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\section*{2-category}

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

\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{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{strict_2categories}{Strict 2-categories}\dotfill \pageref*{strict_2categories} \linebreak
\noindent\hyperlink{Weak}{General 2-categories}\dotfill \pageref*{Weak} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{double_nerve}{Double nerve}\dotfill \pageref*{double_nerve} \linebreak
\noindent\hyperlink{ModelCategoryStructure}{Model category structure}\dotfill \pageref*{ModelCategoryStructure} \linebreak
\noindent\hyperlink{FreeResolution}{Free resolutions}\dotfill \pageref*{FreeResolution} \linebreak
\noindent\hyperlink{2categorical_concepts}{2-categorical concepts}\dotfill \pageref*{2categorical_concepts} \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}

The notion of a \emph{2-category} generalizes that of [[category]]: a 2-category is a [[higher category theory|higher category]], where on top of the objects and morphisms, there are also 2-morphisms.

A 2-category consists of

\begin{itemize}%
\item [[objects]];


\item 1-[[morphisms]] between objects;


\item [[2-morphisms]] between morphisms.



\end{itemize}
The morphisms can be [[composition|composed]] along the objects, while the 2-morphisms can be composed in two different directions: along objects -- called [[horizontal composition]] -- and along morphisms -- called [[vertical composition]]. The composition of morphisms is allowed to be associative only up to [[coherent]] [[associator]] 2-morphisms.

2-Categories are also a [[horizontal categorification]] of [[monoidal categories]]: they are like monoidal categories with many objects.

2-Categories provide the context for discussing

\begin{itemize}%
\item [[adjunction]]s;


\item [[monad]]s.



\end{itemize}
The concept of 2-category generalizes further in [[higher category theory]] to [[n-categories]], which have [[k-morphism]]s for all $k\le n$.

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

\hypertarget{strict_2categories}{}\subsubsection*{{Strict 2-categories}}\label{strict_2categories}

The easiest definition of 2-category is that it is a category [[enriched category|enriched]] over the [[cartesian monoidal category]] [[Cat]]. Thus it has a collection of objects, and for each pair of objects a category $hom(x,y)$. The objects of these hom-categories are the morphisms, and the morphisms of these hom-categories are the 2-morphisms. This produces the classical notion of [[strict 2-category]].

\hypertarget{Weak}{}\subsubsection*{{General 2-categories}}\label{Weak}

For some purposes, strict 2-categories are too strict: one would like to allow composition of morphisms to be associative and unital only up to coherent invertible 2-morphisms. A direct generalization of the above ``enriched'' definition produces the classical notion of [[bicategory]].

One can also obtain notions of 2-category by specialization from the case of higher categories. Specifically, if we fix a meaning of $\infty$-[[infinity-category|category]], however weak or strict we wish, then we can define a \textbf{$2$-category} to be an $\infty$-category such that every 3-morphism is an [[equivalence]], and all parallel pairs of $j$-morphisms are equivalent for $j \geq 3$. It follows that, up to equivalence, there is no point in mentioning anything beyond $2$-morphisms, except whether two given parallel $2$-morphisms are equivalent. In some models of $\infty$-categories, it is possible to make this precise by demanding that all parallel pairs of $j$-morphisms are actually \emph{equal} for $j\geq 3$, producing a simpler notion of 2-category in which we can speak about [[equality]] of 2-morphisms instead of equivalence. (This is the case for both strict $2$-categories and bicategories.)

All of the above definitions produce ``equivalent'' theories of 2-category, although in some cases (such as the fact that every bicategory is equivalent to a strict 2-category) this requires some work to prove. On the nLab, we often use the word ``2-category'' in the general sense of referring to whatever model one may prefer, but usually one in which composition is weak; a [[bicategory]] is an adequate definition. One should beware, however, that in the literature it is common for ``2-category'' to refer only to \emph{strict} 2-categories.

A 2-category in which all 1-morphisms and 2-morphisms are invertible is a [[2-groupoid]].

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

\begin{itemize}%
\item The archetypical 2-category is [[Cat]], the 2-category whose

\begin{itemize}%
\item objects are [[categories]];


\item morphisms are [[functor]]s;


\item 2-morphisms are [[natural transformation]];

horizontal composition of 2-morphisms is the [[Godement product]].



\end{itemize}
This happens to be a [[strict 2-category]].


\item More generally, for $V$ any enriching category (such as a Benabou [[cosmos]]), there is a 2-category $V Cat$ whose

\begin{itemize}%
\item objects are $V$-[[enriched categories]];
\item morphisms are $V$-enriched functors; and
\item 2-morphisms are $V$-natural transformations.

\end{itemize}

\item On the other hand, for any such $V$ we also have a [[bicategory]] $V$-[[Prof]] whose

\begin{itemize}%
\item objects are $V$-[[enriched categories]];
\item morphisms are $V$-[[profunctor]]s; and
\item 2-morphisms are natural transformations between these.

\end{itemize}

\item If $C$ is a category with [[pullbacks]], then there is a bicategory [[Span]]$(C)$ whose

\begin{itemize}%
\item objects are the objects of $C$;
\item morphisms are [[spans]] in $C$; and
\item 2-morphisms are morphisms of spans.

\end{itemize}

\item Every [[monoidal category]] $C$ may be thought of as a [[bicategory]] $\mathbf{B}C$ (its [[delooping]]). This has

\begin{itemize}%
\item a single object $\bullet$;


\item morphisms are the objects of $C$: $(\mathbf{B}C)_1 = C_0$;


\item 2-morphisms are the morphisms of $C$ : $(\mathbf{B}C)_2 = C_1$;



\end{itemize}
[[horizontal composition]] in $\mathbf{B}C$ is the [[tensor product]] in $C$ and [[vertical composition]] in $\mathbf{B}C$ is composition in $C$.

Conversely, every 2-category with a single object comes from a monoidal category this way, so the concepts are effectively equivalent. (Precisely: the 2-category of \emph{pointed} 2-categories with a single object is equivalent to that of monoidal categories). For more on this relation see [[delooping hypothesis]], [[k-tuply monoidal n-category]], and [[periodic table]].


\item Every [[2-groupoid]] is a 2-category. For instance

\begin{itemize}%
\item for $A$ any [[abelian group]], the double [[delooping]] $\mathbf{B}^2 A$ is the strict 2-category with a single object, a single 1-morphisms, set of 2-moprhisms being $A$ and both horizontal composition as well as [[vertical composition]] being the product in $A$.


\item for $G$ any [[2-group]], its single [[delooping]] is a 2-groupoid with a single object.



\end{itemize}

\item Every [[topological space]] has a [[path 2-groupoid]].


\item Every [[(∞,2)-category]] has a \textbf{homotopy 2-category}, obtained by dividing out all 3-morphisms and higher.



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

\hypertarget{double_nerve}{}\subsubsection*{{Double nerve}}\label{double_nerve}

An ordinary [[category]] has a [[nerve]] which is a [[simplicial set]]. For 2-categories one may consider their [[double nerve]] which is a [[bisimplicial set]].

There is also a 2-nerve. (\hyperlink{LackPaoli}{LackPaoli})

(\ldots{})

\hypertarget{ModelCategoryStructure}{}\subsubsection*{{Model category structure}}\label{ModelCategoryStructure}

There is a [[model category]] structure on 2-categories -- sometimes known as the [[folk model structure]] -- that models the [[(2,1)-category]] underlying [[2Cat]] (\hyperlink{LackFolkModel}{Lack}).

For strict 2-categories this is the restriction of the corresponding [[folk model structure]] on [[strict omega-categories]].

\begin{itemize}%
\item The weak equivalences are the [[2-functor]]s that are equivalences of 2-categories.


\item The acyclic fibrations are the [[k-surjective functor]]s for all $k$.



\end{itemize}
\hypertarget{FreeResolution}{}\paragraph*{{Free resolutions}}\label{FreeResolution}

\textbf{Theorem} A strict 2-category $C$ is cofibrant precisely if the underlying 1-category $C_1$ is a [[free category]].

This is theorem 4.8 in (\hyperlink{LackStrict}{LackStrict}). This is a special case of the more general statement that free strict $\omega$-categories are given by [[computad]]s.

\textbf{Example (free resolution of a 1-category).} Let $C$ be an ordinary category (a 1-category) regarded as a strict 2-category. Then the cofibrant resolution $\hat C \stackrel{\simeq}{\to} C$ is the strict 2-category given as follows:

\begin{itemize}%
\item the objects of $\hat C$ are those of $C$;


\item the morphisms of $\hat C$ are finite sequences of composable morphisms of $C$, and composition is concatenation of such sequences

(hence $(\hat C)_1$ is the [[free category]] on the [[quiver]] underlying $C$);


\item the 2-morphisms of $\hat C$ are \emph{generated} from 2-morphisms $c_{f,g}$ of the form

\begin{displaymath}
\itexarray{
     && y
     \\
     & {}^{\mathllap{f}}\nearrow &\Downarrow^{c_{f,g}}& \searrow^{\mathrlap{g}}
     \\
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  }
\end{displaymath}
and their formal inverses

\begin{displaymath}
\itexarray{
     && y
     \\
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     \\
     x && \underset{g \circ_C f }{\to} && z
  }
\end{displaymath}
for all composable $f,g \in Mor(C)$ with composite (in $C$!) $g \circ_C f$;

subject to the \emph{relation} that for all composable triples $f,g,h \in Mor(C)$ the following equation of 2-morphisms holds

\begin{displaymath}
\itexarray{
     y &\to& &\stackrel{g}{\to}& &\to& && z
     \\
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     \\
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     \\
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     \\
     x &\to& &\underset{h \circ (g \circ_C f)}{\to}& &\to& &\to& w
  }
  \;\;\;
  = 
  \;\;\;
      \itexarray{
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     \\
     \uparrow &\searrow& && & & &\swArrow_{c_{g,h}}& \downarrow
     \\
     {}^{\mathllap{f}}\uparrow &&  & \searrow & &&&& \downarrow^{\mathrlap{h}}
     \\
     \uparrow &\Downarrow_{c_{f,(g \circ_C h)}}& && &\searrow& && \downarrow
     \\
     x &\to& &\underset{( h \circ_C g) \circ f}{\to}& &\to& &\to& w
  }
\end{displaymath}


\end{itemize}
\textbf{Observation} Let $D$ be any strict 2-catgeory. Then a [[pseudofunctor]] $C \to D$ is the same as a strict 2-functor $\hat C \to D$.

\hypertarget{2categorical_concepts}{}\subsection*{{2-categorical concepts}}\label{2categorical_concepts}

\textbf{extra properties}

\begin{itemize}%
\item [[regular 2-category]]
\item [[exact 2-category]]
\item [[coherent 2-category]]
\item [[extensive 2-category]]
\item [[2-pretopos]]
\item [[2-topos]]

\end{itemize}
\textbf{types of morphisms}

\begin{itemize}%
\item [[subcategory]]
\item [[faithful morphism]]
\item [[fully faithful morphism]]
\item [[conservative morphism]]
\item [[pseudomonic morphism]]
\item [[discrete morphism]]

\end{itemize}
\textbf{specific versions}

\begin{itemize}%
\item globular [[strict 2-category]]
\item [[bicategory]]

\end{itemize}
\textbf{limit notions}

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

\end{itemize}
\textbf{model structures}

\begin{itemize}%
\item [[canonical model structure]]
\item [[2-trivial model structure]]

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

\begin{itemize}%
\item [[0-category]], [[(0,1)-category]]


\item [[category]]


\item \textbf{2-category}

[[equivalence in a 2-category]]

[[localization of a 2-category]]


\item [[3-category]]


\item [[n-category]]


\item [[(∞,0)-category]]


\item [[(∞,1)-category]]


\item [[(∞,2)-category]]


\item [[(∞,n)-category]]


\item [[(n,r)-category]]



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

A brief account of the definition is in

\begin{itemize}%
\item [[Ross Street]], \emph{Encyclopedia article on 2-categories and bicategories} (\href{http://www.maths.mq.edu.au/~street/Encyclopedia.pdf}{pdf})

\end{itemize}
A more detailed account of the definition, including a discussion of its [[coherence theorem]], is in

\begin{itemize}%
\item [[Tom Leinster]], \emph{Basic bicategories} (\href{http://arxiv.org/abs/math/9810017}{arXiv:9810017})

\end{itemize}
Some 2-category theory, including [[2-limit]]s/2-colimit is discussed in

\begin{itemize}%
\item [[Steve Lack]], \emph{A 2-categories companion} (\href{http://arxiv.org/abs/math/0702535}{arXiv:0702535})

\end{itemize}
and

\begin{itemize}%
\item [[John Power]], \emph{2-Categories} (\href{http://www.brics.dk/NS/98/7/BRICS-NS-98-7.pdf}{pdf})

\end{itemize}
An older reference which introduces some of the basic notions is

\begin{itemize}%
\item [[Max Kelly]] and [[Ross Street]], \emph{Review of the elements of 2-categories}, Sydney Category Seminar 1972/1973, LNM 420

\end{itemize}
A relation between bicategories and Tamsamani weak 2-categories is established in

\begin{itemize}%
\item [[Steve Lack]], [[Simona Paoli]], \emph{2-nerves for bicategories} (\href{http://arxiv.org/abs/math/0607271}{arXiv})

\end{itemize}
The reverse construction is in

\begin{itemize}%
\item [[Simona Paoli]], \emph{From Tamsamani weak 2-categories to bicategories} (\href{http://www.maths.mq.edu.au/~simonap/Bicategories_Rev_4.pdf}{arXiv})

\end{itemize}
There is a [[model category]] structure on 2-categories -- the [[canonical model structure]] -- that models the [[(2,1)-category]] underlying [[2Cat]]:

\begin{itemize}%
\item [[Steve Lack]], \emph{A Quillen Model Structure for 2-Categories}, K-Theory 26: 171--205, 2002. (\href{http://www.maths.usyd.edu.au/u/stevel/papers/qmc2cat.html}{website})

\end{itemize}
\begin{itemize}%
\item [[Steve Lack]], \emph{A Quillen Model Structure for Biategories}, K-Theory 33: 185-197, 2004. (\href{http://www.maths.usyd.edu.au/u/stevel/papers/qmcbicat.html}{website})

\end{itemize}
Discussion of weak 2-categories in the style of [[A-infinity categories]] is (using [[dendroidal sets]] to model the higher [[operads]]) in

\begin{itemize}%
\item Andor Lucacs, \emph{Dendroidal weak 2-categories} (\href{http://de.arxiv.org/abs/1304.4278}{arXiv:1304.4278})


\item [[Jonathan Chiche]], \emph{La th\'e{}orie de l'homotopie des 2-cat\'e{}gories}, thesis, \href{http://arxiv.org/abs/1411.6936}{arXiv}.



\end{itemize}
[[!redirects 2-category]] [[!redirects 2-categories]] [[!redirects weak 2-category]] [[!redirects weak 2-categories]]



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