tiny object

Let $E$ be a locally small category with all small colimits. An object $e$ of $E$ is called **tiny** or **small-projective object** (Kelly, §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.

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 (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.

If $E$ is cartesian closed and the inner hom $(-)^e$ has a right adjoint (and hence preserves all colimits), $e$ is called **atomic** or infinitesimal.

The right adjoint in def. 2 is sometimes called an “amazing right adjoint”, particularly in the context of synthetic differential geometry.

If $E$ is a sheaf topos, then tiny objects and atomic objects coincide, by the adjoint functor theorem.

Any retract of a tiny object is tiny, since splitting of idempotents is an absolute colimit (see also Kelly, prop. 5.25).

The notion of tiny object is clearly highly dependent on the base of enrichment. For example, for a ring $R$, the tiny objects in the category of left $R$-modules $Ab^R$, considered as an Ab-enriched category, are the 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 *Cauchy completion*.

On the other hand, when the category $Ab^R$ is considered as a Set-enriched category, there are *no* tiny objects: If $M$ is a tiny object in a category of modules, the canonical map

$Hom(M,M) \amalg Hom(M,M) \longrightarrow Hom(M,M \oplus M)$

is bijective and in particular surjective. Therefore the diagonal embedding $x \mapsto (x,x)$ can be written as $\iota \circ f$ for some linear map $f : M \to M$, where $\iota$ is one of the inclusions $M \to M \oplus M$. This shows that any element of $M$ is zero, i.e., $M$ would have to be trivial. But then $Hom(M, M)$ is a singleton and the canonical map above cannot be an isomorphism.

In a presheaf category every 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

$Hom(U, \lim_\to F) \simeq (\lim_\to F)(U) \simeq \lim_\to F(U)$

Thus, in a presheaf category, any retract of a representable functor is tiny. In fact the converse also holds:

The tiny objects in a presheaf category are precisely the retracts of representable functors.

This is for instance (BorceuxDejean, prop 2).

Thus, if the domain category is Cauchy complete (has split idempotents), 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.

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

$Set \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}}
[C,Set]$

is an essential geometric morphism of toposes $f : Set \to [C,Set]$; or an **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

$Topos_{ess}(Set,[C,Set])
\simeq
LRFunc([C,Set], Set)
\subset
Func([C,Set], Set)$

for the full subcategory of the functor category on functors that have a left adjoint and a right adjoint.

For $C$ a small category there is an equivalence of categories

$\overline{C} := TinyObjects([C,Set]) \simeq \simeq Topos_{ess}(Set, [C,Set])^{op}$

of the tiny objects of $[C,Set]$ with the category of essential points of $[C,Set]$.

We first exhibit a 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

$f_! S \simeq \coprod_{s \in S} f_!(*)$

is fixed by a choice of copresheaf

$F := f_!(*) \in [C, Set]
\,.$

The $(f_! \dashv f^*)$-adjunction isomorphism then implies that for all $H \in [C,Set]$ we have

$f^* H \simeq Set(*, f^* H) \simeq [C,Set](f_! *, H)
\simeq [C,Set](F,H)
\,.$

naturally in $H$, and hence that

$f^*(-) \simeq [C,Set](F,-) : Set \to [C,Set]
\,.$

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 objects $F \in [C,Set]$ 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 $f : Set \to [C,Set]$.

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 precisely if $X$ is atomic.

This is discuss at local geometric morphism – Local over-toposes.

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 *cohesive (∞,1)-topos -- structures*.

An object $X \in \mathbf{H}$ is called *geometrically contractible* if its shape is contractible, in that $\int X \simeq \ast$.

Over the base (∞,1)-topos ∞Grpd, every atom in a cohesive (∞,1)-topos is geometrically contractible.

By 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

$\left[\int X , S\right] \simeq S
\,.$

Now notice that, by the discussion at *∞-tensoring*, every discrete object here is the homotopy colimit indexed by itself of the (∞,1)-functor constant on the terminal object:

$S \simeq \underset{\rightarrow}{\lim}_S \ast
\,.$

Using this we have

$\begin{aligned}
\left[\int X, S\right]
&\simeq
\left[
X, \flat S
\right]
\\
& \simeq
\left[
X, \flat \underset{\rightarrow}{\lim}_S \ast
\right]
\\
& \simeq
\left[
X, \underset{\rightarrow}{\lim}_S \flat \ast
\right]
\\
& \simeq
\underset{\rightarrow}{\lim}_S
\left[
X, \flat \ast
\right]
\\
& \simeq
\underset{\rightarrow}{\lim}_S
\left[
X, \ast
\right]
\\
& \simeq
\underset{\rightarrow}{\lim}_S
\ast
\\
& \simeq
S
\end{aligned}
\,.$

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.

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.

By the discussion at étale geometric morphism, the slice (∞,1)-topos comes with an adjoint triple of the form

$\mathbf{H}_{/X}
\stackrel{\overset{\sum_X}{\longrightarrow}}{\stackrel{\overset{(-)\times X}{\leftarrow}}{\stackrel{\overset{\prod_X}{\longrightarrow}}{\underset{}{}}}}
\mathbf{H}
\stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{CoDisc}{\leftarrow}}}}
\infty Grpd
\,.$

The bottom composite $\Gamma\circ \prod_X$ has an extra right adjoint by prop 4. The extra left adjoint $\Pi \circ \sum_X$ preserves the terminal object by prop. 5.

The term *small projective object* is used in section 5.5. of

Tiny objects in presheaf categories (Cauchy completion) are discussed in

- Francis Borceux and D. Dejean,
*Cauchy completion in category theory*Cahiers Topologie Géom. Différentielle Catégoriques, 27:133–146, (1986) (numdam)

The term “atomic object” or rather “a.t.o.m” is suggested in

Revised on September 2, 2015 14:24:44
by Urs Schreiber
(195.82.63.198)