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\begin{document}

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\section*{coherent 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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{factorizations}{Factorizations}\dotfill \pageref*{factorizations} \linebreak
\noindent\hyperlink{colimits}{Colimits}\dotfill \pageref*{colimits} \linebreak
\noindent\hyperlink{preservation}{Preservation}\dotfill \pageref*{preservation} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The generalization of the notion of \emph{[[coherent category]]} from [[category theory]] to [[2-category theory]].

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

\begin{defn}
\label{}\hypertarget{}{}
A [[2-category]] is called \textbf{coherent} if 1. it has finite [[2-limits]], 1. finite jointly-[[eso]] families are stable under [[2-pullback]], and 1. every finitary [[2-polycongruence]] which is a kernel can be completed to an exact [[2-polyfork]].

\end{defn}
Here a family $\{f_i:A_i \to B\}$ is said to be \textbf{jointly-eso} if whenever $m:C\to B$ is ff and every $f_i$ factors through $m$ (up to isomorphism), then $m$ is an equivalence.

Likewise, we have \textbf{infinitary coherent} 2-categories in which ``finite'' in the second two conditions is replaced by ``small.''

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

\begin{itemize}%
\item [[Cat]] is coherent.


\item A [[1-category]] is coherent as a 2-category iff it is [[coherent category|coherent]] as a 1-category.


\item A [[(0,1)-category]] (= [[poset]]) is coherent iff it is a [[distributive lattice|distributive lattice]], and infinitary-coherent iff it is a [[frame]].



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

\hypertarget{factorizations}{}\subsubsection*{{Factorizations}}\label{factorizations}

The following are proven just like their unary analogues in a [[regular 2-category]].

\begin{lemma}
\label{}\hypertarget{}{}
\textbf{(Street's Lemma)} In a finitely complete 2-category where finite jointly-eso families are stable under pullback, if $\{e_i:A_i \to B\}$ is finite and jointly-eso and $n:B\to C$ is such that the induced functor $\ker(e_i) \to ker(n e_i)$ is an equivalence, then $n$ is ff.

\end{lemma}
\begin{theorem}
\label{}\hypertarget{}{}
A 2-category is coherent if and only if 1. it has finite limits, 1. finite jointly-eso families are stable under pullback, 1. every finite family $\{f_i\}$ factors as $f_i = m e_i$ where $m$ is ff and $\{e_i\}$ is jointly-eso, and 1. every jointly-eso family is the quotient of its kernel.

\end{theorem}
Of course, there are infinitary versions. In particular, we conclude that in a coherent (resp. infinitary-coherent) 2-category, the posets $Sub(X)$ have finite (resp. small) unions that are preserved by pullback.

\hypertarget{colimits}{}\subsubsection*{{Colimits}}\label{colimits}

\begin{lemma}
\label{}\hypertarget{}{}
A coherent 2-category has a strict [[initial object]]; that is an initial object $0$ such that any morphism $X\to 0$ is an equivalence.

\end{lemma}
\begin{proof}
The empty 2-congruence is the kernel of the empty family (over any object), so it must have a quotient $0$, which is clearly an initial object. The empty family over $0$ is jointly-eso, so for any $X\to 0$ the empty family over $X$ is also jointly-eso; but this clearly makes $X$ initial as well.

\end{proof}
Two [[nLab:ff morphism|ffs]] $m:A\to X$ and $n:B\to X$ are said to be \textbf{disjoint} if the \emph{comma objects} $(m/n)$ and $(n/m)$ are initial objects. If initial objects are strict, then this implies that the pullback $A\times_X B$ is also initial, but it is strictly stronger already in $Pos$.

\begin{lemma}
\label{DisjointUnion}\hypertarget{DisjointUnion}{}
In a coherent 2-category, if $A\to X$ and $B\to X$ are disjoint subobjects, then their union $A\cup B$ in $Sub(X)$ is also their coproduct $A+B$.

\end{lemma}
\begin{proof}
If $A$ and $B$ are disjoint subobjects of $X$, then the kernel of $\{A\to X, B\to X\}$ is the disjoint union of $ker(A)$ and $ker(B)$. Therefore, a quotient of it (which is a union of $A$ and $B$ in $Sub(X)$) will be a coproduct of $A$ and $B$.

\end{proof}
A coproduct $A+B$ in a 2-category is \textbf{disjoint} if $A$ and $B$ are disjoint subobjects of $A+B$. We say a coherent 2-category is \textbf{positive} if any two objects have a disjoint coproduct. By Lemma \ref{DisjointUnion}, this is equivalent to saying that any two objects can be embedded as disjoint subobjects of some other object. Disjoint coproducts in a coherent 2-category are automatically stable under pullback, so a positive coherent 2-category is necessarily [[extensive 2-category|extensive]]. Conversely, we have:

\begin{lemma}
\label{}\hypertarget{}{}
A regular and extensive 2-category is coherent (and positive).

\end{lemma}
\begin{proof}
\end{proof}
\hypertarget{preservation}{}\subsubsection*{{Preservation}}\label{preservation}

If $K$ is coherent, then easily so are $K^{co}$, $disc(K)$, $gpd(K)$, $pos(K)$, and $Sub(1)$. Moreover, we have:

\begin{theorem}
\label{}\hypertarget{}{}
If $K$ is a coherent 2-category, so are the [[fibrational slice]]s $Opf(X)$ and $Fib(X)$ for any $X\in K$.

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

This is due to

\begin{itemize}%
\item [[Michael Shulman]], \emph{[[michaelshulman:coherent 2-category]]}

\end{itemize}
based on

\begin{itemize}%
\item [[Ross Street]], \emph{[[StreetCBS]]}

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



\end{document}