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 object (Kelly, §5.5) if the hom-functor $E(e,-):E\to \mathrm{Set}$ 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.

Remarks

• 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 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 infinitesimal objects coincide, by the adjoint functor theorem.

Properties

General

Observation

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

In presheaf categories

Observation

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 $F:J\to \mathrm{PSh}$ a diagram that

$$\mathrm{Hom}(U,\underrightarrow{\lim }F)\simeq (\underrightarrow{\lim }F)(U)\simeq \underrightarrow{\lim }F(U)$$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).

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

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

is an essential geometric morphism of toposes $f:\mathrm{Set}\to [C,\mathrm{Set}]$; or an essential point of $[C,\mathrm{Set}]$.

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

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

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

of the tiny objects of $[C,\mathrm{Set}]$ with the category of essential points of $[C,\mathrm{Set}]$.

Proof

We first exhibit a full inclusion ${\mathrm{Topos}}_{\mathrm{ess}}(\mathrm{Set},[C,\mathrm{Set}]{)}^{\mathrm{op}}\hookrightarrow \overline{C}$.

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

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

is fixed by a choice of copresheaf

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

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

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

naturally in $H$, and hence that

$$f^{*}(-)\simeq [C,\mathrm{Set}](F,-):\mathrm{Set}\to [C,\mathrm{Set}].$$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 $F\in [C,\mathrm{Set}]$ is a tiny object. By prop. \ref{CauchyComplIsFullSubcatOnTinyObjects} this means that $F$ belongs to $\overline{C}\subset [C,\mathrm{Set}]$.

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

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

Related concepts

• atom (in a poset)

• atomic category

References

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)