently, a short exact sequence
The purpose of this definition is to ensure that any object A admits a projective resolution, i.e., a (long) exact sequence
where the objects are projective.
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).
The coproduct of two projective objects is projective.
The of a projective object is projective.
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 be a ring with identity. Consider the (abelian) category -Mod of left -modules.

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Category of modules

In algebra, given a ring R, the category of left modules over R is the whose are all left modules over R and whose morphisms are all module homomorphisms between left R-modules. For example, when R is the ring of integers Z, it is the same thing as the . The category of right modules is defined in a similar way. One can also define the category of bimodules over a ring R but that category is equivalent to the category of left (or right) modules over the enveloping algebra of R (or over the opposite of that).

Resolution (algebra)

In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of s of an ), which is used to define invariants characterizing the structure of a specific module or object of this category. When, as usually, arrows are oriented to the right, the sequence is supposed to be infinite to the left for (left) resolutions, and to the right for right resolutions.

Injective module

In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if Q is a submodule of some other module, then it is already a direct summand of that module; also, given a submodule of a module Y, any module homomorphism from this submodule to Q can be extended to a homomorphism from all of Y to Q. This concept is to that of projective modules.

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