tiny object




Let EE be a locally small category with all small colimits. An object ee of EE is called tiny or small-projective object (Kelly, §5.5) if the hom-functor E(e,):ESetE(e, -) : E \to Set preserves small colimits.

More generally, for VV a cosmos and EE a VV-enriched category, eEe \in E is called tiny if E(e,):EVE(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 ee is tiny then E(e,)E(e,-) preserves epimorphisms, which is to say that ee 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 EE is cartesian closed and the inner hom () e(-)^e has a right adjoint (and hence preserves all colimits), ee is called atomic or infinitesimal.

(Lawvere 97)


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


If EE 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).

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 RR, the tiny objects in the category of left RR-modules Ab RAb^R, considered as an Ab-enriched category, are the finitely generated projective modules. Certainly f.g. projective modules are tiny because RR is tiny (the forgetful functor hom(R,):Ab RAb\hom(R, -): Ab^R \to Ab preserves AbAb-colimits) and the closure of RR under finite direct sums and retracts, which are absolute AbAb-colimits, comprise finitely generated projective modules. See also Cauchy completion.

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

Hom(M,M)⨿Hom(M,M)Hom(M,MM) Hom(M,M) \amalg Hom(M,M) \longrightarrow Hom(M,M \oplus M)

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

In presheaf categories


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 UU a representable functor and F:JPShF : J \to PSh a diagram that

Hom(U,lim F)(lim F)(U)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:


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 CC small category, that an adjoint triple of functors

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

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

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

Topos ess(Set,[C,Set])LRFunc([C,Set],Set)Func([C,Set],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.


For CC a small category there is an equivalence of categories

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

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


We first exhibit a full inclusion Topos ess(Set,[C,Set]) opC¯Topos_{ess}(Set,[C,Set])^{op} \hookrightarrow \overline{C}.

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

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

is fixed by a choice of copresheaf

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

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

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

naturally in HH, and hence that

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

By assumption this has a further right adjoint f !f_! and hence preserves all colimits. By the discussion at tiny object it follows that F[C,Set]F \in [C,Set] is a tiny object. By prop. \ref{CauchyComplIsFullSubcatOnTinyObjects} this means that FF belongs to C¯[C,Set]\overline{C} \subset [C,Set].

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

The converse inclusion is now immediate by the same arguments: since the objects in C¯\overline{C} are precisely the tiny objects F[C,Set]F \in [C,Set] each of them corresponds to a functor [C,Set](F,):[C,Set]Set[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[C,Set]f : Set \to [C,Set].

In a local topos


The terminal object in any local topos is atomic.

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

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

In a cohesive topos

Let H\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 XHX \in \mathbf{H} is called geometrically contractible if its shape is contractible, in that X*\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 SGrpdHS \in \infty Grpd \hookrightarrow \mathbf{H} we have an equivalence

[X,S]S. \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:

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

Using this we have

[X,S] [X,S] [X,lim S*] [X,lim S*] lim S[X,*] lim S[X,*] lim S* S. \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 XX is atomic, then again the fact that \flat is right adjoint, that *\ast is the terminal object and finally again the \infty-tensoring.


Let H\mathbf{H} be a cohesive (∞,1)-topos over ∞Grpd and let XHX \in \mathbf{H} be an atomic object. Then the slice (∞,1)-topos H /X\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

H /X X()×X XHCoDiscΓDiscΠGrpd. \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 Γ X\Gamma\circ \prod_X has an extra right adjoint by prop 4. The extra left adjoint Π X\Pi \circ \sum_X preserves the terminal object by prop. 5.


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)

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

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