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\begin{document}

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\section*{weak limit}

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

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

[[!include category theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{weak_pullbacks}{Weak pullbacks}\dotfill \pageref*{weak_pullbacks} \linebreak
\noindent\hyperlink{weak_terminal_objects}{Weak terminal objects}\dotfill \pageref*{weak_terminal_objects} \linebreak
\noindent\hyperlink{projective_objects_and_exact_completion}{Projective objects and exact completion}\dotfill \pageref*{projective_objects_and_exact_completion} \linebreak
\noindent\hyperlink{RelationToHomotopyLimits}{Relation to homotopy limits}\dotfill \pageref*{RelationToHomotopyLimits} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \textbf{weak limit} for a [[diagram]] in a [[category]] is a [[cone]] over that diagram which satisfies the existence property of a [[limit]] but not necessarily the uniqueness. The dual concept is a [[weak colimit]].

Beware that ``weak'' here does \textbf{not} correspond to that in ``[[weak n-category]]'', in particular it does \textbf{not} refer to [[homotopy limits]]. Nevertheless, there is a relation, see \hyperlink{RelationToHomotopyLimits}{below}. It is due to this relation that weak limits in [[homotopy categories]] play a key role in the [[Brown representability theorem]].

\hypertarget{weak_pullbacks}{}\subsection*{{Weak pullbacks}}\label{weak_pullbacks}

A \textbf{weak pullback} of a [[cospan]]

\begin{displaymath}
A\overset{f}{\to} C \overset{g}{\leftarrow} B
\end{displaymath}
(in some [[category]]) is a [[commutative diagram|commutative square]]

\begin{displaymath}
\itexarray{ 
    P & \overset{p}{\to} & A
    \\
    {}^{\mathllap{q}} \downarrow && \downarrow^{\mathrlap{f}}
    \\
    B & \overset{g}{\to} & C
  }
\end{displaymath}
such that for every commuting square

\begin{displaymath}
\itexarray{ X & \overset{x}{\to} & A\\
   {}^{\mathllap{y}} \downarrow && \downarrow^{\mathrlap{f}}\\
   B & \overset{g}{\to} & C}
\end{displaymath}
there exists a morphism $h: \colon X\to P$, not necessarily unique, such that $x = h p$ and $y = h q$;

If the actual [[pullback]] $A \underset{C}{\times}B$ exists, then this condition means equivalently that the universal morphism

\begin{displaymath}
P \longrightarrow A \underset{C}{\times}B
\end{displaymath}
is a [[split epimorphism]].

\hypertarget{weak_terminal_objects}{}\subsection*{{Weak terminal objects}}\label{weak_terminal_objects}

Every [[inhabited set]] is a weak [[terminal object]] in [[Set]], since there always exists a [[function]] from any [[set]] to any inhabited set. But only a [[singleton]] is a terminal object.

\hypertarget{projective_objects_and_exact_completion}{}\subsection*{{Projective objects and exact completion}}\label{projective_objects_and_exact_completion}

In any category with finite limits and [[projective object|enough projectives]], the full [[subcategory]] of [[projective object]]s has weak finite limits. For example, given a cospan $A\overset{f}{\to} C \overset{g}{\leftarrow} B$ of projective objects, let $P\to A\times_C B$ be a projective cover of the actual pullback; then any square

\begin{displaymath}
\itexarray{ X & \overset{x}{\to} & A\\
  ^y \downarrow && \downarrow ^f\\
  B & \overset{g}{\to} & C}
\end{displaymath}
with $X$ projective induces a morphism $X\to A\times_C B$, which lifts to a morphism $X\to P$ since $X$ is projective.

Conversely, from any category with weak finite limits one can construct an [[exact category|exact completion]] in which the original category sits as the projective objects, and the exact categories constructible in this way are precisely those having enough projectives.

\hypertarget{RelationToHomotopyLimits}{}\subsection*{{Relation to homotopy limits}}\label{RelationToHomotopyLimits}

Unlike usages of `weak' in terms like [[weak n-category]], a weak limit is not be like a [[homotopy limit]] or a [[2-limit]], which satisfy uniqueness (as well as existence) albeit only up to higher [[homotopy|homotopies]] or [[weak equivalence|equivalences]].

However, some homotopy limits induce the corresponding type of weak limit in the corresponding [[homotopy category]]. For example, suppose that

\begin{displaymath}
\itexarray{ P & \overset{p}{\to} & A\\
  ^q \downarrow && \downarrow ^f\\
  B & \overset{g}{\to} & C}
\end{displaymath}
is a homotopy pullback in some category $M$ having a notion of [[homotopy]], such as a [[model category]]. In particular, this square commutes up to homotopy, and thus it commutes in the homotopy category $Ho(M)$. Then any square

\begin{displaymath}
\itexarray{ X & \overset{x}{\to} & A\\
  ^y \downarrow && \downarrow ^f\\
  B & \overset{g}{\to} & C}
\end{displaymath}
that commutes in $Ho(M)$ commutes up to homotopy in $M$, and thus (by the (``local'') universal property of homotopy pullbacks), there is a map $h:X\to P$ and homotopies $p h \simeq x$ and $q h\simeq y$; thus the given square is a weak pullback in $Ho(M)$. While the universal property of a homotopy pullback means that $h$ is unique up to homotopy, this is only true for a given \emph{choice} of homotopy $f x \simeq g y$, and different such homotopies can induce inequivalent $h$`s. Thus in $Ho(M)$, which remembers only the \emph{existence} of homotopies, we have only a weak pullback.

Note, though, that not \emph{all} homotopy limits produce weak limits in the homotopy category, because in general it will not be possible to lift a cone that commutes in $Ho(M)$ to a cone that commutes up to \emph{coherent} homotopy in $M$. However, in ``simple'' cases such as pullbacks, products, equalizers, sequential inverse limits, and so on, this is always true (and it will be true whenever the diagram category is a [[quiver]]). On the other hand, homotopy products in $M$ give actual (not weak) products in $Ho(M)$, since there are no homotopies necessary.

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

\begin{itemize}%
\item [[weak multilimit]]

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

\begin{itemize}%
\item [[Peter Freyd]], \emph{Representations in abelian categories} 1966 Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965) pp. 95--120 Springer, New York

\end{itemize}
[[!redirects weak limits]]

[[!redirects weak pullback]] [[!redirects weak pullbacks]]

[[!redirects weak pushout]] [[!redirects weak pushouts]]

[[!redirects weak finite limit]] [[!redirects weak finite limits]]



\end{document}