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\begin{document}

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\section*{absolute colimit}

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

\hypertarget{limits_and_colimits}{}\paragraph*{{Limits and colimits}}\label{limits_and_colimits}

[[!include infinity-limits - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{particular_absolute_colimits}{Particular absolute colimits}\dotfill \pageref*{particular_absolute_colimits} \linebreak
\noindent\hyperlink{weights_for_absolute_colimits}{Weights for absolute colimits}\dotfill \pageref*{weights_for_absolute_colimits} \linebreak
\noindent\hyperlink{characterizations}{Characterizations}\dotfill \pageref*{characterizations} \linebreak
\noindent\hyperlink{particular_absolute_colimits_2}{Particular absolute colimits}\dotfill \pageref*{particular_absolute_colimits_2} \linebreak
\noindent\hyperlink{CharacterizationsWeightsForAbsoluteColimits}{Weights for absolute colimits}\dotfill \pageref*{CharacterizationsWeightsForAbsoluteColimits} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{particular_absolute_colimits_3}{Particular absolute colimits}\dotfill \pageref*{particular_absolute_colimits_3} \linebreak
\noindent\hyperlink{WExampleseightsForAbsoluteColimits}{Weights for absolute colimits}\dotfill \pageref*{WExampleseightsForAbsoluteColimits} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

An \textbf{absolute colimit} is a [[colimit]] which is [[preserved limit|preserved]] by any [[functor]] whatsoever. In general this happens because the colimit is a colimit for purely ``diagrammatic'' reasons. The notion is most important in [[enriched category theory]].

Of course, there is a dual notion of \textbf{absolute limit}, but it is used less frequently.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

The term ``absolute colimit'' is actually used for two closely related, but distinct, notions.

\hypertarget{particular_absolute_colimits}{}\subsubsection*{{Particular absolute colimits}}\label{particular_absolute_colimits}

\begin{defn}
\label{DefParticular}\hypertarget{DefParticular}{}
A particular [[colimit]] diagram in a particular [[category]] $C$ is an \textbf{absolute colimit} if it is [[preserved limit|preserved]] by every [[functor]] with domain $C$.

\end{defn}
This definition makes sense also in [[enriched category theory]]: for any $V$, a [[weighted colimit]] in a particular $V$-[[enriched category]] $C$ is an \textbf{absolute colimit} if it is preserved by every $V$-functor with domain $C$.

Note, however, that a [[conical colimit]] in a $V$-category $C$ may be preserved by all $V$-functors without being preserved by all unenriched functors on the underlying ordinary category $C_o$. Thus, for clarity we may speak of a colimit being \textbf{$V$-absolute}.

Generalizing in a different direction, absolute colimits in $Set$-enriched categories can be regarded as the particular case of [[postulated colimit]]s in [[sites]] where the site has the [[trivial topology]].

\hypertarget{weights_for_absolute_colimits}{}\subsubsection*{{Weights for absolute colimits}}\label{weights_for_absolute_colimits}

\begin{defn}
\label{DefWeight}\hypertarget{DefWeight}{}
For a given $V$, a [[weighted colimit|weight]] $\Phi\colon D^{op} \to V$ for colimits is an \textbf{absolute weight}, or a \textbf{weight for absolute colimits}, if $\Phi$-weighted colimits in \emph{all} $V$-categories are preserved by all $V$-functors.

\end{defn}
Absolute colimits of this sort are also called \textbf{Cauchy colimits}. A $V$-[[category]] which admits all absolute colimits --- that is, all $V$-weighted colimits whose weights are absolute -- is called [[Cauchy complete category|Cauchy complete]]. By the characterization below, it is equivalent to admit \emph{limits} weighted by all weights  for absolute limits.

\hypertarget{characterizations}{}\subsection*{{Characterizations}}\label{characterizations}

Both types of absolute colimits admit pleasant characterizations.

\hypertarget{particular_absolute_colimits_2}{}\subsubsection*{{Particular absolute colimits}}\label{particular_absolute_colimits_2}

For a particular [[cocone]] $\mu \colon F \to \Delta A$ under a functor $F\colon I\to C$ (all in the Set-enriched world), the following are equivalent:

\begin{itemize}%
\item $\mu$ is an absolute colimiting cocone.


\item $\mu$ is a colimiting cocone which is is preserved by the [[Yoneda embedding]] $C \hookrightarrow [C^{op},Set]$.


\item $\mu$ is a colimiting cocone which is preserved by the representable functors $C(F(i),-)\colon C\to Set$ (for all $i\in I$) and $C(A,-)\colon C\to Set$.


\item There exists $i_0\in I$ and $d_0\colon A\to F(i_0)$ such that

\begin{enumerate}%
\item For every $i\in I$, $d_0 \circ \mu_i$ and $1_{F(i)}$ are in the same connected component of the [[comma category]] $(F(i) / F)$.
\item $\mu_{i_0} \circ d_0 = 1_{A}$.

\end{enumerate}


\end{itemize}
The equivalence of the first two is basically because the Yoneda embedding is the [[free cocompletion]] of $C$. The third clearly follows from the second. The fourth follows from the third by inspecting exactly what preservation by those representables means in terms of colimits in [[Set]] (as is explained in more detail in the special case of [[absolute coequalizers]]). Finally, it is straightforward to check that the fourth implies that $\mu$ is colimiting, and it is clearly a property preserved by any functor.

Note that in particular, the fourth condition implies that $A$ is a [[retract]] of $F(i_0)$. Also, the first half of the fourth condition by itself characterizes absolute [[weak colimits]].

It is also possible to prove directly that the third condition implies the first two, without extracting the fourth condition along the way. Namely, Let $B$ be the full subcategory of $C$ consisting of the objects $F(i)$ and $A$. Then $F$ defines a functor $I\to B$; call it $F'$. Note that $A$ is also the colimit of $F'$ in $B$. Moreover, by the equivalence of the first two conditions, $A$ is an \emph{absolute} colimit of $F'$, since by hypothesis it is preserved by all representable functors out of $B$. Therefore, this colimit is in particular preserved by the inclusion $B\hookrightarrow C$, along with its composite with any functor out of $C$; so $A$ is an absolute colimit of $F$.

\hypertarget{CharacterizationsWeightsForAbsoluteColimits}{}\subsubsection*{{Weights for absolute colimits}}\label{CharacterizationsWeightsForAbsoluteColimits}

Let $V$ be a B\'e{}nabou [[cosmos]] and $J\colon K &#x21F8; A$ a $V$-[[profunctor]]. Then the following are equivalent:

\begin{itemize}%
\item $J$ is a weight for absolute colimits (i.e. $J$-weighted colimits in any $V$-category are preserved by all $V$-functors)


\item $J$ has a [[right adjoint]] $J^*$ in the [[bicategory]] $V$-[[Prof]].


\item There is a weight $J^*\colon A &#x21F8; K$ such that $J$-weighted colimits coincide naturally with $J^*$-weighted limits.



\end{itemize}
\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\hypertarget{particular_absolute_colimits_3}{}\subsubsection*{{Particular absolute colimits}}\label{particular_absolute_colimits_3}

Of course, every colimit weighted by a weight for absolute colimits is itself a particular absolute colimit. But it may also happen that a particuclar colimit may be absolute without all colimits of that shape being absolute. For example (in ordinary category theory, with $V=Set$):

\begin{itemize}%
\item [[split coequalizers]] are absolute, and figure in Beck's [[monadicity theorem]].
\item More generally (of course), [[absolute coequalizers]] are absolute.
\item [[absolute pushouts]] appear in [[elegant Reedy categories]].

\end{itemize}
We can also say something about non-examples.

\begin{itemize}%
\item Initial objects (in $Set$-enriched categories) are \emph{never} absolute. If $0$ is an initial object, then it is never preserved by the representable functor $C(0,-)\colon C \to Set$.


\item Similarly, coproducts in $Set$-enriched categories are never absolute.



\end{itemize}
\hypertarget{WExampleseightsForAbsoluteColimits}{}\subsubsection*{{Weights for absolute colimits}}\label{WExampleseightsForAbsoluteColimits}

In ordinary $Set$-enriched category theory there are not very many weights for absolute colimits, but we have

\begin{itemize}%
\item [[split idempotents]].

\end{itemize}
In fact, this example is ``universal,'' in that an ordinary category is Cauchy complete iff it has split idempotents, although not every absolute colimit ``is'' the splitting of an idempotent. More precisely, the class of absolute $Set$-limits is the [[saturated class of limits|saturation]] of idempotent-splittings.

In [[enriched category theory]] there can be more types of absolute colimits. For instance:

\begin{itemize}%
\item in categories with [[zero morphisms]] (that is, enriched over [[pointed sets]]), [[initial objects]] are absolute.


\item in [[Ab-enriched categories]] (or, more generally, categories enriched over commutative [[monoids]]), finite [[biproducts]] are absolute. Finite biproducts and splitting of idempotents together are ``universal'' absolute colimits for Ab-enrichment.


\item in [[SupLat]]-enriched categories, arbitrary small biproducts are absolute, and together with splitting of idempotents these generate all absolute colimits.


\item in [[dg-categories]] (or more generally, categories enriched over [[graded set|graded sets]]), \emph{shifts/suspensions} and [[mapping cones]] are absolute.


\item in Lawvere [[metric spaces]], limits of Cauchy sequences are absolute. This is the origin of the name ``Cauchy colimit.''


\item in categories [[bicategory-enriched category|enriched]] over the [[bicategory]] (or [[double category]]) of relations in a [[site]], \emph{gluings} are absolute. In this case the enriched categories can roughly be identified with [[separated presheaves]] and the Cauchy-complete ones with [[sheaves]].


\item in categories enriched over [[rational number|rational]] [[vector spaces]], quotients by finite [[group actions]] are absolute.



\end{itemize}
New kinds of absolute (co)limits also arise in [[higher category theory]].

\begin{itemize}%
\item for [[(∞,1)-categories]] (enriched over $\infty$-groupoids), splitting of idempotents is a universal absolute colimit.


\item in [[stable (∞,1)-categories]] (which are enriched, in the $(\infty,1)$-categorical sense, over the $(\infty,1)$-category of [[spectra]]), initial objects and [[pushouts]] are absolute, and therefore so are all finite colimits.



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

\begin{itemize}%
\item [[Ross Street]], \emph{Absolute colimits in enriched categories}, \href{http://www.numdam.org/numdam-bin/fitem?id=CTGDC_1983__24_4_377_0}{Cahiers 1983}


\item [[Robert Pare]], \emph{On absolute colimits}, J. Alg. 19 (1971), 80-95.



\end{itemize}
[[!redirects absolute colimits]] [[!redirects absolute limit]] [[!redirects absolute limits]] [[!redirects Cauchy colimit]] [[!redirects Cauchy colimits]]



\end{document}