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\begin{document}

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\section*{projective object}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{InAbelianCats}{In abelian categories}\dotfill \pageref*{InAbelianCats} \linebreak
\noindent\hyperlink{EquivalentCharacterizationInAbelianCats}{Equivalent characterizations}\dotfill \pageref*{EquivalentCharacterizationInAbelianCats} \linebreak
\noindent\hyperlink{projective_resolutions}{Projective resolutions}\dotfill \pageref*{projective_resolutions} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{ExistenceOfEnoughProjectives}{Existence of enough projectives}\dotfill \pageref*{ExistenceOfEnoughProjectives} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{defn}
\label{}\hypertarget{}{}
An [[object]] $P$ of a [[category]] $C$ is \textbf{projective} (with respect to [[epimorphisms]]) if it has the [[left lifting property]] against [[epimorphisms]].

\end{defn}
This means that $P$ is projective if for any [[morphism]] $f:P \to B$ and any [[epimorphism]] $q:A \to B$, $f$ factors through $q$ by some morphism $P\to A$.

\begin{displaymath}
\itexarray{
    && A
    \\
    &{}^{\mathllap{\exists}}\nearrow& \downarrow^{\mathrlap{q}} 
    \\
    P &\stackrel{f}{\to}& B
  }
  \,.
\end{displaymath}
Yet another way to say this is that

\begin{defn}
\label{ByCovHomPreservingEpis}\hypertarget{ByCovHomPreservingEpis}{}
An object $P$ is projective precisely if the [[hom-functor]] $Hom(P,-)$ preserves [[epimorphisms]].

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
This generalizes the notion of \emph{[[projective modules]]} over a [[ring]].

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
There are variations of the definition where ``epimorphism'' is replaced by some other type of morphism, such as a [[regular epimorphism]] or [[strong epimorphism]] or the left class in some [[orthogonal factorization system]]. In this case one may speak of \textbf{regular projectives} and so on. In a [[regular category]] ``projective'' almost always means ``regular projective.''

If the ambient category has an [[initial object]] $\emptyset$, then the injective objects $P$ are those for which $\emptyset \stackrel{\exists!}{\to} P$ is called an [[projective morphism]].

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
The [[duality|dual]] notion is that of \emph{[[injective objects]]}.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
There is also a stronger notion of projective object, where $\hom(C, -)$ preserves coequalizers. See more at [[tiny object]], which can be defined as projective [[connected objects]] with ``projective'' used in this stronger sense.

\end{remark}
\begin{defn}
\label{EnoughProjectives}\hypertarget{EnoughProjectives}{}
A category $C$ has \textbf{enough projectives} if for every object $X$ there is an [[epimorphism]] $P\to X$ where $P$ is projective.

Equivalently: if every object admits a \emph{[[projective presentation]]}.

\end{defn}
This terminology refers to the existence of [[projective resolutions]], prop. \ref{EnoughIsEnough} below.

\hypertarget{InAbelianCats}{}\subsubsection*{{In abelian categories}}\label{InAbelianCats}

Projective objects and [[injective objects]] in [[abelian categories]] $\mathcal{A}$ are of central interest in [[homological algebra]]. Here they appear as parts of [[cofibrant resolutions]] and [[fibrant resolutions]], respectively, in the [[category of chain complexes]] $Ch_\bullet(\mathcal{A})$, with respect to one of the two standard [[model structures on chain complexes]].

\hypertarget{EquivalentCharacterizationInAbelianCats}{}\paragraph*{{Equivalent characterizations}}\label{EquivalentCharacterizationInAbelianCats}

\begin{prop}
\label{EquivalenceOfDefinitionsInAbelian}\hypertarget{EquivalenceOfDefinitionsInAbelian}{}
The following are equivalent

\begin{enumerate}%
\item $X \in \mathcal{A}$ is a projective object (in that it has the [[left lifting property]] against [[epimorphisms]], def. \ref{ByCovHomPreservingEpis});


\item The [[hom-functor]] $Hom(X,-) : \mathcal{A} \to$ [[Ab]] is an [[exact functor]].



\end{enumerate}
\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
For every object $X$, the hom-functor $Hom(X,-)$ is a [[left exact functor]]. So the second statement is equivalently that it is also right exact precisely if $X$ is projective.

\end{remark}
\begin{proof}
\textbf{of prop. \ref{EquivalenceOfDefinitionsInAbelian}}

Let

\begin{displaymath}
0 \to A \stackrel{i}{\hookrightarrow} B \stackrel{p}{\to} C \to 0
\end{displaymath}
be a [[short exact sequence]]. The exactness at $A$ and $B$ together is equivalent to the statement that $i = \ker(p)$. Since as remarked above $Hom(X, -)$ is left exact, it preserves [[kernels]] and so $Hom(X, i) = \ker(Hom(X, p))$, giving exactness of the sequence

\begin{displaymath}
0 \to  Hom(X,A) 
    \stackrel{Hom(X,i)}{\to}
  Hom(X,B)
   \stackrel{Hom(X,p)}{\to}
  Hom(X,C)
  \,.
\end{displaymath}
Therefore we are reduced to showing that $Hom(X,p)$ is an [[epimorphism]] precisely if $X$ is projective. But this is def. \ref{ByCovHomPreservingEpis}.

\end{proof}
\hypertarget{projective_resolutions}{}\paragraph*{{Projective resolutions}}\label{projective_resolutions}

Let $\mathcal{A}$ be an [[abelian category]].

\begin{defn}
\label{}\hypertarget{}{}
For $N \in \mathcal{A}$ an object, a \textbf{[[projective resolution]]} of $N$ is a [[chain complex]] $(Q N)_\bullet \in Ch_\bullet(\mathcal{A})$ equipped with a [[chain map]]

\begin{displaymath}
Q N \to N
\end{displaymath}
(with $N$ regarded as a complex concentrated in degree 0) such that

\begin{enumerate}%
\item this morphism is a [[quasi-isomorphism]] (this is what makes it a [[resolution]]), which is equivalent to

\begin{displaymath}
\cdots \to (Q N)_1 \to (Q N)_0 \to N
\end{displaymath}
being an [[exact sequence]];


\item all whose entries $(Q N)_n$ are projective objects.



\end{enumerate}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
This means precisely that $Q N \to N$ is an [[cofibrant resolution]] with respect to the standard [[model structure on chain complexes]] (see \href{model%20structure%20on%20chain%20complexes#StandardQuillenOnBounded}{here}) for which the fibrations are the positive-degreewise epimorphisms. Notice that in this model structure every object is fibrant, so that cofibrant resolutions are the only resolutions that need to be considered.

\end{remark}
\begin{prop}
\label{EnoughIsEnough}\hypertarget{EnoughIsEnough}{}
If $\mathcal{A}$ has \emph{enough projectives} in the sense of def. \ref{EnoughProjectives}, then every object has a projective resolution.

\end{prop}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{remark}
\label{ProjectiveInSetsIsAxiomOfChoice}\hypertarget{ProjectiveInSetsIsAxiomOfChoice}{}
The [[axiom of choice]] can be phrased as ``all objects of [[Set]] are projective.''

See also \emph{[[internally projective object]]} and \emph{[[COSHEP]]}.

\end{remark}
\begin{example}
\label{}\hypertarget{}{}
If $C$ has [[pullback|pullbacks]] and epimorphisms are preserved by pullback, as is the case in a [[pretopos]], then $P$ is projective iff any epimorphism $Q\to P$ is [[split epimorphism|split]].

\end{example}
\begin{example}
\label{ProjectiveObjectsInAbAreFreeGroups}\hypertarget{ProjectiveObjectsInAbAreFreeGroups}{}
An object in [[Ab]], an [[abelian group]], is projective precisely if it is a [[free group]].

\end{example}
\begin{example}
\label{}\hypertarget{}{}
For $R$ a [[commutative ring]], an object in $R$[[Mod]], an $R$-[[module]], is projective (a [[projective module]], see there for more details) precisely if it is a [[direct sum|direct summand]] of a [[free module]]. See at \emph{[[projective module]]} for more on this.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
The projective objects in [[compact topological space|compact]] [[Hausdorff topological spaces]] are precisely the [[extremally disconnected profinite sets]].

\end{example}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{ExistenceOfEnoughProjectives}{}\subsubsection*{{Existence of enough projectives}}\label{ExistenceOfEnoughProjectives}

We list examples of classes of categories that have enough projective, according to def. \ref{EnoughProjectives}.

\begin{prop}
\label{ModHasEnoughProjectives}\hypertarget{ModHasEnoughProjectives}{}
Assuming the [[axiom of choice]], for $R$ a [[ring]] the category $R$[[Mod]] of [[modules]] over $R$ is has enough projectives.

\end{prop}
See at \emph{[[projective module]]} for more.

\begin{lemma}
\label{FreeModulesAreProjective}\hypertarget{FreeModulesAreProjective}{}
Assuming the [[axiom of choice]], a [[free module]] $N \simeq R^{(S)}$ is projective.

\end{lemma}
\begin{proof}
By remark \ref{ProjectiveInSetsIsAxiomOfChoice} and the [[free-forgetful adjunction]].

More explicitly: if $S \in Set$ and $F(S) = R^{(S)}$ is the [[free module]] on $S$, then a module homomorphism $F(S) \to N$ is specified equivalently by a [[function]] $f : S \to U(N)$ from $S$ to the underlying set of $N$, which can be thought of as specifying the images of the unit elements in $R^{(S)} \simeq \oplus_{s \in S} R$ of the ${\vert S\vert}$ copies of $R$.

Accordingly then for $\tilde N \to N$ an epimorphism, the underlying function $U(\tilde N) \to U(N)$ is an epimorphism, and by remark \ref{ProjectiveInSetsIsAxiomOfChoice} the [[axiom of choice]] in [[Set]] says that we have all lifts $\tilde f$ in

\begin{displaymath}
\itexarray{
     && U(\tilde N)
     \\
     & {}^{\tilde f} \nearrow & \downarrow
     \\
     S &\stackrel{f}{\to}& U(N)
  }
  \,.
\end{displaymath}
By [[adjunction]] these are equivalently lifts of module homomorphisms

\begin{displaymath}
\itexarray{
     && \tilde N
     \\
     & \nearrow & \downarrow
     \\
     R^{(S)} &\stackrel{}{\to}& N
  }
  \,.
\end{displaymath}
\end{proof}
\begin{proof}
\textbf{of prop. \ref{ModHasEnoughProjectives}}

For $N \in R Mod$ and $U(N) \in Set$ its underlying set, consider the $R$-linear map

\begin{displaymath}
\left(
    \oplus_{n \in U(n)} R 
  \right)
  \to 
  N
\end{displaymath}
out of the [[direct sum]] of ${\vert U(N)\vert}$ copies of $N$, which sends the unit element $1 \in R_{n}$ of the $n$-labeled copy of $R$ to the corresponding element of $n$ (and is thus fixed on all other elements by $R$-linearity).

This is clearly a [[surjection]] and by lemma \ref{FreeModulesAreProjective} it is a surjection out of a projective object.

\end{proof}
A slightly subtle point is that there is no guarantee that the free module $F U(M)$ is actually projective, unless one assumes some form of the [[axiom of choice]]. Since the axiom of choice is not available in all [[toposes]], one cannot use this procedure in general to construct, say, projective resolutions of [[abelian sheaves]], hence in the abelian category $Ab(E)$ of abelian [[group objects]] in a general [[Grothendieck topos]] $E$ (even though one can construct [[free resolutions]]), such as needed in general in [[abelian sheaf cohomology]].

There are however weak forms of the axiom of choice that hold in many toposes, such as the [[presentation axiom]], aka \emph{[[COSHEP]]}. We have the following result:

\begin{prop}
\label{EnoughWithCOSHEP}\hypertarget{EnoughWithCOSHEP}{}
Let $E$ be a [[W-pretopos]] that satisfies [[COSHEP]]. Then $Ab(E)$ has enough projectives.

\end{prop}
The idea of the proof is that under COSHEP, the underlying object of an abelian group $A$ in $E$ admits an [[epimorphism]] from a projective object $p \colon X \to U(A)$ in $E$. Then the corresponding $F(X) \to A$ is an epimorphism out of a projective in $Ab(E)$, for this map is a composite of epimorphisms

\begin{displaymath}
F(X) \stackrel{F(p)}{\to} F U(A) \stackrel{\varepsilon_A}{\to} A
\end{displaymath}
(the first is epic because left adjoints preserve epis, whereas the second map, the component of the counit $\varepsilon \colon F U \to id$ at $A$, is epic because $U$ is faithful).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[COSHEP]]


\item \textbf{projective object}, [[projective presentation]], [[projective cover]], [[projective resolution]]

\begin{itemize}%
\item [[projective module]]


\item [[internally projective object]]



\end{itemize}

\item [[injective object]], [[injective presentation]], [[injective envelope]], [[injective resolution]]

\begin{itemize}%
\item [[injective module]]

\end{itemize}

\item [[free object]], [[free resolution]]

\begin{itemize}%
\item [[free module]]

\end{itemize}

\item flat object, [[flat resolution]]

\begin{itemize}%
\item [[flat module]]

\end{itemize}


\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

For instance section 4.3 of

\begin{itemize}%
\item [[Pierre Schapira]], \emph{Categories and homological algebra} (2011) (\href{http://people.math.jussieu.fr/~schapira/lectnotes/HomAl.pdf}{pdf})

\end{itemize}
[[!redirects enough projectives]] [[!redirects projective objects]]



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