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\begin{document}

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\section*{finite object}

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

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

[[!include category theory - contents]]

\hypertarget{compact_objects}{}\paragraph*{{Compact objects}}\label{compact_objects}

[[!include compact object - contents]]

\hypertarget{finite_objects}{}\section*{{Finite objects}}\label{finite_objects}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Definitions}{Definitions}\dotfill \pageref*{Definitions} \linebreak
\noindent\hyperlink{ExternalDefinition}{External version}\dotfill \pageref*{ExternalDefinition} \linebreak
\noindent\hyperlink{InternalDefinition}{Internal version}\dotfill \pageref*{InternalDefinition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{closure_of_finite_objects}{Closure of finite objects}\dotfill \pageref*{closure_of_finite_objects} \linebreak
\noindent\hyperlink{subcategories_of_finite_objects}{Subcategories of finite objects}\dotfill \pageref*{subcategories_of_finite_objects} \linebreak
\noindent\hyperlink{relation_to_slice_toposes}{Relation to slice toposes}\dotfill \pageref*{relation_to_slice_toposes} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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}

The notion of \emph{[[finite object]]} in a [[category]] -- notably in a [[topos]] -- is a generalisation of the notion of [[finite set]] in [[Set|the category of sets]].

As there are already at least five distinct notions of [[finite set]] in [[constructive mathematics]], so there must be at least five distinct notions of finite object internal to a [[topos]]. Additionally, the definitions may also be interpreted in an `external' sense, giving even further notions. Only some are mentioned below.

Also beware that in [[category theory]] the term `finite object' is also used in a much more general sense to mean a \emph{[[compact object]]}. Similar finiteness meaning may also be attributed to [[dualizable objects]] in [[monoidal categories]] and to [[perfect complexes]] (of [[abelian sheaves]]) in [[geometry]].

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

Consider an ambient [[topos]] $\mathcal{T}$. Assume that $\mathcal{T}$ is equipped with a [[natural numbers object]] $N$. Write $N_{\lt} \hookrightarrow N\times N$ is its strict order relation.

\hypertarget{ExternalDefinition}{}\subsubsection*{{External version}}\label{ExternalDefinition}

A ``finite set'' in $\mathcal{T}$ in the strictest sense is usually called a \textbf{finite cardinal}. This is an [[object]] $[n] \in \mathcal{C}$ which is the [[pullback]] of $N_{\lt}\to N$ along some [[global element]] $n:1\to N$.

We can then consider [[subobjects]], [[quotient objects]], and [[subquotient objects]] of finite cardinals to obtain external versions of subfinite, finitely indexed, and subfinitely indexed sets.

\hypertarget{InternalDefinition}{}\subsubsection*{{Internal version}}\label{InternalDefinition}

The \emph{internal} version of a ``finite set'' is an object $X$ such that ``$X$ is a finite cardinal'' is true in the [[internal logic]]. This is equivalent to the following

\begin{defn}
\label{KFinite}\hypertarget{KFinite}{}
An object $X \in \mathcal{T}$ is \emph{locally} isomorphic to a finite cardinal, if there is an [[epimorphism]] $U\to 1$ and a [[generalized element]] $n:U\to N$ such that $U\times X \cong n^*(N_\lt)$ over $U$. Equivalently, there is a $U\to 1$ such that $U\times X$ is a finite cardinal in the [[slice topos]] $S/U$.

An \textbf{internally finitely indexed object} is an object $X$ is which is locally a [[quotient]] of a finite cardinal, hence such that there is an [[epimorphism]] $U \to *$, a finite cardinal in the slice topos $n \in \mathcal{T}_{/U}$ and an epimorphism $n \to U \times X$.

\end{defn}
An ``internally finitely indexed'' object is generally called a \textbf{Kuratowski-finite object} or \textbf{$K$-finite object} for short, and an ``internally subfinitely indexed'' one is called a \textbf{$\tilde{K}$-finite object}.

There is a more general definition of K-finite objects that does not need to assume the presence of a [[natural number object]]. See (\hyperlink{Johnstone}{Johnstone, theorem D5.4.13}).

Since it is still provable in the [[internal logic]] that any decidable finitely indexed set is finite, the ``internally finite'' objects (those that are locally isomorphic to a finite cardinal, as above) can be characterized as the \textbf{decidable $K$-finite objects}.

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

\hypertarget{closure_of_finite_objects}{}\subsubsection*{{Closure of finite objects}}\label{closure_of_finite_objects}

The following lists closure properties of K-finite objects, def. \ref{KFinite}.

\begin{prop}
\label{}\hypertarget{}{}
\begin{enumerate}%
\item The [[initial object]] and the [[terminal object]] are K-finite.


\item The [[image]] of a K-finite object under an [[epimorphism]] is K-finite.


\item The [[union]] of two K-finite [[subobjects]] is K-finite.


\item A [[coproduct]] is K-finite precisely if both summands are.


\item A [[subterminal object]] is K-finite precisely if it is a [[complemented subobject]].


\item A [[product]] of two K-finite objects is K-finite.



\end{enumerate}
\end{prop}
This appears in (\hyperlink{Johnstone}{Johnstone}) as lemma D5.4.4, corollary D5.4.5, pro. 5.4.8.

\hypertarget{subcategories_of_finite_objects}{}\subsubsection*{{Subcategories of finite objects}}\label{subcategories_of_finite_objects}

The [[full subcategory]] of finite cardinals in any [[topos]] is again a topos, and it is [[Boolean topos|Boolean]]. Its [[subobject classifier]] is $2=1\sqcup 1$, which in the ambient topos is the classifier only of [[decidable object|decidable]] subobjects. This means that [[classical logic|classically]] valid arguments, including all of finitary combinatorics, can generally be applied easily to finite cardinals, as long as we always interpret ``subset'' to mean ``decidable subset.''

\begin{theorem}
\label{}\hypertarget{}{}
The [[full subcategory]] $\mathcal{T}_{dKf} \hookrightarrow \mathcal{T}$ of decidable $K$-finite objects in a topos $\mathcal{T}$ is a [[Boolean topos]] whose subobject classifier is $2$.

The category of $K$-finite objects is a topos if and only if every $K$-finite object is decidable, and the category of $\tilde{K}$-finite objects is a topos if (but not only if) the subobject classifier is $K$-finite.

\end{theorem}
The first statement appears as (\hyperlink{Johnstone}{Johnstone, theorem 5.4.18}).

\begin{remark}
\label{}\hypertarget{}{}
The full subcategory $\mathcal{T}_{dKf} \hookrightarrow \mathcal{T}$ can be regarded as the ``[[stack]] completion'' of the topos of finite cardinals.

\end{remark}
\hypertarget{relation_to_slice_toposes}{}\subsubsection*{{Relation to slice toposes}}\label{relation_to_slice_toposes}

\begin{prop}
\label{}\hypertarget{}{}
An object $X \in \mathcal{T}$ is K-finite precisely if the [[étale geometric morphism]]

\begin{displaymath}
\mathcal{T}_{/X} \to \mathcal{T}
\end{displaymath}
out of the [[slice topos]] is a [[proper geometric morphism]].

\end{prop}
(\hyperlink{MoerdijkVermeulen}{Moerdijk-Vermeulen, examples III 1.4})

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

\begin{itemize}%
\item In any [[Boolean topos]], all four internal notions coincide. In a [[well-pointed topos]], each internal notion coincides with its external notion. Therefore, in a well-pointed Boolean topos, including the topos [[Set]] as usually conceived, all notions of finiteness coincide.


\item In a [[presheaf]] topos $[C^{op},Set]$, the finite cardinals are the finite-set-valued functors which are constant on each connected component. In particular, if $C$ is a [[group]], then the topos of finite cardinals is equivalent to [[FinSet]].


\item Likewise, in the [[Grothendieck topos]] $Sh(X)$ of [[sheaf|sheaves]] on a space $X$, the finite cardinals are the locally constant functions $X\to N$. So if $X$ is connected, the topos of finite cardinals in $Sh(X)$ is also equivalent to $FinSet$.


\item Examples of such are [[tiny object]]s and [[infinitesimal object]]s in sheaf toposes.


\item By contrast, the $K$-finite objects in $[C^{op},Set]$ are the finite-set-valued functors each of whose transition functions is surjective, and the \emph{decidable} K-finite objects are the finite-set-valued functors each of whose transition functions is bijective.


\item In particular, if $C$ is a groupoid, the topos of decidable $K$-finite objects is equivalent to $[C^{op},FinSet]$. Since the topos of presheaves on a groupoid is Boolean, this gives an example of a Boolean topos in which the finite cardinals (``externally finite objects'') and the (decidable) $K$-finite objects (``internally finite objects'') fail to coincide.


\item In the [[category of sheaves]] $Sh(X)$ over a [[topological space]], the decidable K-finite objects are those that are ``locally finite;'' i.e. there is an [[open cover]] of $X$ such that over each open in the cover, the sheaf is a [[locally constant function]] to $N$. These are essentially the same as [[covering space|covering spaces]] of $X$ with finite fibres.



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

\begin{itemize}%
\item [[finite set]], [[hereditarily finite set]]


\item [[finite category]], [[finite limit]]


\item [[decidable object]], [[theory of objects]]


\item [[finite homotopy type]], [[finite spectrum]]


\item [[finite (infinity,1)-category]], [[finite (infinity,1)-limit]]



\end{itemize}
[[!include finite objects -- table]]

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

In

\begin{itemize}%
\item [[Peter Johnstone]], \emph{[[Sketches of an elephant]]}

\end{itemize}
finite cardinal objects are discussed in section D5.2, Kuratowski finite objects in section D5.4

See also

\begin{itemize}%
\item O. Acu\~n{}a-Ortega, [[Fred Linton]], \emph{Finiteness and decidability: I} , Springer Lecture Notes in Mathematics, (1979), Volume \textbf{753}, pp.80-100, (DOI: 10.1007/BFb0061813)


\item [[Peter Johnstone]], [[Fred Linton]], \emph{Finiteness and decidability: II} , Cambridge Philosophical Society Mathematical Proceedings of the Cambridge Philosophical Society (1978).


\item B. P. Chisala, M.-M. Mawanda, \emph{Counting Measure for Kuratowski Finite Parts and Decidability} , Cah.Top.G\'e{}om.Diff.Cat. \textbf{XXXII} 4 (1991) pp.345-353. (\href{archive.numdam.org/article/CTGDC_1991__32_4_345_0.pdf}{pdf})


\item S. J. Henry, \emph{Classifying Topoi and Preservation of Higher Order Logic by Geometric Morphisms} , PhD University of Michigan (2013). (\href{http://arxiv.org/abs/1305.3254}{arxiv})


\item C. Kuratowski, \emph{Sur la notion d'ensemble fini} , Fund. Math. \textbf{1} (1920) pp.129-131. (\href{http://matwbn.icm.edu.pl/ksiazki/fm/fm1/fm1117.pdf}{pdf})


\item [[Ieke Moerdijk]], J. Vermeulen, \emph{Relative compactness conditions for toposes} (\href{http://igitur-archive.library.uu.nl/math/2001-0702-142944/1039.pdf}{pdf}) and \emph{Proper maps of toposes} , American Mathematical Society (2000)


\item L. N. Stout, \emph{Dedekind finiteness in topoi} , JPAA \textbf{49} (1987) pp.219-225.


\item T. Streicher, P. Freyd, F. Linton, P.Johnstone, W. Lawvere, \emph{catlist discussion `finiteness in toposes'}, January 1997. (\href{http://www.mta.ca/~cat-dist/catlist/1999/finite-topos}{link})


\item [[Alfred Tarski|A. Tarski]], \emph{Sur les ensembles finis} , Fund. Math. \textbf{3} (1924) pp.45-95. (\href{http://matwbn.icm.edu.pl/ksiazki/fm/fm6/fm619.pdf}{pdf})


\item H. Volger, \emph{Ultrafilters, ultrapowers and finiteness in a topos} , JPAA \textbf{6} (1975) pp.345-356.



\end{itemize}
[[!redirects finite object]] [[!redirects finite objects]] [[!redirects finite]] [[!redirects finiteness]] [[!redirects B-finite object]] [[!redirects B-finite objects]] [[!redirects B-finite]] [[!redirects B-finiteness]] [[!redirects F-finite object]] [[!redirects F-finite objects]] [[!redirects F-finite]] [[!redirects F-finiteness]]

[[!redirects subfinite object]] [[!redirects subfinite objects]] [[!redirects subfinite]] [[!redirects subfiniteness]] [[!redirects B-tilde-finite object]] [[!redirects B-tilde-finite objects]] [[!redirects B-tilde-finite]] [[!redirects B-tilde-finiteness]] [[!redirects F-tilde-finite object]] [[!redirects F-tilde-finite objects]] [[!redirects F-tilde-finite]] [[!redirects F-tilde-finiteness]]

[[!redirects finitely indexed object]] [[!redirects finitely-indexed object]] [[!redirects finitely indexed objects]] [[!redirects finitely-indexed objects]] [[!redirects finitely indexed]] [[!redirects finitely-indexed]] [[!redirects Kuratowski-finite object]] [[!redirects Kuratowski finite object]] [[!redirects Kuratowski-finite objects]] [[!redirects Kuratowski finite objects]] [[!redirects Kuratowski-finite]] [[!redirects Kuratowski finite]] [[!redirects Kuratowski finiteness]] [[!redirects Kuratowski-finiteness]] [[!redirects K-finite object]] [[!redirects K-finite objects]] [[!redirects K-finite]] [[!redirects K-finiteness]]

[[!redirects subfinitely indexed object]] [[!redirects subfinitely-indexed object]] [[!redirects subfinitely indexed objects]] [[!redirects subfinitely-indexed objects]] [[!redirects subfinitely indexed]] [[!redirects subfinitely-indexed]] [[!redirects Kuratowski-subfinite object]] [[!redirects Kuratowski subfinite object]] [[!redirects Kuratowski-subfinite objects]] [[!redirects Kuratowski subfinite objects]] [[!redirects Kuratowski-subfinite]] [[!redirects Kuratowski subfinite]] [[!redirects Kuratowski subfiniteness]] [[!redirects Kuratowski-subfiniteness]] [[!redirects K-tilde-finite object]] [[!redirects K-tilde-finite objects]] [[!redirects K-tilde-finite]] [[!redirects K-tilde-finiteness]]

[[!redirects Dedekind-finite object]] [[!redirects Dedekind finite object]] [[!redirects Dedekind-finite objects]] [[!redirects Dedekind finite objects]] [[!redirects Dedekind-finite]] [[!redirects Dedekind finite]] [[!redirects Dedekind finiteness]] [[!redirects Dedekind-finiteness]] [[!redirects D-finite object]] [[!redirects D-finite objects]] [[!redirects D-finite]] [[!redirects D-finiteness]]



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