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\section*{strict 2-category}

\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{strict_categories}{}\section*{{Strict $2$-categories}}\label{strict_categories}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \linebreak
\noindent\hyperlink{detailedDefn}{More details}\dotfill \pageref*{detailedDefn} \linebreak
\noindent\hyperlink{2categories_as_sesquicategories}{2-categories as sesquicategories}\dotfill \pageref*{2categories_as_sesquicategories} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{history}{History}\dotfill \pageref*{history} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\begin{itemize}%
\item A strict 2-category is a [[directed n-graph|directed 2-graph]] equipped with a composition operation on adjacent 1-cells and 2-cells which is strictly unital and associative.


\item The concept of a strict 2-category is the simplest generalization of a [[category]] to a [[higher category theory|higher category]]. It is the one-step [[vertical categorification|categorification]] of the concept of a category.



\end{itemize}
The term \emph{2-category} implicitly refers to a [[geometric shapes for higher structures|globular]] structure. By contrast, [[double category|double categories]] are based on [[cubes]] instead. The two notions are closely related, however: every strict 2-category gives rise to several strict double categories, and every double category has several underlying 2-categories.

Notice that \emph{[[double category]]} is another term for \emph{[[n-fold category|2-fold category]]}. Strict 2-categories may be identified with those strict 2-fold/double categories whose category of vertical morphisms is [[discrete category|discrete]], or those whose category of horizontal morphisms is discrete.

(And similarly, strict globular [[n-category|n-categories]] may be identified with those [[n-fold category|n-fold categories]] for which all cube faces ``in one direction'' are discrete. A similar statement for weak $n$-categories is to be expected, but little seems to be known about this.)

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

A \emph{strict 2-category}, often called simply a \emph{2-category}, is a [[enriched category|category enriched over]] [[Cat]], where $Cat$ is treated as the [[1-category]] of [[strict categories]].

Similarly, a [[strict 2-groupoid]] is a groupoid enriched over groupoids. This is also called a [[geometric shapes for higher structures|globular]] strict 2-groupoid, to emphasise the underlying geometry. The category of strict 2-groupoids is equivalent to the category of [[crossed module]]s over groupoids. It is also equivalent to the category of (strict) double groupoids with connections.

They are also special cases of strict globular omega-groupoids, and the category of these is equivalent to the category of [[crossed complex]]es.

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

Working out the meaning of `$Cat$-enriched category', we find that a strict 2-category $K$ is given by

\begin{itemize}%
\item a collection $ob K$ of objects $a,b,c,\ldots$, together with
\item a hom-[[category]] $K(a,b)$ for each $a,b$, and
\item a [[functor]] $1_a : \mathbf{1} \to K(a,a)$ and a functor $comp : K(b,c) \times K(a,b) \to K(a,c)$ for each $a,b,c$ satisfying associativity and identity axioms (given [[enriched category|here]]).

\end{itemize}
As for ordinary ($Set$-enriched) categories, an object $f \in K(a,b)$ is called a \emph{morphism} or \emph{1-cell} from $a$ to $b$ and written $f:a\to b$ as usual. But given $f,g:a\to b$, it is now possible to have non-trivial arrows $\alpha:f\to g \in K(a,b)$, called \emph{2-cells} from $f$ to $g$ and written as $\alpha : f \Rightarrow g$. Because the hom-objects $K(a,b)$ are by definition categories, 2-cells carry an associative and unital operation called \emph{vertical composition}. The identities for this operation, of course, are the identity 2-cells $1_f$ given by the category structure on $K(a,b)$.

The functor $comp$ gives us an operation of \emph{horizontal} composition on 2-cells. Functoriality of $comp$ then says that given $\alpha : f \Rightarrow g : a\to b$ and $\beta : f' \Rightarrow g' : b\to c$, the composite $\comp(\beta,\alpha)$ is a 2-cell $\beta \alpha : f'f \Rightarrow g'g : a \to c$. Note that the boundaries of the composite 2-cell are the composites of the boundaries of the components.

We also have the \emph{interchange law} (also called \emph{Godement law} or \emph{middle 4 interchange law}): because $comp$ is a functor it commutes with composition in the hom-categories, so we have (writing vertical composition with $\circ$ and horizontal as juxtaposition):

\begin{displaymath}
(\beta' \circ \beta)(\alpha' \circ \alpha) = (\beta' \alpha') \circ (\beta \alpha)
\end{displaymath}
The axioms for associativity and unitality of $comp$ ensure that horizontal composition behaves just like composition of 1-cells in a 1-category. In particular, the action of $comp$ on objects $f,g$ of hom-categories (i.e. 1-cells of $K$) is the usual composite of morphisms.

\hypertarget{detailedDefn}{}\subsubsection*{{More details}}\label{detailedDefn}

(See also the section below on [[sesquicategories]], which provide a conceptual package for the [[stuff, structure, property|stuff and structure]] described below.)

In even more detail, a strict $2$-category $K$ consists of \emph{stuff}:

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

\end{itemize}
that is equipped with \emph{structure}:

\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} $1_f\colon f \Rightarrow f$ or $\Id_f\colon f \Rightarrow f$,
\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} $\eta \triangleleft f\colon g \circ f \Rightarrow h \circ f$, and
\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} $h \triangleright \eta \colon h \circ f \Rightarrow h \circ g$,

\end{itemize}
satisfying the following properties:

\begin{enumerate}%
\item for each $f\colon a \to b$, the composites $f \circ \id_a$ and $\id_b \circ f$ each equal $f$,
\item for each $a \overset{f}\to b \overset{g}\to c \overset{h}\to d$, the composites $h \circ (g \circ f)$ and $(h \circ g) \circ f$ are equal,
\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 $\eta\colon f \Rightarrow g\colon a \to b$, the whiskerings $\eta \triangleleft \id_a$ and $\id_b \triangleright \eta$ equal $\eta$,
\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 $a \overset{f}\to b \overset{g}\to c$ and $\eta\colon h \Rightarrow i\colon c \to d$, the left whiskerings $\eta \triangleleft (g \circ f)$ and $(\eta \triangleleft g) \triangleleft f$ 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 right whiskerings $i \triangleright (h \triangleright \eta)$ and $(i \circ h) \triangleright \eta$ 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 whiskerings $i \triangleright (\eta \triangleleft f)$ and $(i \triangleright \eta) \triangleleft f$ are equal, and
\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.

\end{enumerate}
The construction in the last axiom is the \emph{horizontal composite} $\theta \circ \eta\colon h \circ f \to i \circ g$. It is possible (and probably more common) to take the horizontal composite as basic and the whiskerings as derived operations. This results in fewer, but more complicated, axioms.

\hypertarget{2categories_as_sesquicategories}{}\subsection*{{2-categories as sesquicategories}}\label{2categories_as_sesquicategories}

The fine-grained description in the \href{/nlab/show/strict+2-category#detailedDefn}{previous subsection} can be concisely repackaged by saying that a 2-category is a [[sesquicategory]] that satisfies the interchange axiom, i.e., the last axiom (12) which gives the horizontal composite construction. This description is essentially patterned after the ``five rules of functorial calculus'' introduced by \hyperlink{Godement}{Godement} for the special case [[Cat]].

So to say it again, but a little differently: a sesquicategory consists of a category $K$ (giving the $0$-cells and $1$-cells) together with a functor

\begin{displaymath}
K(-, -): K^{op} \times K \to Cat
\end{displaymath}
such that composing $K(-, -)$ with the functor $ob: Cat \to Set$ (the one sending a category to its set of objects) gives $\hom_K: K^{op} \times K \to Set$, the hom-functor for the category $K$. So: for $0$-cells $a, b$, the objects of the category $K(a, b)$ are $1$-cells $f \in \hom_K(a, b)$. The morphisms of $K(a, b)$ are $2$-cells (with $0$-source $a$ and $0$-target $b$). Composition within the category $K(a, b)$ corresponds to vertical composition.

For each object $a$ of $K$ and each morphism $h: b \to c$ of $K$, there is a functor $K(a, h): K(a, b) \to K(a, c)$. This is \emph{right whiskering}; it sends a 2-cell $\eta$ (a morphism of $K(a, b)$) to a morphism $h \triangleright \eta$ of $K(b, c)$. Similarly, for each object $c$ and morphism $f: a \to b$, there is a functor $K(f, c): K(b, c) \to K(a, c)$. This is \emph{left whiskering}; it sends a 2-cell $\eta$ (a morphism of $K(b, c)$) to a morphism $\eta \triangleleft f$ of $K(a, c)$.

The long list of compatibility properties enumerated in the previous subsection, all except the last, are concisely summarized in the definition of sesquicategory as recalled above. For example, property (8) just says that left whiskering preserves vertical composition, as it must since it is a [[functor]] (a morphism in $Cat$).

In summary, a sesquicategory consists of ``stuff'' and structure as described in the \href{/nlab/show/strict+2-category#detailedDefn}{previous subsection}, satisfying properties 1-11. A 2-category is then a sesquicategory that further satisfies the interchange axiom (12). Some further illumination of this point of view can be obtained by contemplating [[string diagrams]] for 2-categories, where the interchange axiom corresponds to isotopies of (planar, progressive) string diagrams during which the relative heights of nodes labeled by 2-cells are interchanged.

\hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks}

\begin{itemize}%
\item A strict 2-category is the same as a [[strict omega-category]] which is trivial in degree $n \geq 3$.


\item This is to be contrasted with a \emph{weak 2-category} called a [[bicategory]]. In a strict 2-category composition of 1-morphisms is strictly associative and composition with [[identity morphism]]s strictly satisfies the required identity law. In a weak 2-category these laws may hold only up to coherent 2-morphisms.



\end{itemize}
\hypertarget{history}{}\subsection*{{History}}\label{history}

As intimated above, the essential rules which abstractly govern the behavior of functors and natural transformations and their various compositions were made explicit by \hyperlink{Godement}{Godement}, in his ``five rules of functorial calculus''. He did not however go as far as use these rules to define the abstract notion of 2-category; this step was taken a few years later by \hyperlink{Ehresmann}{Ehresmann}, who in fact defined [[double categories]], and 2-categories as a special case. In any event, the primitive compositional operations in Godement's account were what we call vertical compositions and whiskerings, with horizontal composition of natural transformations being a derived operation (made unambiguous in the presence of the interchange axiom). Indeed, horizontal composition is often called the \emph{Godement product}.

A few years after that, B\'e{}nabou introduced the notion of [[bicategory]].

Literature references for the abstract notion of [[sesquicategory]], a structure in which vertical compositions and whiskerings are primitive, do not seem to be abundant, but they are mentioned for example in \hyperlink{Street}{Street} together with the observation that 2-categories are special types of sesquicategories (page 535).

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

\begin{itemize}%
\item [[Roger Godement]], \emph{Topologie alg\'e{}brique et theorie des faisceaux}, Hermann, Paris, 1958.

\end{itemize}
\begin{itemize}%
\item [[Charles Ehresmann]], \emph{Cat\'e{}gories double et cat\'e{}gories structur\'e{}es}, C.R. Acad. Paris 256 (1963), 1198-1201. (\href{https://www.researchgate.net/publication/266276121_Categories_doubles_et_categories_structurees}{ResearchGate link})

\end{itemize}
\begin{itemize}%
\item [[Ross Street]], \emph{Categorical Structures}, in Handbook of Algebra Vol. 1 (ed. M. Hazewinkel), Elsevier Science, Amsterdam 1996. (\href{http://maths.mq.edu.au/~street/45.pdf}{pdf})

\end{itemize}
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