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\section*{2-limit}

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

\hypertarget{limits}{}\paragraph*{{$\infty$-Limits}}\label{limits}

[[!include infinity-limits - contents]]


\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{2category_theory}{}\paragraph*{{2-Category theory}}\label{2category_theory}

[[!include 2-category theory - 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{Terminology}{Strictness and terminology}\dotfill \pageref*{Terminology} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{2limits_over_diagrams_of_special_shape}{2-limits over diagrams of special shape}\dotfill \pageref*{2limits_over_diagrams_of_special_shape} \linebreak
\noindent\hyperlink{finite_2limits}{Finite 2-Limits}\dotfill \pageref*{finite_2limits} \linebreak
\noindent\hyperlink{(2,1)limit}{$(2,1)$-limits}\dotfill \pageref*{(2,1)limit} \linebreak
\noindent\hyperlink{lax}{Lax limits}\dotfill \pageref*{lax} \linebreak
\noindent\hyperlink{2ColimitsInCat}{2-Colimits in $Cat$}\dotfill \pageref*{2ColimitsInCat} \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{2-limit} is the type of [[limit]] that is appropriate in a (weak) [[2-category]]. (Since general 2-categories are often called \emph{[[bicategories]]}, 2-limits are often called \emph{[[bilimits]]}.)

There are three notable changes when passing from ordinary 1-limits to 2-limits:

\begin{enumerate}%
\item In order to satisfy the [[principle of equivalence]], the ``cones'' in a 2-limit are required to commute only up to [[2-morphism|2-isomorphism]].


\item The [[universal property]] of the limit is expressed by an [[equivalence of categories]] rather than a [[bijection]] of [[sets]]. This means that

\begin{enumerate}%
\item every other cone over the diagram that commutes up to isomorphism factors through the limit, up to isomorphism, and


\item every transformation \emph{between} cones also factors through a 2-cell in the limit. We will give some examples below.



\end{enumerate}

\item Since 2-categories are [[enriched category|enriched]] over [[Cat]] (this is precise in the [[strict 2-category|strict]] case, and [[bicategory|weakly]] true otherwise), $Cat$-[[weighted limit]]s become important. This means that both the diagrams we take limits of and the shape of ``cones'' that limits represent can involve $2$-cells as well as $1$-cells.



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

Let $K$ and $D$ be [[2-categories]], and $J\colon D\to Cat$ and $F\colon D\to K$ be [[2-functors]]. A \textbf{$J$-weighted (2-)limit of $F$} is an object $L\in K$ equipped with a [[pseudonatural equivalence]]

\begin{displaymath}
K(X,L) \simeq [D,Cat](J,K(X,F-)).
\end{displaymath}
where $[D,Cat]$ denotes the 2-category of [[2-functors]] $D\to Cat$, [[pseudonatural transformations]] between them, and [[modifications]] between those.

A 2-limit in the [[opposite 2-category]] $K^{op}$ is called a \textbf{2-colimit} in $K$. Everything below applies dually to 2-colimits, the higher analogues of [[colimits]]. (But somebody might want to make a separate page that gives appropriate examples of these.)

\hypertarget{Terminology}{}\subsubsection*{{Strictness and terminology}}\label{Terminology}

If $K$ and $D$ are [[strict 2-categories]], $J$ and $F$ are [[strict 2-functors]], and if we replace this pseudonatural equivalence by a (strictly 2-natural) isomorphism \emph{and} the 2-category $[D,Cat]$ by the 2-category $[D,Cat]_{strict}$ of strict 2-functors and strict 2-natural transformations, then we obtain the definition of a \textbf{[[strict 2-limit]]}. This is precisely a Cat-weighted limit in the sense of ordinary [[enriched category]] theory. See [[strict 2-limit]] for details.

On the other hand, if $K$, $D$, $J$, and $F$ are strict as above, and we replace the equivalence by an isomorphism but keep the weak meaning of $[D,Cat]$, then we obtain the notion of a \textbf{strict pseudolimit}. Strict pseudolimits are, in particular, 2-limits, whereas strict 2-limits are not always (although some, such as [[PIE-limits]] and [[flexible limits]], are). In a strict 2-category, these types of strict limits are often technically useful in constructing the ``up-to-isomorphism'' 2-limits we consider here.

When we know we are working in a (weak) 2-category, the only type of limit that makes sense is a (non-strict) 2-limit. Therefore, we usually call these simply ``limits.'' To emphasize the distinction with the strict 2-limits in a strict 2-category, the ``up-to-isomorphism'' 2-limits were historically often called \emph{bilimits} (by analogy with [[bicategory]] for ``weak 2-category''). However, this terminology is somewhat unfortunate, not only because it doesn't generalize well to $n$, but because it leads to words like ``biproduct,'' which also has the [[biproduct|completely unrelated meaning]] of an object that is both a product and a coproduct (which is common in [[additive category|additive categories]]).

Unfortunately, we probably shouldn't use ``weak limit'' to emphasize the ``up-to-isomorphism'' nature of these limits, because that also has the [[weak limit|completely unrelated meaning]] of an object in a 1-category satisfying the existence, but not the uniqueness property of an ordinary limit.

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

\hypertarget{2limits_over_diagrams_of_special_shape}{}\subsubsection*{{2-limits over diagrams of special shape}}\label{2limits_over_diagrams_of_special_shape}

Any ordinary type of limit can be ``2-ified'' by boosting its ordinary universal property up to a 2-categorical one. In the following examples we work in a 2-category $K$.

\begin{itemize}%
\item A \textbf{[[terminal object]]} in $K$ is an object 1 such that $K(X,1)$ is equivalent to the [[terminal category]] for any object $X$. This means that for any $X$ there is a morphism $X\to 1$ and for any two morphisms $f,g:X\to 1$ there is a unique morphism $f\to g$, and this morphism is an isomorphism.


\item A \textbf{[[product]]} of two objects $A,B$ in $K$ is an object $A\times B$ together with a natural equivalence of categories $K(X,A\times B) \simeq K(X,A)\times K(X,B)$. This means that we have projections $p:A\times B\to A$ and $q:A\times B\to B$ such that (1) for any $f:X\to A$ and $g:X\to B$, there exists an $h:X\to A\times B$ and isomorphisms $p h\cong f$ and $q h\cong g$, and (2) for any $h,k:X\to A\times B$ and 2-cells $\alpha:p h \to p k$ and $\beta: q h \to q k$, there exists a unique $\gamma:h \to k$ such that $p \gamma = \alpha$ and $q \gamma = \beta$.


\item A \textbf{[[pullback]]} of a [[co-span]] $A \overset{f}{\to} C \overset{g}{\leftarrow} B$ consists of an object $A\times_C B$ and projections $p:A\times_C B\to A$ and $q:A\times_C B\to B$ together with an isomorphism $\phi:f p \cong g q$, such that (1) for any $m:X\to A$ and $n:X\to B$ with an isomorphism $\psi:f m \cong g n$, there exists an $h:X\to A\times_C B$ and isomorphisms $\alpha:p h \cong m$ and $\beta:q h \cong n$ such that $g\beta . \phi h . f \alpha^{-1} = \psi$, and (2) given any two morphisms $h,k:X\to A\times_C B$ and 2-cells $\mu:p h \to p k$ and $\nu:q h \to q k$ such that $f \mu = g \nu$ (modulo the given isomorphism $f p \cong g q$), i.e., $\phi k . f\mu = g\nu . \phi h$, there exists a unique 2-cell $\gamma:h\to k$ such that $p \gamma = \mu$ and $q \gamma = \nu$. This is sometimes called the \emph{pseudo-pullback} but that term more properly refers to a particular [[strict 2-limit]].


\item An \textbf{[[equalizer]]} of $f,g:A\to B$ consists of an object $E$ and a morphism $e:E\to A$ together with an isomorphism $f e \cong g e$, which is universal in a sense the reader should now be able to write down. This is sometimes called the \emph{pseudo-equalizer} but that term more properly refers to a particular [[strict 2-limit]]. Note that frequently, such as in the construction of all limits from basic ones, equalizers need to be replaced by [[descent object]]s.



\end{itemize}
There are also various important types of 2-limits that involve 2-cells in the diagram shape or in the weight, and hence are not just ``boosted-up'' versions of 1-limits.

\begin{itemize}%
\item The \textbf{[[comma object]]} of a cospan $A \overset{f}{\to} C \overset{g}{\leftarrow} B$ is a universal object $(f/g)$ and projections $p:(f/g)\to A$ and $q:(f/g)\to B$ together with a transformation (not an isomorphism) $f p \to g q$. In [[Cat]], [[comma objects]] are [[comma category|comma categories]]. Comma objects are sometimes called \emph{lax pullbacks} but that term more properly refers to the lax version of a pullback; see ``lax limits'' below.


\item The \textbf{[[inserter]]} of a pair of parallel arrows $f,g:A \;\rightrightarrows\; B$ is a universal object $I$ equipped with a map $i:I\to A$ and a 2-cell $f i \to g i$.


\item The \textbf{[[equifier]]} of a pair of parallel 2-cells $\alpha,\beta: f\to g: A\to B$ is a universal object $E$ equipped with a map $e:E\to A$ such that $\alpha e = \beta e$.


\item The \textbf{[[inverter]]} of a 2-cell $\alpha:f\to g:A\to B$ is a universal object $V$ with a map $v:V\to A$ such that $\alpha v$ is invertible.


\item The \textbf{[[power]]} of an object $A$ by a category $C$ is a universal object $A^C$ equipped with a functor $C\to K(A^C,A)$. Of particular importance is the case when $C$ is the [[walking arrow]] $\mathbf{2}$.



\end{itemize}
\hypertarget{finite_2limits}{}\subsubsection*{{Finite 2-Limits}}\label{finite_2limits}

A 2-limit is called \textbf{finite} if its diagram shape and its weight are both ``finitely presentable'' in a suitable sense (defined in terms of [[computads]]; see \hyperlink{StreetLimitsIndexed}{Street's article} \emph{Limits indexed by category-valued 2-functors} ). Pullbacks, comma objects, inserters, equifiers, and so on are all finite limits, as are powers by any finitely presented category. All finite limits can be constructed from pullbacks, a terminal object, and powers with $\mathbf{2}$.

\hypertarget{(2,1)limit}{}\subsubsection*{{$(2,1)$-limits}}\label{(2,1)limit}

If the ambient [[2-category]] is in fact a [[(2,1)-category]] in that all [[2-morphism]]s are invertible then there is a rich set of tools available for handling the 2-limits in this context. We may say \textbf{$(2,1)$-limits} and \textbf{$(2,1)$-colimits} in this case.

These are then a special case of the more general [[(∞,1)-limit]]s and [[(∞,1)-colimit]]s in a [[(∞,1)-category]]. A [[(2,1)-category]] is a special case of an [[(∞,1)-category]].

Traditionally, [[(∞,1)-limit]]s are best known in terms of the presentation of $(\infty,1)$-catgeories by [[categories with weak equivalences]] in general and [[model categories]] in particular. (2,1)-limits can often also be viewed in this way. The corresponding presentation of the $(\infty,1)$-limits / $(2,1)$-limits is called \textbf{[[homotopy limit]]s} and \textbf{[[homotopy colimit]]s}.

For instance 2-limits in the [[(2,1)-category]] [[Grpd]] of [[groupoid]]s, [[functor]]s and (necessarily) [[natural isomorphism]]s. Are equivalently computed as [[homotopy limit]]s in the [[model structure on simplicial sets]] $sSet_{Quillen}$ of diagrams of [[1-truncated]] [[Kan complex]]es. (The equivalence of homotopy limits with $(\infty,1)$-limits is discussed at [[(∞,1)-limit]]).

Or for instance, more generally, the 2-limits in any [[(2,1)-sheaf]](=[[stack]]) [[(2,1)-topos]] may be computed as [[homotopy limit]]s in a [[model structure on simplicial presheaves]] over the given [[(2,1)-site]] of diagrams of [[1-truncated]] [[simplicial presheaves]]. This includes as examples [[big topos|big (2,1)-toposes]] such as those over the large sites [[Top]] or [[SmoothMfd]] where computations with [[topological groupoid]]s/[[topological stack]]s, [[Lie groupoid]]s/[[differentiable stack]]s etc. take place.

\hypertarget{lax}{}\subsubsection*{{Lax limits}}\label{lax}

A \textbf{lax limit} can be defined like a 2-limit, except that the triangles in the definition of a cone are required only to commute up to a specified \emph{transformation}, not necessarily an isomorphism. In other words, in place of the 2-category $[D,Cat]$ we use the 2-category $[D,Cat]_l$ whose morphisms are [[lax natural transformations]]; thus the lax limit $L$ of a diagram $F$ weighted by $J$ is equipped with a universal lax natural transformation $J\to K(L,F-)$.

This may look like a different concept, but in fact, for any weight $J$ there is another weight $Q_l(J)$ such that lax $J$-weighted limits are the same as $Q_l(J)$-weighted 2-limits. Here $Q_l$ is the [[lax morphism classifier]] for 2-functors. Therefore, lax limits are really a special case of 2-limits. Similarly, \textbf{oplax limits}, in which we use oplax natural transformations, are also a special case of 2-limits.

There is a further simplification of lax limits in the case of ``conical'' lax limits where the weight $J=\Delta 1$ is constant at the [[terminal category]]. In this case, it is easy to check that a lax natural transformation $\Delta 1 \to K(X,F-)$ is the same as a lax natural transformation $\Delta X \to F$. Thus, a conical lax limit of $F$ is a representing object for such lax transformations.

Here are some examples.

\begin{itemize}%
\item Lax terminal objects and lax products are the same as ordinary ones, since there are no commutativity conditions on the cones.


\item The \textbf{lax limit of an arrow} $f:A\to B$ is a universal object $L$ equipped with projections $p:L\to A$ and $q:L\to B$ and a 2-cell $f p \to q$. Note that this is equivalent to a comma object $(f/1_B)$.


\item The \textbf{lax pullback} of a cospan $A \overset{f}{\to} C \overset{g}{\leftarrow} B$ is a universal object $P$ equipped with projections $p:P\to A$, $q:P\to B$, $r:P\to C$, and 2-cells $f p \to r$ and $g q \to r$.



\end{itemize}
Note that lax pullbacks are \emph{not} the same as [[comma objects]]. In general comma objects are much more useful, but there are 2-categories that admit all lax limits but do not admit comma objects, so using ``lax pullback'' to mean ``comma object'' can be misleading.

A \textbf{lax colimit} in $K$ is, of course, a lax limit in $K^{op}$. Thus, it is a representing object for lax natural transformations $J \to K(F-,L)$. There is a subtlety here, however, because in the case $J=\Delta 1$, a lax natural transformation $\Delta 1 \to K(F-,L)$ is the same as an \emph{oplax} natural transformation $F \to \Delta L$. Thus, it is easy to mistakenly say ``lax colimit'' when one really means ``oplax colimit'' and vice versa.

\begin{uremark}
With this in mind, one might be tempted to switch the meanings of ``lax colimit'' and ``oplax colimit'', but there is a good reason not to. Recall that a lax $J$-weighted limit is the same as an ordinary $Q_l(J)$-weighted limit. Standard terminology in enriched category theory is that a $W$-weighted colimit in an enriched category $K$ is the same as a $W$-weighted limit in $K^{op}$, and indeed in that generality there is no other option. Thus, a lax $J$-weighted colimit in $K$ should be an ordinary $Q_l(J)$-weighted colimit in $K$, hence a $Q_l(J)$-weighted limit in $K^{op}$, and thus a lax $J$-weighted limit in $K^{op}$.

\end{uremark}
Here are some examples of lax and oplax colimits:

\begin{itemize}%
\item A [[Kleisli object]] is a lax colimit of a [[monad]], regarded as a diagram in a 2-category.


\item The [[collage]] of a [[profunctor]] is its lax colimit, regarded as a diagram in the 2-category [[Prof]].


\item When $C$ is a category, the [[Grothendieck construction]] of a functor $C\to Cat$ is the same as its \emph{oplax} colimit; see \href{http://ncatlab.org/nlab/show/Grothendieck+construction#AsALaxColimit}{here}.



\end{itemize}
\hypertarget{2ColimitsInCat}{}\subsubsection*{{2-Colimits in $Cat$}}\label{2ColimitsInCat}

As shown \href{http://ncatlab.org/nlab/show/Grothendieck+construction#AsALaxColimit}{here}, if $C$ is an ordinary category and $F \colon C \to Cat$ is a [[pseudofunctor]], then the [[oplax colimit]] of $F$ is given by the [[Grothendieck construction]] $\int F$ --- and its [[pseudo-colimit]] is given by [[localization|formally inverting]] the [[cartesian morphism|opcartesian]] morphisms in $\int F$. This yields a construction of certain pseudo 2-colimits in $Cat$.

Moreover, a similar result holds more generally when $C$ is a [[bicategory]]. In this case, $\int F$ is also a bicategory: a 2-cell from $(m \colon c \to d, f \colon m_*x \to y)$ to $(n \colon c \to d, g \colon n_*x \to y)$ is given by a 2-cell $\mu \colon m \Rightarrow n$ in $C$ such that $\mu_* x$ is a morphism $f \to g$ over $y$.

Let $\pi_*$ denote the functor that sends a bicategory $K$ to the category whose objects are those of $K$ and whose hom-sets are the [[connected category|connected components]] of the hom-categories of $K$; let also $d_*$ denote the functor that sends a category $X$ to the corresponding locally discrete bicategory. Then there is an equivalence of categories

\begin{displaymath}
[K, d_* X] \simeq [\pi_* K, X]
\end{displaymath}
It is straightforward to check that the first of the above facts extends to the bicategorical case:

\begin{displaymath}
Lax(F, \Delta X) \simeq [{\textstyle \int} F, d_* X]
\end{displaymath}
as does the fact that a lax transformation on the left is pseudo if and only if the corresponding functor on the right inverts the opcartesian morphisms in $\int F$. It is almost trivial that the adjunction $\pi_* \dashv d_*$ holds when restricted to the functor $[-, -]_{S^{-1}}$ that takes two categories or bicategories to the full subcategory of functors that invert the class $S$ of morphisms. Taking $S$ to be the opcartesian morphisms in $\int F$, then, we have

\begin{displaymath}
Ps(F, \Delta X) \simeq [{\textstyle \int} F, d_* X]_{S^{-1}}
  \simeq [\pi_* {\textstyle \int} F, X]_{S^{-1}}
  \simeq [(\pi_* {\textstyle \int} F)[S^{-1}], X]
\end{displaymath}
Hence the pseudo colimit of $F$ is got by taking its bicategory of elements, applying the `local $\pi_0$' functor, and then inverting the (images of the) opcartesian morphisms as usual.

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

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


\item \textbf{2-limit}


\item [[(∞,1)-limit]]

\begin{itemize}%
\item [[homotopy limit]]


\item [[lax (∞,1)-colimit]]



\end{itemize}

\item [[coherence for bicategories with finite limits]]



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

\begin{itemize}%
\item [[Ross Street]], \emph{Limits indexed by category-valued 2-functors} Journal of Pure and Applied Algebra \textbf{8}, Issue 2 (1976) pp 149-181. doi:\href{https://doi.org/10.1016/0022-4049(76%2990013-X}{10.1016/0022-4049(76)90013-X}


\item [[Max Kelly]], \emph{Elementary observations on 2-categorical limits}, Bulletin of the Australian Mathematical Society (1989), 39: 301-317, doi:\href{https://doi.org/10.1017/S0004972700002781}{10.1017/S0004972700002781}


\item [[Ross Street]], \emph{Fibrations in Bicategories}, Cahiers de Topologie et G\'e{}om\'e{}trie Diff\'e{}rentielle Cat\'e{}goriques, Volume \textbf{21} (1980) no. 2, pp 111-160. \href{http://www.numdam.org/item?id=CTGDC_1980__21_2_111_0}{Numdam} and correction, Cahiers de Topologie et G\'e{}om\'e{}trie Diff\'e{}rentielle Cat\'e{}goriques, Volume \textbf{28} (1987) no. 1, pp 53-56 \href{http://www.numdam.org/item?id=CTGDC_1987__28_1_53_0}{Numdam}



\end{itemize}
Section 6, page 37 in

\begin{itemize}%
\item [[Steve Lack]], \emph{A 2-categories companion}. In: Baez J., May J. (eds) Towards Higher Categories. The IMA Volumes in Mathematics and its Applications, vol \textbf{152} 2010 Springer, New York, NY. doi:\href{https://doi.org/10.1007/978-1-4419-1524-5_4}{10.1007/978-1-4419-1524-5\_4}, arXiv:\href{http://arxiv.org/abs/math.CT/0702535}{math.CT/0702535}.


\item G. J. Bird, [[Max Kelly]], [[John Power]], [[Ross Street]], \emph{Flexible limits for 2-categories}, Journal of Pure and Applied Algebra \textbf{61} Issue 1 (1989) pp 1-27. doi:\href{http://dx.doi.org/10.1016/0022-4049(89%2990065-0}{10.1016/0022-4049(89)90065-0} Chapters 3,4,5 in


\item [[Thomas Fiore]], \emph{Pseudo Limits, Biadjoints, and Pseudo Algebras: Categorical Foundations of Conformal Field Theory}, Mem. Amer. Math. Soc. \textbf{182} (2006), no. 860 (\href{http://arxiv.org/abs/math/0408298}{arXiv:math/0408298}) (\href{http://bookstore.ams.org/memo-182-860}{AMS page}, \href{https://books.google.com.au/books?id=y_DUCQAAQBAJ}{Google Books})



\end{itemize}
[[!redirects 2-limits]] [[!redirects 2-colimit]] [[!redirects 2-colimits]] [[!redirects 2-categorical limit]] [[!redirects 2-categorical limits]] [[!redirects 2-categorial limit]] [[!redirects 2-categorial limits]] [[!redirects 2-categorical colimit]] [[!redirects 2-categorical colimits]] [[!redirects 2-categorial colimit]] [[!redirects 2-categorial colimits]]

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[[!redirects pseudolimit]] [[!redirects pseudolimits]] [[!redirects pseudo limit]] [[!redirects pseudo limits]] [[!redirects pseudo-limit]] [[!redirects pseudo-limits]] [[!redirects pseudocolimit]] [[!redirects pseudocolimits]] [[!redirects pseudo colimit]] [[!redirects pseudo colimits]] [[!redirects pseudo-colimit]] [[!redirects pseudo-colimits]]

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[[!redirects (2,1)-limit]] [[!redirects (2,1)-limits]] [[!redirects (2,1)-colimit]] [[!redirects (2,1)-colimits]]



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