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\section*{extensive 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{extensive_categories}{}\section*{{Extensive categories}}\label{extensive_categories}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{superextensive_sites}{Superextensive sites}\dotfill \pageref*{superextensive_sites} \linebreak
\noindent\hyperlink{prelextensive_categories}{Pre-lextensive categories}\dotfill \pageref*{prelextensive_categories} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{discussions}{Discussions}\dotfill \pageref*{discussions} \linebreak


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

A [[category]] is \emph{extensive} if it has [[coproducts]] that interact well with [[pullbacks]]. Variations (some only terminological) include \emph{lextensive}, \emph{disjunctive}, and \emph{positive} categories. All of these come in \emph{finitary} and \emph{infinitary} versions (and, more generally, $\kappa$-ary versions for any [[arity class]] $\kappa$).

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

A \textbf{finitely extensive category} (or \textbf{finitary extensive category}) is a category $E$ with finite [[coproduct|coproducts]] such that one, and hence all, of the following equivalent conditions holds:

\begin{enumerate}%
\item Pullbacks of finite-coproduct injections along arbitrary morphisms exist and finite coproducts are [[disjoint coproduct|disjoint]] and [[pullback stability|stable under pullback]].
\item For any objects $a,b$ the coproduct functor $E/a \times E/b \to E/(a+b)$ is an [[equivalence of categories]].
\item In any commutative diagram\begin{displaymath}
\itexarray{x & \to & z & \leftarrow & y\\
\downarrow  & &   \downarrow  &  &  \downarrow \\
a & \to & a+b & \leftarrow & b}
\end{displaymath}
the two squares are pullbacks if and only if the top row is a coproduct diagram.


\item Finite coproducts are [[van Kampen colimits]]. By definition, this is to say one of the previous two conditions (depending on the definition chosen).

\end{enumerate}
An \textbf{infinitary extensive category} is a category $E$ with all (small) [[coproduct]]s such that the following analogous equivalent conditions hold:

\begin{enumerate}%
\item Pullbacks of coproduct injections along arbitrary morphisms exist and finite coproducts are disjoint and stable under pullback.
\item For any small family $(a_i)$ of objects, the coproduct functor $\prod_i (E/a_i) \to E/_{(\coprod_i a_i)}$ is an equivalence of categories.
\item For any family of commutative squares\begin{displaymath}
\itexarray{ x_i & \to & z \\\downarrow &&\downarrow^f \\ a_i & \to & \coprod_i a_i }
\end{displaymath}
in which the bottom family of morphisms is the coproduct injections and the right-hand morphism is always the same, the top family are the injections of a coproduct diagram (hence $z = \coprod_i x_i$) if and only if all the squares are pullbacks.


\item All small coproducts are [[van Kampen colimits]].

\end{enumerate}
In between, a \textbf{$\kappa$-ary extensive category} (for $\kappa$ a [[cardinal number]] or an [[arity class]]) is one with disjoint and stable coproducts of fewer than $\kappa$ objects. The unqualified term \textbf{extensive category} can refer to either the finitary or infinitary version, depending on the author; the more usual meaning is the finitary version.

Extensive categories are also called \textbf{positive categories}, especially if they are also [[coherent category|coherent]]. Note that any disjoint coproduct in a coherent category is automatically pullback-stable. A positive coherent category which is also [[exact category|exact]] is called a [[pretopos]]. Infinitary pretoposes encapsulate all the exactness conditions of Giraud's theorem characterizing [[Grothendieck topos|Grothendieck toposes]] (the remaining condition is the existence of a small [[generating set]]).

If an extensive category also has [[finitely complete category|finite limits]], it is called \textbf{lextensive} or \textbf{disjunctive}. (Note that the more usual default meaning of `disjunctive', unlike the other terms, is the infinitary case.)

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

\begin{itemize}%
\item The alternative definitions of finitary disjunctive refer only to binary coproducts, but they obviously imply analogous statements for $n$-ary coproducts for all finite $n \ge 1$. Less obviously, they also imply the analogous statement for $0$-ary coproducts (that is, [[initial object]]s). In this case, the statement is that the initial object 0 is \emph{strict} (any map $a\to 0$ is an isomorphism).


\item Furthermore, if binary coproducts are disjoint, then (at least assuming [[classical logic]]) any infinitary coproducts that exist are also disjoint, since

\begin{displaymath}
\bigsqcup_{a\in A} X_a \cong
X_{a_0} \sqcup \bigsqcup_{a\neq a_0} X_a \cong
X_{a_0} \sqcup X_{a_1} \sqcup \bigsqcup_{a\neq a_0,a_1} X_a
\end{displaymath}
for any $a_0, a_1\in A$. Therefore, if a finitary-extensive category has infinitary pullback-stable coproducts, it is necessarily infinitary-extensive. In particular, a cocomplete [[locally cartesian closed category]] is finitary extensive if and only if it is infinitary extensive.


\item As a further special case of the preceding, since an [[elementary topos]] is finitary extensive, any cocomplete elementary topos is infinitary extensive. However, in this case, one of the arguments for finitary extensivity applies directly to the infinitary case and does \emph{not} require classical logic; see [[toposes are extensive]].


\item See [[familial regularity and exactness]] for a generalization of extensivity and its relationship to [[exact category|exactness]].


\item Any extensive category with finite [[products]] is automatically a [[distributive category]].



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

\begin{enumerate}%
\item An [[elementary topos]] is finitary lextensive; a [[Grothendieck topos]] (or, more generally, any [[cocomplete category|cocomplete]] elementary topos) is infinitary lextensive.


\item A [[quasitopos]] with disjoint coproducts, or more generally a [[locally cartesian closed category]] with disjoint coproducts, is extensive. (Of course not all quasitoposes have disjoint coproducts, one example being a [[complete Heyting algebra]].)


\item The category [[Top]] of [[topological spaces]] is infinitary lextensive. The category [[Diff]] of smooth [[manifolds]] is infinitary extensive, though it lacks all pullbacks.


\item The category of [[schemes]] is infinitary lextensive. In more detail: the category of functors $CRing \to Set$ is infinitary lextensive (since finite limits and small coproducts are computed pointwise in $Set$), then sheaves with respect to the Zariski topology on $CRing^{op}$ form an infinitary lextensive category (since finite limits and small coproducts are reflected back from $[CRing, Set]$ by applying a left exact reflection to the inclusion of sheaves in presheaves). Finally, the category of schemes, as a full subcategory of the Zariski sheaves, are closed under finite limits and small coproducts. (Some discussion of these points can be found at the \href{http://nforum.mathforge.org/discussion/5149/extensive-category/}{nForum}, particularly in comment \#18.)


\item The category of affine schemes (opposite to the category of commutative [[ring]]s with identity) is lextensive, but (perhaps contrary to geometric intuition) \emph{not} infinitary lextensive. Some details may be found \href{http://ncatlab.org/toddtrimble/published/Dippy+disproof+of+infinitary+extensivity+of+affine+schemes}{here}.


\item The category [[Cat]] is infinitary lextensive.


\item The category [[Vect]] is not even finitely extensive.



\end{enumerate}
\hypertarget{superextensive_sites}{}\subsection*{{Superextensive sites}}\label{superextensive_sites}

Any extensive category admits a [[Grothendieck topology]] whose [[cover|covering families]] are (generated by) the families of inclusions into a [[coproduct]] (finite or small, as appropriate). We call this the \textbf{extensive [[coverage]]} or \textbf{extensive topology}. The [[codomain fibration]] of any extensive category is a [[stack]] for its extensive topology.

In general, we call a [[site]] \textbf{superextensive} if its underlying category is extensive, its covering families are generated by (finite or small) families, and its coverage includes the extensive one. See [[superextensive site]] for more details.

\hypertarget{prelextensive_categories}{}\subsection*{{Pre-lextensive categories}}\label{prelextensive_categories}

Extensivity is an ``exactness'' condition, analogous to being a [[exact category]] or a [[pretopos]] (a pretopos being precisely a category that is exact and finitary-extensive). The corresponding ``regularity'' condition analogous to being a [[regular category]] or a [[coherent category]] is not well-known, but is not hard to write down.

Let us say (without making any assertion that this is good or permanent terminology) that a category is \textbf{pre-lextensive} if

\begin{enumerate}%
\item it has a [[strict initial object]] $0$ (equivalently, its [[subobject]] [[preorders]] have pullback-stable bottom elements), and
\item whenever $A\rightarrowtail X$ and $B\rightarrowtail X$ are disjoint subobjects (i.e. $A\cap B=0$), they have a pullback-stable union (which is then automatically a disjoint and stable coproduct).

\end{enumerate}
This is intended to complete the table of analogies:

\begin{tabular}{l|l}
some&all\\
\hline 
[[regular category]]&[[exact category]]\\
[[coherent category]]&[[pretopos]]\\
pre-lextensive category&lextensive category\\
\end{tabular}

Regular/exact categories have quotients of (some) [[congruences]]. Exact categories have quotients of all congruences, while regular ones have quotients only of congruences that are [[kernel pairs]]. Another way to say that is that in a regular category, you can conclude that the quotient of some congruence exists if you can exhibit another object of which the quotient would be a [[subobject]] if it existed. Similarly, pre-/lextensive categories have [[disjoint unions]]. Lextensive categories have all disjoint unions (= [[coproducts]]), while in a pre-lextensive category you can conclude that a pair of objects $X,Y$ have a disjoint union if you can exhibit another object in which $X$ and $Y$ can be embedded disjointly. Finally, coherent categories/pretoposes have both quotients and disjoint unions, or equivalently quotients and not-necessarily-disjoint unions, with the same sort of relationship between the two.

Evidently a pre-lextensive category is lextensive as soon as any two objects can be embedded disjointly in a third. Pre-lextensive categories also suffice for the interpretation of [[internal logic|disjunctive logic]].

Being pre-lextensive is also sufficient to define the extensive topology and show that it is subcanonical, since it implies that whatever disjoint coproducts exist must be pullback-stable. The codomain fibration of a pre-lextensive category is not necessarily a stack for its extensive topology, but this condition is weaker than extensivity. It is true, however, that if $C$ is a pre-lextensive category whose codomain fibration is a stack for its extensive topology, and in which the disjoint coproduct $1+1$ exists, then $C$ is extensive, for arbitrary disjoint (binary) coproducts can then be obtained by descent along the covering family $(1\to 1+1, 1\to 1+1)$.

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

\begin{itemize}%
\item [[extensive 2-category]]


\item [[disjunctive (∞,1)-category]]



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

Carboni, Aurelio and Lack, Stephen and Walters, R. F. C., \emph{Introduction to extensive and distributive categories}, JPAA 84 no. 2

\hypertarget{discussions}{}\subsection*{{Discussions}}\label{discussions}

While creating this page, we had the following discussion regarding ``finitely'' versus ``infinitary.''

Can we say `small-extensive'? Or even redefine `extensive' to have this meaning, using `finitely extensive' for the first version? ---Toby

I think ``extensive'' is pretty well established for the finite version, and I would be reluctant to try to change it. I wouldn't object too much to ``small-extensive'' for the infinitary version in principle, but $\infty$-positive is used in the Elephant and possibly elsewhere. I think the topos theorists think by analogy with $\infty$-pretopos, which I don't think we have much hope of changing, despite the unfortunate clash with ``$\infty$-topos.'' But you can use ``finitary disjunctive'' and ``disjunctive'' in the lex case, which most examples are. -Mike

\emph{Mike}: Okay, I just ran across one paper that uses ``(infinitary) extensive'' for the infinitary version the first time it was introduced, and then dropped the parenthetical for the rest of the paper. I also recall seeing ``extensive fibration'' used for a fibration having disjoint and stable indexed coproducts, which is certainly a (potentially) infinitary notion. So perhaps there is no real consensus on whether ``extensive'' definitely implies the finite version or the infinitary one.

\emph{Toby}: It would be nice to not overload the prefix `$\infty$-' so much. It's like `continuous'; default to small.

\emph{Mike}: I agree that it would be nice to avoid $\infty$-. What if we do what we did for [[omega-category]]? That is, if you want to be unambiguous, say either ``finitary extensive'' or ``infinitary extensive,'' and in any particular context you are allowed to define ``extensive'' at the beginning to be one of the two and use it without prefix in what follows.

\emph{Toby}: Sure. Of course, the general concept is $\kappa$-extensive, where $\kappa$ is any cardinal (which we may assume to be regular).

[[!redirects extensive category]] [[!redirects extensive categories]]

[[!redirects extensive topology]] [[!redirects extensive topologies]] [[!redirects extensive coverage]] [[!redirects extensive coverages]]

[[!redirects lextensive category]] [[!redirects lextensive categories]]

[[!redirects disjunctive category]] [[!redirects disjunctive categories]]

[[!redirects infinitary extensive category]] [[!redirects infinitary extensive categories]]

[[!redirects positive category]] [[!redirects positive categories]] [[!redirects positive coherent category]] [[!redirects positive coherent categories]]



\end{document}