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\begin{document}

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\section*{bicategory}

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

\hypertarget{2category_theory}{}\paragraph*{{2-Category theory}}\label{2category_theory}

[[!include 2-category theory - contents]]

\hypertarget{higher_category_theory}{}\paragraph*{{Higher category theory}}\label{higher_category_theory}

[[!include higher category theory - contents]]

\hypertarget{bicategories}{}\section*{{Bicategories}}\label{bicategories}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{detailedDefn}{Details}\dotfill \pageref*{detailedDefn} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{Coherence}{Coherence theorems}\dotfill \pageref*{Coherence} \linebreak
\noindent\hyperlink{terminology}{Terminology}\dotfill \pageref*{terminology} \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{bicategory} is a particular [[algebraic definition of higher category|algebraic]] notion of \emph{weak [[2-category]]} (in fact, the earliest to be formulated, and still the one in most common use). The idea is that a bicategory is a category \emph{[[weak enrichment|weakly enriched]]} over [[Cat]]: the [[hom-objects]] of a bicategory are [[hom-category|hom-categories]], but the associativity and unity laws of [[enriched category|enriched categories]] hold only up to coherent isomorphism.

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

A \textbf{bicategory} $B$ consists of

\begin{itemize}%
\item A [[collection]] of \textbf{[[objects]]} $x,y,z,\dots$, also called \textbf{$0$-cells};
\item For each pair of $0$-cells $x,y$, a [[category]] $B(x,y)$, whose objects are called \textbf{[[morphisms]]} or \textbf{$1$-cells} and whose morphisms are called \textbf{[[2-morphisms]]} or \textbf{$2$-cells};
\item For each $0$-cell $x$, a distinguished $1$-cell $1_x\in B(x,x)$ called the \textbf{[[identity morphism]]} or \textbf{identity $1$-cell} at $x$;
\item For each triple of $0$-cells $x,y,z$, a functor ${\circ}\colon B(y,z)\times B(x,y) \to B(x,z)$ called \textbf{[[horizontal composition]]};
\item For each pair of $0$-cells $x,y$, [[natural isomorphisms]] called \textbf{[[unitors]]}: $\left(
\begin{array}{rcl}
f&\mapsto&f \circ 1_x\\
\theta&\mapsto&\theta \circ 1_{1_x}
\end{array}
\right)  \cong id_{B(x,y)} \cong \left(
\begin{array}{rcl}
f&\mapsto&1_y\circ f\\
\theta&\mapsto&1_{1_y} \circ \theta
\end{array}
\right):B(x,y)\rightarrow B(x,y)$
\item For each quadruple of $0$-cells $w,x,y,z$, a natural isomorphism called the \textbf{[[associator]]} between the two functors from $B_{y,z} \times B_{x,y} \times B_{w,x}$ to $B_{w,z}$ built out of ${\circ}$; such that
\item Such that the [[pentagon identity]] is satisfied by the [[associators]].
\item And such that the triangle identity is satisfied by the [[unitors]].

\end{itemize}
If there is exactly one $0$-cell, say $*$, then the definition is exactly the same as a monoidal structure on the category $B(*,*)$. This is one of the motivating examples behind the [[delooping hypothesis]] and the general notion of [[k-tuply monoidal n-category]].

\hypertarget{detailedDefn}{}\subsubsection*{{Details}}\label{detailedDefn}

Here we spell out the above definition in full detail. Compare to the \href{/nlab/show/strict+2-category#detailedDefn}{detailed definition of strict $2$-category}, which is written in the same style but is simpler.

A bicategory $B$ consists of

\begin{itemize}%
\item a collection $Ob B$ or $Ob_B$ of \emph{objects} or \emph{$0$-cells},
\item for each object $a$ and object $b$, a collection $B(a,b)$ or $Hom_B(a,b)$ of \emph{morphisms} or \emph{$1$-cells} $a \to b$, and
\item for each object $a$, object $b$, morphism $f\colon a \to b$, and morphism $g\colon a \to b$, a collection $B(f,g)$ or $2Hom_B(f,g)$ of \emph{$2$-morphisms} or \emph{$2$-cells} $f \Rightarrow g$ or $f \Rightarrow g\colon a \to b$,

\end{itemize}
equipped with

\begin{itemize}%
\item for each object $a$, an \emph{identity} $1_a\colon a \to a$ or $\id_a\colon a \to a$,
\item for each $a,b,c$, $f\colon a \to b$, and $g\colon b \to c$, a \emph{composite} $f ; g\colon a \to c$ or $g \circ f\colon a \to c$,
\item for each $f\colon a \to b$, an \emph{identity} or \emph{$2$-identity} $1_f\colon f \Rightarrow f$ or $\Id_f\colon f \to f$,
\item for each $f,g,h\colon a \to b$, $\eta\colon f \Rightarrow g$, and $\theta\colon g \Rightarrow h$, a \emph{vertical composite} $\theta \bullet \eta\colon f \Rightarrow h$,
\item for each $a,b,c$, $f\colon a \to b$, $g,h\colon b \to c$, and $\eta\colon g \Rightarrow h$, a \emph{left whiskering} $\eta \triangleleft f\colon g \circ f \Rightarrow h \circ f$,
\item for each $a,b,c$, $f,g\colon a \to b$, $h\colon b \to c$, and $\eta\colon f \Rightarrow g$, a \emph{right whiskering} $h \triangleright \eta \colon h \circ f \Rightarrow h \circ g$,
\item for each $f\colon a \to b$, a \emph{left unitor} $\lambda_f\colon f \circ \id_a \Rightarrow f$ and an \emph{inverse left unitor} $\bar{\lambda}_f\colon f \Rightarrow f \circ \id_a$,
\item for each $f\colon a \to b$, a \emph{right unitor} $\rho_f\colon \id_b \circ f \Rightarrow f$, and an \emph{inverse right unitor} $\bar{\rho}_f\colon f \Rightarrow \id_b \circ f$, and
\item for each $a \overset{f}\to b \overset{g}\to c \overset{h}\to d$, an \emph{associator} $\alpha_{f,g,h}\colon h \circ (g \circ f) \Rightarrow (h \circ g) \circ f$ and an \emph{inverse associator} $\bar{\alpha}_{f,g,h}\colon (h \circ g) \circ f \Rightarrow h \circ (g \circ f)$,

\end{itemize}
such that

\begin{itemize}%
\item for each $\eta\colon f \Rightarrow g\colon a \to b$, the vertical composites $\eta \bullet \Id_f$ and $\Id_g \bullet \eta$ both equal $\eta$,
\item for each $f \overset{\eta}\Rightarrow g \overset{\theta}\Rightarrow h \overset{\iota}\Rightarrow i\colon a \to b$, the vertical composites $\iota \bullet (\theta \bullet \eta)$ and $(\iota \bullet \theta) \bullet \eta$ are equal,
\item for each $a \overset{f}\to b \overset{g}\to c$, the whiskerings $\Id_g \triangleleft f$ and $g \triangleright \Id_f$ both equal $\Id_{g \circ f }$,
\item for each $f\colon a \to b$ and $g \overset{\eta}\Rightarrow h \overset{\theta}\Rightarrow i\colon b \to c$, the vertical composite $(\theta \triangleleft f) \bullet (\eta \triangleleft f)$ equals the whiskering $(\theta \bullet \eta) \triangleleft f$,
\item for each $f \overset{\eta}\Rightarrow g \overset{\theta}\Rightarrow h\colon a \to b$ and $i\colon b \to c$, the vertical composite $(i \triangleright \theta) \bullet (i \triangleright \eta)$ equals the whiskering $i \triangleright (\theta \bullet \eta)$,
\item for each $\eta\colon f \Rightarrow g\colon a \to b$, the vertical composites $\lambda_g \bullet (\eta \triangleleft \id_a)$ and $\eta \bullet \lambda_f$ are equal,
\item for each $\eta\colon f \Rightarrow g\colon a \to b$, the vertical composites $\rho_g \bullet (\id_b \triangleright \eta)$ and $\eta \bullet \rho_f$ are equal,
\item for each $a \overset{f}\to b \overset{g}\to c$ and $\eta\colon h \Rightarrow i\colon c \to d$, the vertical composites $\alpha_{f,g,i} \bullet (\eta \triangleleft (g \circ f))$ and $((\eta \triangleleft g) \triangleleft f) \bullet \alpha_{f,g,h}$ are equal,
\item for each $f\colon a \to b$, $\eta\colon g \Rightarrow h\colon b \to c$, and $i\colon c \to d$, the vertical composites $\alpha_{f,h,i} \bullet (i \triangleright (\eta \triangleleft f))$ and $((i \triangleright \eta) \triangleleft f) \bullet \alpha_{f,g,i}$ are equal,
\item for each $\eta\colon f \Rightarrow g\colon a \to b$ and $b \overset{h}\to c \overset{i}\to d$, the vertical composites $\alpha_{g,h,i} \bullet (i \triangleright (h \triangleright \eta))$ and $((i \circ h) \triangleright \eta) \bullet \alpha_{f,h,i}$ are equal,
\item for each $\eta\colon f \Rightarrow g\colon a \to b$ and $\theta\colon h \Rightarrow i\colon b \to c$, the vertical composites $(i \triangleright \eta) \bullet (\theta \triangleleft f)$ and $(\theta \triangleleft g) \bullet (h \triangleright \eta)$ are equal,
\item for each $f\colon a \to b$, the vertical composites $\lambda_f \bullet \bar{\lambda}_f\colon f \Rightarrow f$ and $\bar{\lambda}_f \bullet \lambda_f\colon f \circ \id_a \Rightarrow f \circ \id_a$ equal the appropriate identity $2$-morphisms,
\item for each $f\colon a \to b$, the vertical composites $\rho_f \bullet \bar{\rho}_f\colon f \Rightarrow f$ and $\bar{\rho}_f \bullet \rho_f\colon \id_b \circ f \Rightarrow \id_b \circ f$ equal the appropriate identity $2$-morphisms,
\item for each $a \overset{f}\to b \overset{g}\to c \overset{h}\to d$, the vertical composites $\alpha_{f,g,h} \bullet \bar{\alpha}_{f,g,h}\colon (h \circ g) \circ f \Rightarrow (h \circ g) \circ f$ and $\bar{\alpha}_{f,g,h} \bullet \alpha_{f,g,h}\colon h \circ (g \circ f) \Rightarrow h \circ (g \circ f)$ equal the appropriate identity $2$-morphisms,
\item for each $a \overset{f}\to b \overset{g}\to c$, the vertical composite $(\lambda_g \triangleleft f) \bullet \alpha_{f,\id_b,g}$ equals the whiskering $g \triangleright \rho_f$, and
\item for each $a \overset{f}\to b \overset{g}\to c \overset{h}\to d \overset{i}\to e$, the vertical composites $((\alpha_{g,h,i} \triangleleft f) \bullet \alpha_{f,h \circ g,i}) \bullet (i \triangleright \alpha_{f,g,h})$ and $\alpha_{f,g,i \circ h}\bullet \alpha_{g \circ f,h,i}$ are equal.

\end{itemize}
It is quite possible that there are errors or omissions in this list, although they should be easy to correct. The point is not that one would \emph{want} to write out the definition in such elementary terms (although apparently I just did anyway) but rather that one \emph{can}.

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

\begin{itemize}%
\item Any [[strict 2-category]] is a bicategory in which the unitors and associator are identities. This includes [[Cat]], [[MonCat]], the algebras for any strict [[2-monad]], and so on, at least as classically conceived.


\item Categories, [[anafunctor]]s, and natural transformations, which is a more appropriate definition of [[Cat]] in the absence of the [[axiom of choice]], form a bicategory that is not a strict 2-category. Indeed, without the axiom of choice, the proper notion of bicategory is [[anabicategory]].


\item [[ring|Rings]], [[bimodule]]s, and bimodule homomorphisms are the prototype for many similar examples. Notably, we can generalize from rings to [[enriched category|enriched categories]].


\item Objects, [[span]]s, and morphisms of spans in any category with [[pullback]]s also form a bicategory.


\item The [[fundamental 2-groupoid]] of a space is a bicategory which is not necessarily strict (although it can be made strict fairly easily when the space is Hausdorff by quotienting by [[thin homotopy]], see [[path groupoid]] and [[fundamental infinity-groupoid]]). When the space is a CW-complex, there are easier and more computationally amenable equivalent strict 2-categories, such as that arising from the fundamental [[crossed complex]].



\end{itemize}
\hypertarget{Coherence}{}\subsection*{{Coherence theorems}}\label{Coherence}

One way to state the [[coherence theorem]] for bicategories is that every bicategory is [[equivalence of categories|equivalent]] to a strict $2$-category. This ``strictification'' is not obtained naively by forcing composition to be associative, but (at least in one construction) by freely adding new composites which are strictly associative. Another way to state the coherence theorem is that every formal diagram of the constraints (associators and unitors) commutes.

Note that $n=2$ is the greatest value of $n$ for which every weak $n$-category is equivalent to a fully strict one; see [[semi-strict infinity-category]] and [[Gray-category]].

The proof of the coherence theorem is basically the same as the proof of the [[coherence theorem for monoidal categories]]. An abstract approach can be found in [[John Power|Power]]`s paper ``A general coherence result.''

The strictification [[adjunction]] between bicategories and strict 2-categories can be expressed in terms of [[3-categories]]; see \hyperlink{Campbell18}{Campbell}.

\hypertarget{terminology}{}\subsection*{{Terminology}}\label{terminology}

Classically, ``2-category'' meant [[strict 2-category]], with ``bicategory'' used for the weak notion. This led to the more general use of the prefix ``2-'' for strict (that is, strictly [[Cat]]-enriched) notions and ``bi-'' for weak ones. For example, classically a ``2-adjunction'' means a Cat-enriched adjunction, consisting of two strict 2-functors $F,G$ and a strictly Cat-natural isomorphism of categories $D(F X, Y)\cong C(X, G Y)$, while a ``biadjunction'' means the weak version, consisting of two weak 2-functors and a pseudo natural equivalence $D(F X, Y)\simeq C(X, G Y)$. Similarly for ``2-equivalence'' and ``biequivalence,'' and ``2-limit'' and ``bilimit.''

We often use ``2-category'' to mean a strict or weak 2-category without prejudice, although we do still use ``bicategory'' to refer to the particular classical algebraic notion of weak 2-category. We try to avoid the more general use of ``bi-'' meaning ``weak,'' however. For one thing, is it confusing; a ``biproduct'' could mean a weak [[2-limit]], but it could also mean an object which is both a product and a coproduct (which happens quite frequently in [[additive category|additive categories]]).

Moreover, in most cases the prefix is unnecessary, since once we know we are working in a bicategory, there is usually no point in considering strict notions at all. Fully weak limits are really the only sensible ones to ask for in a bicategory, and likewise for fully weak adjunctions and equivalences. Even in a strict 2-category, while we might need to say ``strict'' sometimes to be clear, we don't need to say ``$2$-'', since we know that we are not working in a mere category. (Max Kelly pushed this point.)

When we do have a strict 2-category, however, other strict notions can be quite technically useful, even if our ultimate interest is in the weak ones. This is somewhat analogous to the use of strict structures to model weak ones in [[homotopy theory]]; see \href{http://arxiv.org/abs/math/0702535}{here} and \href{http://arxiv.org/abs/math/0607646}{here} for good introductions to this sort of thing.

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

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


\item \textbf{bicategory}

[[(infinity,2)-category]]


\item [[tricategory]]


\item [[tetracategory]]


\item [[n-category]]


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



\end{itemize}
Discussion about the use of the term ``weak enrichment'' above is at \emph{[[weak enrichment]]}.

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

\begin{itemize}%
\item [[Jean Bénabou]], Introduction to Bicategories, \emph{Lecture Notes in Mathematics 47}, Springer (1967), pp.1-77. (\href{http://dx.doi.org/10.1007/BFb0074299}{doi})

\end{itemize}
See also the references at [[2-category]].

\begin{itemize}%
\item Power, A. J. \emph{A general coherence result.} J. Pure Appl. Algebra 57 (1989), no. 2, 165--173. \href{http://dx.doi.org/10.1016/0022-4049%2889%2990113-8}{doi:10.1016/0022-4049(89)90113-8} \href{http://www.ams.org/mathscinet-getitem?mr=985657}{MR0985657}


\item [[Alexander Campbell]], \emph{How strict is strictification?}, \href{https://arxiv.org/abs/1802.07538}{arxiv}



\end{itemize}
Formalization in [[homotopy type theory]] (see also at [[internal category in homotopy type theory]]):

\begin{itemize}%
\item [[Benedikt Ahrens]], Dan Frumin, Marco Maggesi, Niels van der Weide, \emph{Bicategories in Univalent Foundations} (\href{https://arxiv.org/abs/1903.01152}{arXiv:1903.01152})

\end{itemize}
[[!redirects bicategory]] [[!redirects bicategories]]

[[!redirects coherence theorem for bicategories]]



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