Let be a locally small category with all small colimits. An object of is called tiny or small-projective (Kelly 1982, §5.5) if the hom-functor preserves small colimits. It is called absolutely presentable if the functor preserves all colimits.
More generally, for a cosmos and a -enriched category, is called tiny if 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 is tiny then preserves epimorphisms, which is to say that 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 is cartesian closed and the inner hom has a right adjoint (and hence preserves all colimits), is called (internally) atomic or infinitesimal.
(See for instance Lawvere 97.)
The right adjoint in def. is sometimes called an “amazing right adjoint”, particularly in the context of synthetic differential geometry.
Various terminological discrepancies in the literature hinge on the distinction between internal notions and external notions. Thus, if is a cartesian closed category with small colimits, we may say is internally tiny if the functor preserves small colimits. Relatedly, the word “atomic” has been used in both an external sense where has a right adjoint, as in Bunge’s thesis, and in an internal sense, as when Lawvere refers to as an a.t.o.m. (“amazingly tiny object model”) if has a right adjoint. But under certain hypotheses, the two notions coincide; see for instance Proposition .
If is a sheaf topos, then (externally) tiny objects and externally atomic objects coincide.
Clearly any externally atomic object is tiny. For the converse, use a dual form of the special adjoint functor theorem (SAFT): is locally small, cocomplete, and co-well-powered (because for any object , equivalence classes of epimorphisms with domain are in natural bijection with internal equivalence relations on , and there is a small set of these because they are contained in a set isomorphic to ), and finally has a generating set (namely, the set of associated sheaves of representables coming from a small site presentation for ). Under these conditions, the SAFT guarantees that any cocontinuous functor has a right adjoint, provided that is locally small; then apply this to the case .
Clearly the statement and the proof of Proposition carry over when “external” is replaced by “internal” throughout.
Any retract of a tiny object is tiny, since splitting of idempotents is an absolute colimit (see also Kelly, prop. 5.25).
The notion of tiny object is clearly highly dependent on the base of enrichment. For example, for a ring , the tiny objects in the category of left -modules , considered as an Ab-enriched category, are the finitely generated projective modules. Certainly f.g. projective modules are tiny because is tiny (the forgetful functor preserves -colimits) and the closure of under finite direct sums and retracts, which are absolute -colimits, comprise finitely generated projective modules. See also Cauchy completion.
On the other hand, when the category 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 be a tiny object. The morphism induces a map . This map has empty codomain (since preserves the zero object, as an empty colimit). Thus in contradiction to .
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 a representable functor and a diagram that
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). For instance, the only tiny object in G-set is 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.
For presheaves on a category with finite products, the notions of externally tiny object and internally tiny object coincide.
Without loss of generality, we may assume 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 .
Let denote the presheaf category. Given that the empty product is tiny, if is internally tiny, then the composite
is cocontinuous, hence is externally tiny.
In the other direction, recall how exponentials in are constructed: we have the formula
In particular, if is externally tiny, hence a representable , we have
where the last isomorphism is by the Yoneda lemma. Since colimits in are computed pointwise, whereby evaluation functors preserve colimits, we see that preserves colimits, so that is internally tiny. The amazing right adjoint in this case takes a presheaf to the presheaf that takes an object to the set .
(Compare the result here.)
In the context of topos theory we say, for small category, that an adjoint triple of functors
is an essential geometric morphism of toposes ; or an essential point of .
By the adjoint functor theorem this is equivalently simply a single functor that preserves all small limits and colimits. Write
for the full subcategory of the functor category on functors that have a left adjoint and a right adjoint.
For a small category there is an equivalence of categories
of the tiny objects of with the category of essential points of .
We first exhibit a full inclusion .
So let be an essential geometric morphism. Then because is left adjoint and thus preserves all small colimits and because every set is the colimit over itself of the singleton set we have that
is fixed by a choice of copresheaf
The -adjunction isomorphism then implies that for all we have
naturally in , and hence that
By assumption this has a further right adjoint and hence preserves all colimits. By the discussion at tiny object it follows that is a tiny object. By prop. this means that belongs to .
A morphism between geometric morphisms is a geometric transformation, which is a natural transformation , hence by the above a natural transformation . By the Yoneda lemma these are in bijection with morphisms in . This gives the full inclusion .
The converse inclusion is now immediate by the same arguments: since the objects in are precisely the tiny objects each of them corresponds to a functor that has a right adjoint. Since this generally also has a left adjoint, it is the inverse image of an essential geometric morphism .
The terminal object in any local topos is atomic.
In particular for a topos and an object, the slice topos is local precisely if is atomic.
This is discuss at local geometric morphism – Local over-toposes.
Let be a cohesive (∞,1)-topos. Write for its adjoint triple of shape modality flat modality sharp modality. Consider the following basic notion from cohesive (∞,1)-topos – structures.
An object is called geometrically contractible if its shape is contractible, in that .
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 we have an equivalence
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:
Using this we have
where we applied, in order of appearance: the -adjunction, the -tensoring, the fact that is also left adjoint (hence the existence of the sharp modality), the assumption that is atomic, then again the fact that is right adjoint, that is the terminal object and finally again the -tensoring.
Let be a cohesive (∞,1)-topos over ∞Grpd and let be an atomic object. Then the slice (∞,1)-topos 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
The bottom composite has an extra right adjoint by prop . The extra left adjoint preserves the terminal object by prop. .
The term small projective object is used in:
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
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]
Last revised on July 14, 2024 at 12:59:42. See the history of this page for a list of all contributions to it.