projective object#Enough projectives

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

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That is, every morphism $P\backslash to\; X$ factors through every epimorphism $E\backslash twoheadrightarrow\; X$.{{harvtxt|Awodey|2010|loc=§2.1}}

If C is locally small, i.e., in particular $\backslash operatorname\{Hom\}\_C(P,\; X)$ 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)

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

preserves epimorphisms.{{harvtxt|Mac Lane|1978|loc=p. 118}}

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

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

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

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

:$0\; \backslash to\; K\; \backslash to\; P\; \backslash to\; A\; \backslash to\; 0.$

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

:$\backslash dots\; P\_2\; \backslash to\; P\_1\; \backslash to\; P\_0\; \backslash to\; A\; \backslash to\; 0$

where the objects $P\_0,\; P\_1,\; \backslash dots$ are projective.

{{harvtxt|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).

• The coproduct of two projective objects is projective.{{harvtxt|Awodey|2010|loc=p. 72}}
• The retract of a projective object is projective.{{harvtxt|Awodey|2010|loc=p. 33}}

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 identity. Consider the (abelian) category $R$-Mod of left $R$-modules. The projective objects in $R$-Mod are precisely the projective left R-modules. Consequently, $R$ is itself a projective object in $R$-Mod. Dually, the injective objects in $R$-Mod 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 $M$ (for example we can take $X$ to be $M$). Then the canonical projection $\backslash pi\backslash colon\; F\backslash to\; M$ 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 {{harvtxt|Gleason|1958}}, with a simplified proof given by {{harvtxt|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. {{harvtxt|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 $l^1$-spaces.{{harvtxt|Semadeni|1963}}

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{{nlab|id=projective+object|title=projective object}}

Category:Homological algebra

Category:Objects (category theory)