nLab
tiny object

Contents

Definition

Definition

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.

Remark

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.

Definition

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)

Remark

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

Remark

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

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 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 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:

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 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.

Proposition

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].

Proof

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

Proposition

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.

Definition

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

Proposition

Let H\mathbf{H} be a cohesive (∞,1)-topos over ∞Grpd and let XHX \in \mathbf{H} be an atomic object. Then also the slice (∞,1)-topos H /X\mathbf{H}_{/X} is cohesive over ∞Grpd, except that the shape modality may fail to preserve binary products.

Proof

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.

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)

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

Revised on November 28, 2013 02:00:04 by Urs Schreiber (77.251.114.72)