# nLab tiny object

Contents

### Context

#### Compact objects

objects $d \in C$ such that $C(d,-)$ commutes with certain colimits

# Contents

## Definition

###### Definition

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.

###### Remark

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.

###### Definition

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

(See for instance Lawvere 97.)

###### Remark

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

###### Remark

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 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 .

###### Proposition

If $E$ is a sheaf topos, then (externally) tiny objects and externally atomic objects coincide.

###### 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, cocomplete, and 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 $C = Set$.

Clearly the statement and the proof of Proposition carry over when “external” is replaced by “internal” throughout.

## Properties

### General

###### Proposition

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

### In categories of modules over rings

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. 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 $id_X \in Hom(X,X)$.

### In presheaf categories

###### Example

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

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

where now the last colimit is in Set.

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

###### Proposition

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

This is for instance (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 (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.

###### Proposition

For presheaves on a category $C$ with finite products, the notions of externally tiny object and internally tiny object coincide.

###### 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

$E(e, -): E \to Set = \left(E \stackrel{(-)^e}{\to} E \stackrel{E(1,-)}{\to} Set \right)$

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

$G^F(c) = E(C(-, c) \times F, G).$

In particular, if $F$ is externally tiny, hence a representable $C(-, c')$, we have

$G^F(c) = E(C(-, c) \times C(-, c'), G) \cong E(C(-, c \times c'), G) \cong G(c \times c')$

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 $(R H)(d) = E(C(- \times c', d), H)$.

(Compare the result here.)

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.

###### Proposition

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

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

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

###### Proof

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. 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 $f : Set \to [C,Set]$.

### In a local topos

###### Proposition

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.

### In a cohesive topos

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.

###### Definition

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

###### Proposition

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

###### Proof

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.

###### Proposition

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.

###### Proof

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 . The extra left adjoint $\Pi \circ \sum_X$ preserves the terminal object by prop. .

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

• Max Kelly, Basic Concepts of Enriched Category Theory (pdf)

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)