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 (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 EE is a cartesian closed category with small colimits, we may say eOb(E)e \in Ob(E) is internally tiny if the functor () e:EE(-)^e: E \to E preserves small colimits. Relatedly, the word “atomic” has been used in both an external sense where E(e,):ESetE(e, -): E \to Set has a right adjoint, as in Bunge’s thesis, and in an internal sense, as when Lawvere refers to ee as an a.t.o.m. (“amazingly tiny object model”) if () e:EE(-)^e: E \to E has a right adjoint. But under certain hypotheses, the two notions coincide; see for instance Proposition .

Proposition

If EE 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): EE is locally small, cocomplete, and co-well-powered (because for any object XX, equivalence classes of epimorphisms with domain XX are in natural bijection with internal equivalence relations on XX, and there is a small set of these because they are contained in a set isomorphic to hom(X×X,Ω)\hom(X \times X, \Omega)), and finally EE has a generating set (namely, the set of associated sheaves of representables coming from a small site presentation for EE). Under these conditions, the SAFT guarantees that any cocontinuous functor ECE \to C has a right adjoint, provided that CC is locally small; then apply this to the case C=SetC = 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 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. In fact this is true for any Set-enriched category with a zero object: Let XX be a tiny object. The morphism X0X \to 0 induces a map Hom(X,X)Hom(X,0)Hom(X,X) \to Hom(X,0). This map has empty codomain (since Hom(X,)Hom(X,-) preserves the zero object, as an empty colimit). Thus Hom(X,X)=Hom(X,X) = \emptyset in contradiction to id XHom(X,X)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 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.

Proposition

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

Proof

Without loss of generality, we may assume CC 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)C(-, c).

Let EE denote the presheaf category. Given that the empty product 11 is tiny, if eOb(E)e \in Ob(E) is internally tiny, then the composite

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

is cocontinuous, hence ee is externally tiny.

In the other direction, recall how exponentials G FG^F in E=PSh(C)E = PSh(C) are constructed: we have the formula

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

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

G F(c)=E(C(,c)×C(,c),G)E(C(,c×c),G)G(c×c)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)PSh(C) are computed pointwise, whereby evaluation functors ev c:PSh(C)Setev_c: PSh(C) \to Set preserve colimits, we see that () C(,c):GG(×c)(-)^{C(-, c')}: G \mapsto G(- \times c') preserves colimits, so that F=C(,c)F = C(-, c') is internally tiny. The amazing right adjoint RR in this case takes a presheaf HH to the presheaf RHR H that takes an object dd to the set (RH)(d)=E(C(×c,d),H)(R H)(d) = E(C(- \times c', d), H).

(Compare the result here.)

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 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. 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 the slice (∞,1)-topos H /X\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

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

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)

  • David Yetter, On right adjoints to exponential functors link

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

Last revised on March 24, 2018 at 10:59:57. See the history of this page for a list of all contributions to it.