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\begin{document}

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\section*{final functor}

\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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{generalizations}{Generalizations}\dotfill \pageref*{generalizations} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \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 [[functor]] $F : C \to D$ is \textbf{final}, if we can restrict [[diagram]]s on $D$ to diagrams on $C$ along $F$ without changing their [[colimit]].

Dually, a functor is \textbf{initial} if pulling back diagrams along it does not change the [[limit]]s of these diagrams.

Beware that this property is pretty much unrelated to that of a functor being an [[initial object]] or [[terminal object]] in the [[functor category]] $[C,D]$. The terminology comes instead from the fact that an object $d\in D$ is initial (resp. terminal) just when the corresponding functor $d:1\to D$ is initial (resp. final).

\begin{description}

\item[\textbf{Warning:} In older references (and also some others like [[Higher Topos Theory|HTT]]), final functors are sometimes called \emph{cofinal}, the terminology having been imported from order theory (e.g. [[cofinality]]). However, this is confusing in category theory because usually the prefix ``co-'' denotes dualization. In at least one place (\hyperlink{fb}{Borceux}) this non-dualization was treated as a dualization and the word ``final'' used for the \emph{dual} concept, but in general it seems that the consensus is to use ``final'' for what used to be called ``cofinal'', and ``initial'' for the dual concept (since ``co-final'' would be ambiguous). For example, Johnstone in [[Sketches of an Elephant]] before Proposition B2.5.12 says:] Traditionally, final functors were called `cofinal functors'; but this use of `co' is potentially misleading as it has nothing to do with dualization — it is derived from the Latin `cum' rather than `contra' — and so it is now generally omitted. 
\end{description}
\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{}\hypertarget{}{}
A [[functor]] $F : C \to D$ is \textbf{final} if for every [[object]] $d \in D$ the [[comma category]] $(d/F)$ is non-empty and [[connected category|connected]].

A [[functor]] $F : C \to D$ is \textbf{initial} if the [[opposite category|opposite]] $F^{op} : C^{op} \to D^{op}$ is final, i.e. if for every [[object]] $d \in D$ the [[comma category]] $(F/d)$ is non-empty and [[connected category|connected]].

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

\begin{prop}
\label{}\hypertarget{}{}
Let $F : C \to D$ be a [[functor]]

The following conditions are equivalent.

\begin{enumerate}%
\item $F$ is final.


\item For all functors $G : D \to Set$ the natural [[function]] between [[colimit]]s

\begin{displaymath}
\lim_\to G \circ F \to \lim_{\to} G
\end{displaymath}
is a [[bijection]].


\item For all categories $E$ and all functors $G : D \to E$ the natural [[morphism]] between [[colimit]]s

\begin{displaymath}
\lim_\to G \circ F \to \lim_{\to} G
\end{displaymath}
is a [[isomorphism]].


\item For all functors $G : D^{op} \to Set$ the natural [[function]] between [[limit]]s

\begin{displaymath}
\lim_\leftarrow G \to \lim_\leftarrow G \circ F^{op}
\end{displaymath}
is a [[bijection]].


\item For all categories $E$ and all functors $G : D^{op} \to E$ the natural [[morphism]]

\begin{displaymath}
\lim_\leftarrow G \to \lim_\leftarrow G \circ F^{op}
\end{displaymath}
is an [[isomorphism]].


\item For all $d \in D$

\begin{displaymath}
{\lim_\to}_{c \in C} Hom_D(d,F(c)) \simeq *
  \,.
\end{displaymath}


\end{enumerate}
\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
If $F : C \to D$ is final then $C$ is connected precisely if $D$ is.

\end{prop}
\begin{prop}
\label{Stability}\hypertarget{Stability}{}
If $F_1$ and $F_2$ are final, then so is their composite $F_1 \circ F_2$.

If $F_2$ and the composite $F_1 \circ F_2$ are final, then so is $F_1$.

If $F_1$ is a [[full and faithful functor]] and the composite is final, then both functors seperately are final.

\end{prop}
The first two statements of Proposition \ref{Stability} in fact follow from the stability properties of orthogonal factorization systems:

\begin{prop}
\label{}\hypertarget{}{}
Final functors and \href{http://ncatlab.org/nlab/show/discrete+fibration}{discrete fibrations} form an \href{https://ncatlab.org/nlab/show/orthogonal+factorization+system}{orthogonal factorization system}.

\end{prop}
\hypertarget{generalizations}{}\subsection*{{Generalizations}}\label{generalizations}

The generalization of the notion of final functor from [[category theory]] to [[(∞,1)-category|(∞,1)]]-[[higher category theory]] is described at

\begin{itemize}%
\item [[final (∞,1)-functor]].

\end{itemize}
The characterization of final functors is also a special case of the characterization of [[exact squares]].

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

\begin{example}
\label{}\hypertarget{}{}
If $D$ has a [[terminal object]] then the functor $F : {*} \to D$ that picks that terminal object is final: for every $d \in D$ the [[comma category]] $d/F$ is equivalent to $*$. The converse is also true: if a functor $*\to D$ is final, then its image is a terminal object.

In this case the statement about preservation of colimits states that the colimit over a category with a terminal object is the value of the diagram at that object. Which is also readily checked directly.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
Every [[right adjoint|right]] [[adjoint functor]] is final.

\end{example}
\begin{proof}
Let $(L \dashv R) : C \to D$ be a pair of [[adjoint functors]].To see that $R$ is final, we may for instance check that for all $d \in D$ the comma category $d / R$ is non-empty and connected:

It is non-empty because it contains the [[unit of an adjunction|adjunction unit]] $(L(d), d \to R L (d))$. Similarly, for

\begin{displaymath}
\itexarray{
    && d
    \\
    & {}^{\mathllap{f}}\swarrow && \searrow^{\mathrlap{g}}
    \\
    R(a) &&&& R(b)
  }
\end{displaymath}
two objects, they are connected by a zig-zag going through the unit, by the \hyperlink{http://ncatlab.org/nlab/show/adjoint%20functor#UniversalArrows}{universal factorization property} of adjunctions

\begin{displaymath}
\itexarray{
    && d
    \\
    & \swarrow &\downarrow& \searrow
    \\
    R(a) &\stackrel{R \bar f}{\leftarrow}& R L (d)& \stackrel{R(\bar g)}{\to} & R(b)
  }
  \,.
\end{displaymath}
\end{proof}
\begin{example}
\label{}\hypertarget{}{}
The inclusion $\mathcal{C} \to \tilde \mathcal{C}$ of any category into its [[idempotent completion]] is final.

\end{example}
See at \emph{[[idempotent completion]]} in the section on \emph{\href{Karoubi+envelope#Finality}{Finality}}.

\begin{example}
\label{CoconeUnderCospan}\hypertarget{CoconeUnderCospan}{}
The inclusion of the [[cospan]] [[diagram]] into its [[cocone]]

\begin{displaymath}
\left(
    \itexarray{
       a
       \\
       \downarrow
       \\
       c
       \\
       \uparrow
       \\
       b
    }
  \right)
  \hookrightarrow
  \left(
    \itexarray{
       a
       \\
       \downarrow & \searrow
       \\
       c &\longrightarrow & p  
       \\
       \uparrow & \nearrow
       \\
       b
    }
  \right)
\end{displaymath}
is initial.

\end{example}
\begin{remark}
\label{}\hypertarget{}{}
By the characterization (\href{overcategory#LimitsInSliceViaLimitsOfCoconedDiagram}{here}) of limits in a [[slice category]], this implies that [[fiber products]] in a [[slice category]] are computed as fiber products in the underlying category, or in other words that [[dependent sum]] to the point preserves fiber products.

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

\begin{itemize}%
\item \textbf{final functor}, [[cofinal diagram]]


\item [[homotopy final functor]]


\item [[final (∞,1)-functor]]



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

Section 2.5 of

\begin{itemize}%
\item Kashiwara, Shapira, \emph{[[Categories and Sheaves]]}

\end{itemize}
Section 2.11 of

\begin{itemize}%
\item [[Francis Borceux]], \emph{Handbook of categorical algebra 1, Basic category theory}

\end{itemize}
Notice that this says ``final functor'' for the version under which limits are invariant.

Section IX.3 of

\begin{itemize}%
\item [[Saunders Mac Lane]], \emph{[[Categories for the Working Mathematician]]}

\end{itemize}
[[!redirects cofinal functor]] [[!redirects cofinal functors]] [[!redirects final functor]] [[!redirects final functors]] [[!redirects initial functor]] [[!redirects initial functors]]



\end{document}