Projective object of a category

A concept formalizing a property of retracts (or direct summands) of free groups, free modules, etc. An object $ P $ of a category $ \mathfrak K $ is said to be projective if for every epimorphism $ \nu : A \rightarrow B $ and every morphism $ \gamma : P \rightarrow B $ there is a morphism $ \gamma ^ \prime : P \rightarrow A $ such that $ \gamma = \gamma ^ \prime \nu $. In other words, an object $ P $ is projective if the representable functor $ H _ {P} ( X) = \mathop{\rm Hom} ( P , X ) $ from $ \mathfrak K $ to the category $ \mathfrak S $ of sets takes epimorphisms of $ \mathfrak K $ to epimorphisms of $ \mathfrak S $, i.e. to surjective mappings.

Examples. 1) In the category of sets every object is projective. 2) In the category of groups, only free groups are projective. 3) In the category $ {} _ \Lambda \mathfrak M $ of left modules over an associative ring $ \Lambda $ with a unit, a module is projective if and only if it is a direct summand of a free module. The description of the rings over which every projective module is free constitutes the content of the Serre problem. 4) In the category $ {} _ \Lambda \mathfrak M $ all modules are projective if and only if the ring $ \Lambda $ is classically semi-simple. 5) In the category $ {\mathcal F} ( \mathfrak D , \mathfrak S ) $ of functions from a small category $ \mathfrak D $ to the category $ \mathfrak S $ of sets, every object is projective if and only if $ \mathfrak D $ is a discrete category.

In the definition of projective objects it is sometimes supposed that the functor $ H _ {P} $ takes not all the epimorphisms but only the morphisms of a distinguished class $ \mathfrak C $ to surjective mappings. In particular, if $ \mathfrak C $ is the class of admissible epimorphisms of a bicategory $ ( \mathfrak K , \mathfrak C , \mathfrak M ) $, then $ P $ is called an admissible projective object. For instance, in some group varieties, the free groups of that variety are admissible projective objects with respect to the class of all surjective homomorphisms but are not projective objects since there exist non-surjective epimorphisms.

Dual to the concept of a projective object is that of an injective object. The fundamental role of projective and injective objects was first observed in the development of homological algebra. In categories of modules every module is representable as a quotient of a projective module. This property allows one to construct the so-called projective resolutions and to study various notions of homological dimension.

References

[1]	H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956)
[2]	S. MacLane, "Homology" , Springer (1963)

Comments

The assertion that every object is projective in the category of sets (Example 1) is one way of formulating the axiom of choice, and most of the other assertions above about projectives in particular categories involve the axiom of choice in some way. For example, the assertion that free Abelian groups are projective has been shown to be equivalent to the axiom of choice [a1], though the assertion that every Abelian group is a quotient of a projective is weaker.

References

[a1]	A.R. Blass, "Injectivity, projectivity and the axiom of choice" Trans. Amer. Math. Soc. , 255 (1979) pp. 31–59