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\begin{document}

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\section*{weighted limit}

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

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

[[!include category theory - contents]]

\hypertarget{enriched_category_theory}{}\paragraph*{{Enriched category theory}}\label{enriched_category_theory}

[[!include enriched category theory contents]]

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

[[!include infinity-limits - contents]]

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

\noindent\hyperlink{contents}{Contents}\dotfill \pageref*{contents} \linebreak
\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{homotopy_limits}{Homotopy limits}\dotfill \pageref*{homotopy_limits} \linebreak
\noindent\hyperlink{homotopy_pullback}{Homotopy pullback}\dotfill \pageref*{homotopy_pullback} \linebreak
\noindent\hyperlink{references_for_homotopy_limits_in_terms_of_weighted_limits}{References for homotopy limits in terms of weighted limits}\dotfill \pageref*{references_for_homotopy_limits_in_terms_of_weighted_limits} \linebreak
\noindent\hyperlink{related_pages}{Related pages}\dotfill \pageref*{related_pages} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The notion of \emph{weighted limit} is naturally understood from the point of view on [[limits]] as described at [[representable functor]].

Weighted limits make sense and are considered in the general context of $V$-[[enriched category theory]], but restrict attention to $V=$ [[Set]] for the moment, in order to motivate the concept.

Let $K$ denote the small category which indexes [[diagrams]] over which we want to consider limits and eventually weighted limits. Notice that for

\begin{displaymath}
F : K \to Set
\end{displaymath}
a [[Set]]-valued functor on $K$, the limit of $F$ is canonically identified simply with the set of [[cones]] with tip the singleton set $pt = \{\bullet\}$:

\begin{displaymath}
lim F = [K,Set](\Delta pt, F)
  \,.
\end{displaymath}
This means, more generally, that for

\begin{displaymath}
F : K \to C
\end{displaymath}
a functor with values in an arbitrary category $C$, the object-wise limit of the functor $F$ under the [[Yoneda embedding]]

\begin{displaymath}
C(-,F(-)) : K \stackrel{F}{\to} C \stackrel{Y}{\to} Set^{C^{op}}
\end{displaymath}
which appears in the discussion in example 1 at [[representable functor]] can be expressed by the right side of

\begin{displaymath}
lim C(-,F(-)) = [K,Set](\Delta pt, C(-,F(-)))
  \,.
\end{displaymath}
(Recall that this is the limit over the diagram $C(-,F(-)) : K \to Set^{C^{op}}$ which, if [[representable functor|representable]] defines the desired limit of $F$.)

The \textbf{idea} of weighted limits is to

\begin{enumerate}%
\item allow in the formula above the particular functor $\Delta pt$ to be replaced by any other functor $W : K \to Set$;


\item to generalize everything straightforwardly from the [[Set]]-[[enriched category|enriched]] context to arbitrary $V$-enriched contexts.



\end{enumerate}
The idea is that the weight $W : K \to V$ encodes the way in which one generalizes the concept of a [[cone]] over a diagram $F$ (that is, something with just a tip from which morphisms are emanating down to $F$) to a more intricate structure over the diagram $F$. For instance in the application to [[homotopy limits]] discussed below with $V$ set to [[SimpSet]] the weight is such that it ensures that not only 1-morphisms are emanating from the tip, but that any triangle formed by these is filled by a 2-cell, every tetrahedron by a 3-cell, etc.

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

Let $V$ be a [[closed category|closed]] [[symmetric monoidal category]]. All categories in the following are $V$-[[enriched category|enriched categories]], all functors are $V$-functors.

A \textbf{weighted limit} over a functor

\begin{displaymath}
F : K \to C
\end{displaymath}
with respect to a \emph{weight} or \emph{indexing type} functor

\begin{displaymath}
W : K \to V
\end{displaymath}
is, if it exists, the object $lim^W F \in C$ which [[representable functor|represents]] the functor (in $c \in C$)

\begin{displaymath}
[K,V](W, C(c,F(-))) : C^{op} \to V
  \,,
\end{displaymath}
i.e. such that for all objects $c \in C$ there is an isomorphism

\begin{displaymath}
C(c, lim^W F)
  \simeq
  [K,V](W(-), C(c,F(-)))
\end{displaymath}
natural in $c$.

(Here $[K,V]$ is the $V$-[[enriched functor category]], as usual.)

In particular, if $C = V$ itself, then we get the direct formula

\begin{displaymath}
lim^W F \simeq [K,V](W,F)
  \,.
\end{displaymath}
This follows from the above by the [[coend]] manipulation

\begin{displaymath}
\begin{aligned}
    [K,V](W(-),C(c,F(-)))
    &:=
    \int_{k \in K} V(W(k),V(c,F(k)))
    \\
    &  \simeq 
      \int_{k \in K} V(c,V(W(k),F(k))
    \\
    &  \simeq 
      V(c, \int_{k \in K} V(W(k),F(k))   
    \\
    & =:
      V(c, [K,V](W,F))   
    \,.
  \end{aligned}
\end{displaymath}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{homotopy_limits}{}\subsubsection*{{Homotopy limits}}\label{homotopy_limits}

For $V$ some category of higher structures, the \emph{local} definition of [[homotopy limit]] over a diagram $F : K \to C$ replaces the ordinary notion of [[cone]] over $F$ by a higher cone in which all triangles of 1-morphisms are filled by 2-cells, all tetrahedra by 3-cells, etc.

One can convince oneself that for the choice of [[SimpSet]] for $V$ this is realized in terms of the weighted limit $lim^W F$ with the weight $W$ taken to be

\begin{displaymath}
W : K \to \Simp\Set
\end{displaymath}
\begin{displaymath}
W : k \mapsto N(K/k)
   \,,
\end{displaymath}
where $K/k$ denotes the [[over category]] of $K$ over $k$ and $N(K/k)$ denotes its [[nerve]].

This leads to the classical definition of homotopy limits in $\Simp\Set$-enriched categories due to

\begin{itemize}%
\item A.K. Bousfield and D.M. Kan, \emph{Homotopy limits, completions, and localizations}

\end{itemize}
See for instance also

\begin{itemize}%
\item Nicola Gambino, \emph{Weighted limits in simplicial homotopy theory} (\href{http://www.crm.cat/Publications/08/Pr790.pdf}{pdf} or \href{http://www.math.unipa.it/%7Engambino/Research/Papers/weighted.pdf}{pdf})

\end{itemize}
In some nice cases the weight $N(K/-)$ can be replaced by a simpler weight; an example is discussed at [[Bousfield-Kan map]].

\hypertarget{homotopy_pullback}{}\subsubsection*{{Homotopy pullback}}\label{homotopy_pullback}

For instance in the case that $K = \{r \to t \leftarrow s\}$ is the [[pullback]] diagram we have

\begin{displaymath}
W(r) = \{r\}
\end{displaymath}
\begin{displaymath}
W(s) = \{s\}
\end{displaymath}
\begin{displaymath}
W(t) = N( \{r \to t \leftarrow s\} )
\end{displaymath}
and $W(r \to t) : \{r\} \to \{r \to t \leftarrow s\}$ injects the vertex $r$ into $\{r \to t \leftarrow s\}$ and similarly for $W(s \to t)$.

This implies that for $F : K \to C$ a pullback diagram in the [[SimpSet]]-eriched category $C$, a $W$-weighted [[cone]] over $F$ with tip some object $c  \in C$, i.e. a natural transformation

\begin{displaymath}
W \Rightarrow C(c, F(-))
\end{displaymath}
is

\begin{itemize}%
\item over $r$ a ``morphism'' from the tip $c$ to $F(r)$ (i.e. a vertex in the Hom-simplicial set $C(c,F(r))$);


\item similarly over $s$;


\item over $t$ three ``morphisms'' from $c$ to $F(t)$ together with 2-cells between them (i.e. a 2-[[horn]] in the Hom-simplicial set $C(c,F(t))$)


\item such that the two outer morphisms over $t$ are identified with the morphisms over $r$ and $s$, respectively, postcompoised with the morphisms $F(r \to t)$ and $F(s \to t)$, respectively.



\end{itemize}
So in total such a $W$-weighted cone looks like

\begin{displaymath}
\itexarray{
     &&& c
     \\
     & \swarrow &\Rightarrow& \downarrow    
      &\Leftarrow& \searrow
     \\
     F(r) 
      &&
     \stackrel{F(r \to t)}{\to}
      & 
     F(t)
      & 
     \stackrel{F(s \to t)}{\leftarrow}
     &&
     F(s)
  }
\end{displaymath}
as one would expect for a ``homotopy cone''.

\hypertarget{references_for_homotopy_limits_in_terms_of_weighted_limits}{}\subsection*{{References for homotopy limits in terms of weighted limits}}\label{references_for_homotopy_limits_in_terms_of_weighted_limits}

Details of this are discussed for instance in the book

\begin{itemize}%
\item Hirschhorn, \emph{Model categories and their localization}

\end{itemize}
To compare with the above discussion notice that

\begin{itemize}%
\item The functor

\begin{displaymath}
W := N(K/-)
\end{displaymath}
is discussed there in definition 14.7.8 on p. 269.


\item the $V$-enriched hom-category $[K,V]$ which on $V$-functors $S,T$ is the [[end]] $[K,V](S,T) = \int_{k \in K} V(S(k), T(k))$ appears as $hom^K(S,T)$ in definition 18.3.1 (see bottom of the page).


\item for $V$ set to [[SimpSet]] the above definition of homotopy limit appears in example 18.3.6 (2).



\end{itemize}
\hypertarget{related_pages}{}\section*{{Related pages}}\label{related_pages}

\begin{itemize}%
\item [[strict 2-limit]]


\item [[saturated class of limits]]


\item [[weighted colimit]]



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

A standard reference is

\begin{itemize}%
\item [[Max Kelly]], \emph{Basic concepts of enriched category theory}, \href{http://www.emis.de/journals/TAC/reprints/articles/10/tr10.pdf#page=37}{section 3.1, p. 37}

\end{itemize}
In

\begin{itemize}%
\item [[Emily Riehl]], \emph{Weighted limits and colimits} (2008) (\href{http://www.math.jhu.edu/~eriehl/weighted.pdf}{pdf}) is given an account of lectures by [[Mike Shulman]] on the subject. The definition appears there as \href{http://www.math.jhu.edu/~eriehl/weighted.pdf#page=4}{definition 3.1, p. 4} (in a form a bit more general than the one above).

\end{itemize}
The analogous notion of weighted [[(infinity,1)-limit]] is discussed in

\begin{itemize}%
\item Martina Rovelli, \emph{Weighted limits in an (∞,1)-category}, 2019, \href{https://arxiv.org/abs/1902.00805}{arxiv:1902.00805}

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



\end{document}