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\begin{document}

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\section*{tiny object}

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

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

[[!include compact object - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{InCategoriesOfModulesOverRings}{In categories of modules over rings}\dotfill \pageref*{InCategoriesOfModulesOverRings} \linebreak
\noindent\hyperlink{in_presheaf_categories}{In presheaf categories}\dotfill \pageref*{in_presheaf_categories} \linebreak
\noindent\hyperlink{AtomsInALocalTopos}{In a local topos}\dotfill \pageref*{AtomsInALocalTopos} \linebreak
\noindent\hyperlink{AtomsInACohesiveTopos}{In a cohesive topos}\dotfill \pageref*{AtomsInACohesiveTopos} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\begin{defn}
\label{}\hypertarget{}{}
Let $E$ be a [[locally small category]] with all small [[colimits]]. An object $e$ of $E$ is called \textbf{tiny} or \textbf{small-[[projective object]]} (\hyperlink{Kelly}{Kelly, \S{}5.5}) if the [[hom-functor]] $E(e, -) : E \to Set$ preserves small colimits.

More generally, for $V$ a [[cosmos]] and $E$ a $V$-[[enriched category]], $e \in E$ is called tiny if $E(e,-) : E \to V$ preserves all small colimits.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Since being an [[epimorphism]] is a ``colimit-property'' (a morphism is epic iff its [[pushout]] with itself consists of identities), if $e$ is tiny then $E(e,-)$ preserves epimorphisms, which is to say that $e$ is [[projective object|projective]] (with respect to epimorphisms). This is presumably the origin of the term ``small-projective'', i.e. the corepresentable functor preserves small colimits instead of just a certain type of finite one.

\end{remark}
\begin{defn}
\label{Atomic}\hypertarget{Atomic}{}
If $E$ is [[cartesian closed category|cartesian closed]] and the [[inner hom]] $(-)^e$ has a [[right adjoint]] (and hence preserves all colimits), $e$ is called (internally) \textbf{atomic} or [[infinitesimal object|infinitesimal]].

\end{defn}
(See for instance \hyperlink{Lawvere97}{Lawvere 97}.)

\begin{remark}
\label{}\hypertarget{}{}
The [[right adjoint]] in def. \ref{Atomic} is sometimes called an ``[[amazing right adjoint]]'', particularly in the context of [[synthetic differential geometry]].

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
Various terminological discrepancies in the literature hinge on the distinction between internal notions and external notions. Thus, if $E$ is a [[cartesian closed category]] with small colimits, we may say $e \in Ob(E)$ is \emph{internally} tiny if the functor $(-)^e: E \to E$ preserves small colimits. Relatedly, the word ``atomic'' has been used in both an external sense where $E(e, -): E \to Set$ has a right adjoint, as in Bunge's thesis, and in an internal sense, as when Lawvere refers to $e$ as an a.t.o.m. (``amazingly tiny object model'') if $(-)^e: E \to E$ has a right adjoint. But under certain hypotheses, the two notions coincide; see for instance Proposition \ref{coincide}.

\end{remark}
\begin{prop}
\label{InSheafToposTinyImpliesAtomic}\hypertarget{InSheafToposTinyImpliesAtomic}{}
If $E$ is a [[sheaf topos]], then (externally) tiny objects and externally atomic objects coincide.

\end{prop}
\begin{proof}
Clearly any externally atomic object is tiny. For the converse, use a dual form of the special [[adjoint functor theorem]] (SAFT): $E$ is [[locally small category|locally small]], [[cocomplete category|cocomplete]], and [[well-powered category|co-well-powered]] (because for any object $X$, equivalence classes of [[epimorphisms]] with domain $X$ are in natural bijection with [[internal equivalence relations]] on $X$, and there is a small set of these because they are contained in a set isomorphic to $\hom(X \times X, \Omega)$), and finally $E$ has a [[generating set]] (namely, the set of associated [[sheaves]] of [[representables]] coming from a small [[site]] presentation for $E$). Under these conditions, the SAFT guarantees that any [[cocontinuous functor]] $E \to C$ has a right adjoint, provided that $C$ is locally small; then apply this to the case $C = Set$.

\end{proof}
Clearly the statement and the proof of Proposition \ref{InSheafToposTinyImpliesAtomic} carry over when ``external'' is replaced by ``internal'' throughout.

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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{prop}
\label{}\hypertarget{}{}
Any [[retract]] of a tiny object is tiny, since [[split idempotent|splitting of idempotents]] is an [[absolute colimit]] (see also \hyperlink{Kelly}{Kelly, prop. 5.25}).

\end{prop}
\hypertarget{InCategoriesOfModulesOverRings}{}\subsubsection*{{In categories of modules over rings}}\label{InCategoriesOfModulesOverRings}

The notion of tiny object is clearly highly dependent on the base of [[enriched category|enrichment]]. For example, for a [[ring]] $R$, the tiny objects in the category of left $R$-[[category of modules|modules]] $Ab^R$, considered as an [[Ab]]-[[enriched category]], are the [[finitely generated module|finitely generated]] [[projective modules]]. Certainly f.g. projective modules are tiny because $R$ is tiny (the [[forgetful functor]] $\hom(R, -): Ab^R \to Ab$ preserves $Ab$-[[colimits]]) and the closure of $R$ under finite direct sums and retracts, which are absolute $Ab$-[[colimits]], comprise finitely generated projective modules. See also \emph{[[Cauchy completion]]}.

On the other hand, when the category $Ab^R$ is considered as a [[Set]]-[[enriched category]], there are \emph{no} tiny objects. In fact this is true for any Set-enriched category with a [[zero object]]: Let $X$ be a tiny object. The morphism $X \to 0$ induces a map $Hom(X,X) \to Hom(X,0)$. This map has empty codomain (since $Hom(X,-)$ preserves the zero object, as an empty colimit). Thus $Hom(X,X) = \emptyset$ in contradiction to $id_X \in Hom(X,X)$.

\hypertarget{in_presheaf_categories}{}\subsubsection*{{In presheaf categories}}\label{in_presheaf_categories}

\begin{example}
\label{}\hypertarget{}{}
In a [[presheaf category]] every [[representable functor|representable]] is a tiny object:

since colimits of presheaves are computed objectwise (see [[limits and colimits by example]]) and using the [[Yoneda lemma]] we have for $U$ a [[representable]] functor and $F : J \to PSh$ a diagram that

\begin{displaymath}
Hom(U, \lim_\to F) \simeq (\lim_\to F)(U) \simeq \lim_\to F(U)
\end{displaymath}
where now the last [[colimit]] is in [[Set]].

\end{example}
Thus, in a presheaf category, any [[retract]] of a representable functor is tiny. In fact the converse also holds:

\begin{prop}
\label{}\hypertarget{}{}
The tiny objects in a [[presheaf category]] are precisely the [[retracts]] of [[representable functor]]s.

\end{prop}
This is for instance (\hyperlink{BorceuxDejean}{BorceuxDejean, prop 2}). For instance, the only tiny object in [[G-set]] is $G$ itself with its regular action.

Thus, if the domain category is [[Cauchy complete category|Cauchy complete]] (has [[split idempotent]]s), then every tiny presheaf is representable; and more generally the Cauchy completion or [[Karoubi envelope]] of a category can be defined to consist of the tiny presheaves on it. See [[Cauchy complete category]] for more on this.

\begin{prop}
\label{coincide}\hypertarget{coincide}{}
For presheaves on a category $C$ with finite products, the notions of externally tiny object and internally tiny object coincide.

\end{prop}
\begin{proof}
Without loss of generality, we may assume $C$ is Cauchy complete (note that the Cauchy completion of a category with finite products again has finite products), so that tiny presheaves coincide with representable functors $C(-, c)$.

Let $E$ denote the presheaf category. Given that the empty product $1$ is tiny, if $e \in Ob(E)$ is internally tiny, then the composite

\begin{displaymath}
E(e, -): E \to Set = \left(E \stackrel{(-)^e}{\to} E \stackrel{E(1,-)}{\to} Set \right)
\end{displaymath}
is cocontinuous, hence $e$ is externally tiny.

In the other direction, recall how exponentials $G^F$ in $E = PSh(C)$ are constructed: we have the formula

\begin{displaymath}
G^F(c) = E(C(-, c) \times F, G).
\end{displaymath}
In particular, if $F$ is externally tiny, hence a representable $C(-, c')$, we have

\begin{displaymath}
G^F(c) = E(C(-, c) \times C(-, c'), G) \cong E(C(-, c \times c'), G) \cong G(c \times c')
\end{displaymath}
where the last isomorphism is by the [[Yoneda lemma]]. Since colimits in $PSh(C)$ are computed pointwise, whereby evaluation functors $ev_c: PSh(C) \to Set$ preserve colimits, we see that $(-)^{C(-, c')}: G \mapsto G(- \times c')$ preserves colimits, so that $F = C(-, c')$ is internally tiny. The amazing right adjoint $R$ in this case takes a presheaf $H$ to the presheaf $R H$ that takes an object $d$ to the set $(R H)(d) = E(C(- \times c', d), H)$.

\end{proof}
(Compare the result \href{https://ncatlab.org/nlab/show/internally+projective+object#enough}{here}.)

In the context of [[topos theory]] we say, for $C$ [[small category]], that an [[adjoint triple]] of [[functors]]

\begin{displaymath}
Set \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}}
  [C,Set]
\end{displaymath}
is an [[essential geometric morphism]] of [[topos]]es $f : Set \to [C,Set]$; or an \textbf{[[point of a topos|essential point]]} of $[C,Set]$.

By the [[adjoint functor theorem]] this is equivalently simply a single functor $f^* : [C, Set] \to Set$ that preserves all small [[limits]] and [[colimits]]. Write

\begin{displaymath}
Topos_{ess}(Set,[C,Set]) 
   \simeq 
  LRFunc([C,Set], Set)
   \subset 
  Func([C,Set], Set)
\end{displaymath}
for the [[full subcategory]] of the [[functor category]] on functors that have a [[left adjoint]] and a [[right adjoint]].

\begin{prop}
\label{}\hypertarget{}{}
For $C$ a [[small category]] there is an [[equivalence of categories]]

\begin{displaymath}
\overline{C} := TinyObjects([C,Set]) \simeq Topos_{ess}(Set, [C,Set])^{op}
\end{displaymath}
of the tiny objects of $[C,Set]$ with the category of essential points of $[C,Set]$.

\end{prop}
\begin{proof}
We first exhibit a [[full subcategory|full inclusion]] $Topos_{ess}(Set,[C,Set])^{op} \hookrightarrow \overline{C}$.

So let $Set \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}} [C,Set]$ be an [[essential geometric morphism]]. Then because $f_!$ is [[left adjoint]] and thus preserves all small [[colimits]] and because every [[set]] $S \in Set$ is the colimit over itself of the singleton set we have that

\begin{displaymath}
f_! S \simeq \coprod_{s \in S} f_!(*)
\end{displaymath}
is fixed by a choice of [[copresheaf]]

\begin{displaymath}
F := f_!(*) \in [C, Set]
  \,.
\end{displaymath}
The $(f_! \dashv f^*)$-[[adjunction]] [[isomorphism]] then implies that for all $H \in [C,Set]$ we have

\begin{displaymath}
f^* H \simeq Set(*, f^* H) \simeq [C,Set](f_! *, H)
  \simeq [C,Set](F,H)
  \,.
\end{displaymath}
naturally in $H$, and hence that

\begin{displaymath}
f^*(-) \simeq [C,Set](F,-) : Set \to [C,Set]
  \,.
\end{displaymath}
By assumption this has a further right adjoint $f_!$ and hence preserves all [[colimits]]. By the discussion at [[tiny object]] it follows that $F \in [C,Set]$ is a tiny object. By prop. \ref{CauchyComplIsFullSubcatOnTinyObjects} this means that $F$ belongs to $\overline{C} \subset [C,Set]$.

A morphism $f \Rightarrow g$ between [[geometric morphisms]] $f,g : Set \to [C,Set]$ is a [[geometric transformation]], which is a [[natural transformation]] $f^* \Rightarrow g^*$, hence by the above a natural transformation $[C,Set](F,-) \Rightarrow [C,Set](G,-)$. By the [[Yoneda lemma]] these are in bijection with morphisms $G \to H$ in $[C,Set]$. This gives the full inclusion $Topos_{ess}(Set,[C,Set])^{op} \subset \overline{C}$.

The converse inclusion is now immediate by the same arguments: since the objects in $\overline{C}$ are precisely the [[tiny object]]s $F \in [C,Set]$ each of them corresponds to a functor $[C,Set](F,-) : [C,Set] \to Set$ that has a [[right adjoint]]. Since this generally also has a left adjoint, it is the [[inverse image]] of an essential geometric morphism $f : Set \to [C,Set]$.

\end{proof}
\hypertarget{AtomsInALocalTopos}{}\subsubsection*{{In a local topos}}\label{AtomsInALocalTopos}

\begin{prop}
\label{SliceOverAtomicObject}\hypertarget{SliceOverAtomicObject}{}
The [[terminal object]] in any [[local topos]] is atomic.

In particular for $\mathbf{H}$ a [[topos]] and $X \in \mathbf{H}$ an object, the [[slice topos]] $\mathbf{H}_{/X}$ is [[local topos|local]] precisely if $X$ is atomic.

\end{prop}
This is discuss at \href{local+geometric+morphism#LocalOverTopoes}{local geometric morphism -- Local over-toposes}.

\hypertarget{AtomsInACohesiveTopos}{}\subsubsection*{{In a cohesive topos}}\label{AtomsInACohesiveTopos}

Let $\mathbf{H}$ be a [[cohesive (∞,1)-topos]]. Write $(\int \dashv \flat \dashv \sharp)$ for its [[adjoint triple]] of [[shape modality]] $\dashv$ [[flat modality]] $\dashv$ [[sharp modality]]. Consider the following basic notion from \emph{[[cohesive (∞,1)-topos -- structures]]}.

\begin{defn}
\label{}\hypertarget{}{}
An [[object]] $X \in \mathbf{H}$ is called \emph{geometrically contractible} if its [[shape modality|shape]] is [[contractible]], in that $\int X \simeq \ast$.

\end{defn}
\begin{prop}
\label{InCohesionAtomicObjectIsGeometricallyContractible}\hypertarget{InCohesionAtomicObjectIsGeometricallyContractible}{}
Over the [[base (∞,1)-topos]] [[∞Grpd]], every atom in a [[cohesive (∞,1)-topos]] is geometrically contractible.

\end{prop}
\begin{proof}
By [[reflective sub-(∞,1)-category|reflection]] of the [[discrete objects]] it will be sufficient to show that for all [[discrete objects]] $S \in \infty Grpd \hookrightarrow \mathbf{H}$ we have an [[equivalence]]

\begin{displaymath}
\left[\int X , S\right] \simeq S
  \,.
\end{displaymath}
Now notice that, by the discussion at \emph{\href{limit+in+a+quasi-category#Tensoring}{∞-tensoring}}, every [[discrete object]] here is the [[homotopy colimit]] indexed by itself of the [[(∞,1)-functor]] constant on the [[terminal object]]:

\begin{displaymath}
S \simeq \underset{\rightarrow}{\lim}_S \ast
  \,.
\end{displaymath}
Using this we have

\begin{displaymath}
\begin{aligned}
    \left[\int X, S\right]
    &\simeq
    \left[
       X, \flat S
    \right]
    \\
    & \simeq 
    \left[
       X, \flat \underset{\rightarrow}{\lim}_S \ast
    \right]
    \\
    & \simeq 
    \left[
       X, \underset{\rightarrow}{\lim}_S \flat  \ast
    \right]
    \\
    & \simeq 
    \underset{\rightarrow}{\lim}_S
    \left[
       X, \flat  \ast
    \right]
    \\
    & \simeq 
    \underset{\rightarrow}{\lim}_S
    \left[
       X, \ast
    \right]
    \\
    & \simeq 
    \underset{\rightarrow}{\lim}_S
    \ast
    \\
    & \simeq 
    S
  \end{aligned}
  \,.
\end{displaymath}
where we applied, in order of appearance: the $(\int \dashv \flat)$-[[adjunction]], the $\infty$-[[tensoring]], the fact that $\flat$ is also [[left adjoint]] (hence the existence of the [[sharp modality]]), the assumption that $X$ is atomic, then again the fact that $\flat$ is right adjoint, that $\ast$ is the terminal object and finally again the $\infty$-tensoring.

\end{proof}
\begin{prop}
\label{SliceOfCohesionOverAtomIsCohesive}\hypertarget{SliceOfCohesionOverAtomIsCohesive}{}
Let $\mathbf{H}$ be a [[cohesive (∞,1)-topos]] over [[∞Grpd]] and let $X \in \mathbf{H}$ be an atomic object. Then the [[slice (∞,1)-topos]] $\mathbf{H}_{/X}$ sits by an [[adjoint quadruple]] over [[∞Grpd]] whose leftmost adjoint preserves the terminal object.

\end{prop}
\begin{proof}
By the discussion at [[étale geometric morphism]], the [[slice (∞,1)-topos]] comes with an [[adjoint triple]] of the form

\begin{displaymath}
\mathbf{H}_{/X}
   \stackrel{\overset{\sum_X}{\longrightarrow}}{\stackrel{\overset{(-)\times X}{\leftarrow}}{\stackrel{\overset{\prod_X}{\longrightarrow}}{\underset{}{}}}}
  \mathbf{H}
    \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{CoDisc}{\leftarrow}}}}
  \infty Grpd
  \,.
\end{displaymath}
The bottom composite $\Gamma\circ \prod_X$ has an extra right adjoint by prop \ref{SliceOverAtomicObject}. The extra left adjoint $\Pi \circ \sum_X$ preserves the terminal object by prop. \ref{InCohesionAtomicObjectIsGeometricallyContractible}.

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

\begin{itemize}%
\item [[atom]] (in a [[poset]])


\item [[atomic category]]



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

The term \emph{small projective object} is used in section 5.5. of

\begin{itemize}%
\item [[Max Kelly]], \emph{Basic Concepts of Enriched Category Theory} (\href{http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf}{pdf})

\end{itemize}
Tiny objects in presheaf categories ([[Cauchy complete categories|Cauchy completion]]) are discussed in

\begin{itemize}%
\item [[Francis Borceux]] and D. Dejean, \emph{Cauchy completion in category theory} Cahiers Topologie G\'e{}om. Diff\'e{}rentielle Cat\'e{}goriques, 27:133--146, (1986) (\href{http://www.numdam.org/item?id=CTGDC_1986__27_2_133_0}{numdam})


\item David Yetter, \emph{On right adjoints to exponential functors} \href{https://www.sciencedirect.com/science/article/pii/0022404987900776}{link}



\end{itemize}
The term ``atomic object'' or rather ``a.t.o.m'' is suggested in

\begin{itemize}%
\item [[William Lawvere]], \emph{[[Toposes of laws of motion]]}, 1997

\end{itemize}
[[!redirects small-projective object]] [[!redirects small-projective objects]] [[!redirects small-projective]]

[[!redirects tiny objects]] [[!redirects tiny]]

[[!redirects atomic object]] [[!redirects atomic objects]]



\end{document}