nLab tiny object

Definition

Definition

Let $E$ be a locally small category with all small colimits. An object $e$ of $E$ is called tiny or small-projective (Kelly 1982, §5.5) if the hom-functor $E(e, -) \colon E \to Set$ preserves small colimits. It is called absolutely presentable if the functor preserves all colimits.

More generally, for $V$ a cosmos and $E$ a $V$-enriched category, $e \in E$ is called tiny if $E(e,-) \colon E \to V$ 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. .

Related concepts

• atom (in a poset)

• atomic category

References

The term small projective object is used in:

• Max Kelly, section 5.5. of: Basic concepts of enriched category theory, London Math. Soc. Lec. Note Series 64, Cambridge Univ. Press (1982), Reprints in Theory and Applications of Categories 10 (2005) 1-136 [ISBN:9780521287029, tac:tr10, 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:CTGDC_1986__27_2_133_0]

• David Yetter: On right adjoints to exponential functors, Journal of Pure and Applied Algebra 45 3 (1987) 287-304 [doi:10.1016/0022-4049(87)90077-6]

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

• William Lawvere, Toposes of laws of motion (1997)

A modal type theory for tiny objects:

• Mitchell Riley, A Type Theory with a Tiny Object [arXiv:2403.01939]

• Mitchell Riley: Tiny Objects in Type Theory, talk at Running HoTT 2024, CQTS @ NYU Abu Dhabi (Apr 2024) [slides:pdf, video: kt]