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\section*{κ-ary exact category}

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

\hypertarget{regular_and_exact_categories}{}\paragraph*{{Regular and Exact categories}}\label{regular_and_exact_categories}

[[!include regular and exact categories - contents]]

\hypertarget{category_theory}{}\paragraph*{{Category Theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{ary_regular_and_exact_categories}{}\section*{{$\kappa$-ary regular and exact categories}}\label{ary_regular_and_exact_categories}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{sinks_and_relations}{Sinks and relations}\dotfill \pageref*{sinks_and_relations} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{ary_regularity_and_exactness}{$\kappa$-ary regularity and exactness}\dotfill \pageref*{ary_regularity_and_exactness} \linebreak
\noindent\hyperlink{examples_2}{Examples}\dotfill \pageref*{examples_2} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{the_2category_of_ary_exact_categories}{The 2-category of $\kappa$-ary exact categories}\dotfill \pageref*{the_2category_of_ary_exact_categories} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The notions of [[regular category]], [[exact category]], [[coherent category]], [[extensive category]], [[pretopos]], and [[Grothendieck topos]] can be nicely unified in a theory of ``familial regularity and exactness.'' This was apparently first noticed by [[Ross Street]], and expanded by [[Mike Shulman]] with a generalized theory of [[exact completion]].

\hypertarget{sinks_and_relations}{}\subsection*{{Sinks and relations}}\label{sinks_and_relations}

Let $C$ be a [[finitely complete category]]. By a [[sink]] in $C$ we mean a family $\{f_i\colon A_i \to B\}_{i\in I}$ of [[morphism]]s with common [[target]]. A sink $\{f_i\colon A_i \to B\}$ is \textbf{[[extremal epimorphism|extremal epic]]} if it doesn't factor through any proper [[subobject]] of $B$. The \emph{[[pullback]]} of a sink along a morphism $B' \to B$ is defined in the evident way.

By a (many-object) \textbf{[[relation]]} in $C$ we will mean a family of objects $\{A_i\}_{i\in I}$ together with, for every $i,j\in I$, a monic span $A_i \leftarrow R_{i j} \to A_j$ (that is, a [[subobject]] $R_{i j}$ of $A_i \times A_j$. We say such a relation is:

\begin{itemize}%
\item [[reflexive relation|reflexive]] if $R_{i i}$ contains the diagonal $A_i \to A_i \times A_i$, for all $i$,
\item [[transitive relation|transitive]] if the pullback $R_{i j} \times_{A_j} R_{j k}$ factors through $R_{i k}$, for all $i,j,k$,
\item [[symmetric relation|symmetric]] if $R_{i j}$ contains, hence is equal to, the transpose of $R_{j i}$ for all $i,j$, and
\item a [[congruence]] if it is reflexive, transitive, and symmetric; this is an internal notion of (many-object) [[equivalence relation]].

\end{itemize}
Abstractly, reflexive and transitive relations can be identified with categories [[enriched category|enriched]] in a suitable [[bicategory]]; see (Street 1984). Congruences can be identified with enriched $\dagger$-[[dagger-categories|categories]].

A \textbf{quotient} for a relation is a [[colimit]] for the diagram consisting of all the $A_i$ and all the spans $A_i \leftarrow R_{i j} \to A_j$. And the \textbf{kernel} of a sink $\{f_i\colon A_i\to B\}$ is the relation on $\{A_i\}$ with $R_{i j} = A_i \times_B A_j$. It is evidently a congruence.

Finally, a sink is called \textbf{[[effective epimorphism|effective-epic]]} if it is the quotient of its kernel. It is called \textbf{universally effective-epic} if any pullback of it is effective-epic.

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

\begin{itemize}%
\item If ${|I|} = 1$, a congruence is the same as the ordinary internal notion of [[congruence]]. In this case [[quotient object|quotients]] and [[kernel pair|kernels]] reduce to the usual notions.


\item If ${|I|} = 0$, a congruence contains no data and a sink is just an object in $C$. The empty congruence is, trivially, the kernel of the empty sink with any target $B$, and a quotient for the empty congruence is an [[initial object]].


\item Given a family of objects $\{A_i\}$, define a congruence by $R_{i i}=A_i$ and $R_{i j}=0$ (an initial object) if $i \neq j$. Call a congruence of this sort \emph{trivial} (empty congruences are always trivial). Then a quotient for a trivial congruence is a [[coproduct]] of the objects $A_i$, and the kernel of a sink $\{f_i\colon A_i\to B\}$ is trivial iff the $f_i$ are disjoint monomorphisms.



\end{itemize}
\hypertarget{ary_regularity_and_exactness}{}\subsection*{{$\kappa$-ary regularity and exactness}}\label{ary_regularity_and_exactness}

Let $\kappa$ be an [[arity class]]. We call a sink or relation \textbf{$\kappa$-ary} if the cardinality ${|I|}$ is $\kappa$-small. As usual for arity classes, the cases of most interest have special names:

\begin{itemize}%
\item When $\kappa = \{1\}$ we say \textbf{unary}.
\item When $\kappa = \omega$ is the set of finite cardinals, we say \textbf{finitary}.
\item When $\kappa$ is the class of all cardinal numbers, we say \textbf{infinitary}.

\end{itemize}
\begin{theorem}
\label{KRegular}\hypertarget{KRegular}{}
For a category $C$, the following are equivalent:

\begin{enumerate}%
\item $C$ has finite limits, every $\kappa$-ary sink in $C$ factors as an extremal epic sink followed by a monomorphism, and the pullback of any extremal epic $\kappa$-ary sink is extremal epic.


\item $C$ has finite limits, and the kernel of any $\kappa$-ary sink in $C$ is also the kernel of some universally effective-epic sink.


\item $C$ is a [[regular category]] and has pullback-stable [[joins]] of $\kappa$-small families of [[subobjects]].



\end{enumerate}
\end{theorem}
When these conditions hold, we say $C$ is \textbf{$\kappa$-ary regular}, or alternatively \textbf{$\kappa$-ary coherent}. There are also some other more technical characterizations; see \hyperlink{Shulman}{Shulman}.

\begin{theorem}
\label{KExact}\hypertarget{KExact}{}
For a category $C$, the following are equivalent:

\begin{enumerate}%
\item $C$ has finite limits, and every $\kappa$-ary congruence is the kernel of some universally effective-epic sink.


\item $C$ is $\kappa$-ary regular, and every $\kappa$-ary congruence is the kernel of some sink.


\item $C$ is both [[exact category|exact]] and $\kappa$-ary [[extensive category|extensive]].



\end{enumerate}
\end{theorem}
When these conditions hold, we say that $C$ is \textbf{$\kappa$-ary exact}, or alternatively a \textbf{$\kappa$-ary pretopos}.

\hypertarget{examples_2}{}\subsection*{{Examples}}\label{examples_2}

\begin{enumerate}%
\item $C$ is [[regular category|regular]] iff it is unary regular.
\item $C$ is [[coherent category|coherent]] iff it is finitary regular.
\item $C$ is [[coherent category|infinitary-coherent]] iff it is [[well-powered category|well-powered]] and infinitary regular.
\item $C$ is [[exact category|exact]] iff it is unary exact.
\item $C$ is a [[pretopos]] iff it is finitary exact.
\item $C$ is an [[pretopos|infinitary pretopos]] iff it is [[well-powered category|well-powered]] and infinitary exact.

\end{enumerate}
Some other sorts of [[exactness properties]] (especially [[lex-colimits]]) can also be characterized in terms of congruences, kernels, and quotients. For instance:

\begin{enumerate}%
\item $C$ is $\kappa$-ary [[extensive category|lextensive]] iff every $\kappa$-ary \emph{trivial} congruence has a pullback-stable quotient of which it is the kernel.

\end{enumerate}
In \hyperlink{Street}{Street}, there is also a version of regularity and exactness that applies even to some \emph{large} sinks and congruences, and implies some small-generation properties of the category as well.

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

In a $\kappa$-ary regular category,

\begin{itemize}%
\item Every extremal-epic $\kappa$-ary sink is the quotient of its kernel.
\item Any $\kappa$-ary congruence that is a kernel has a quotient.

\end{itemize}
Thus, in a $\kappa$-ary exact category,

\begin{itemize}%
\item Every $\kappa$-ary congruence has a quotient.

\end{itemize}
In a $\kappa$-ary regular category, the class of all $\kappa$-small and effective-epic families generates a [[topology]], called its $\kappa$-canonical topology. This topology makes it a [[∞-ary site]].

\hypertarget{the_2category_of_ary_exact_categories}{}\subsection*{{The 2-category of $\kappa$-ary exact categories}}\label{the_2category_of_ary_exact_categories}

A functor $F:C\to D$ between $\kappa$-ary exact categories is called \textbf{$\kappa$-ary exact} if it preserves finite limits and $\kappa$-small effective-epic (or equivalently extremal-epic) families.

The resulting 2-category $EX_\kappa$ is a full [[reflective subcategory|reflective]] sub-2-category of the 2-category $SITE_\kappa$ of [[∞-ary sites]]. The reflector is called [[exact completion]].

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

\begin{itemize}%
\item [[Ross Street]], ``The family approach to total cocompleteness and toposes.'' Transactions of the AMS 284 no. 1, 1984

\end{itemize}
\begin{itemize}%
\item [[Michael Shulman]], ``Exact completions and small sheaves''. \emph{Theory and Applications of Categories}, Vol. 27, 2012, No. 7, pp 97-173. \href{http://www.tac.mta.ca/tac/volumes/27/7/27-07abs.html}{Free online}

\end{itemize}
[[!redirects familial regularity and exactness]] [[!redirects familial regularity]] [[!redirects familial exactness]] [[!redirects familially exact category]] [[!redirects familially regular category]] [[!redirects ∞-ary exact category]] [[!redirects ∞-ary exact categories]] [[!redirects k-ary exact category]] [[!redirects k-ary exact categories]] [[!redirects ∞-ary exactness]] [[!redirects k-ary exactness]] [[!redirects ∞-ary regular category]] [[!redirects ∞-ary regular categories]] [[!redirects k-ary regular category]] [[!redirects k-ary regular categories]] [[!redirects ∞-ary regularity]] [[!redirects k-ary regularity]] [[!redirects ∞-ary pretopos]] [[!redirects ∞-ary pretoposes]] [[!redirects ∞-ary pretopoi]] [[!redirects k-ary pretopos]] [[!redirects k-ary pretoposes]] [[!redirects k-ary pretopoi]] [[!redirects ∞-ary coherent category]] [[!redirects ∞-ary coherent categories]] [[!redirects k-ary coherent category]] [[!redirects k-ary coherent categories]]

[[!redirects ∞-ary regular and exact categories]]



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