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 $E(e, -) : E \to Set$ 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.
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 (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 $E$ is a cartesian closed category with small colimits, we may say $e \in Ob(E)$ is internally tiny if the functor $(-)^e: E \to E$ preserves small colimits. Relatedly, the word “atomic” has been used in both an external sense where $E(e, -): E \to Set$ has a right adjoint, as in Bunge’s thesis, and in an internal sense, as when Lawvere refers to $e$ as an a.t.o.m. (“amazingly tiny object model”) if $(-)^e: E \to E$ has a right adjoint. But under certain hypotheses, the two notions coincide; see for instance Proposition .
If $E$ 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): $E$ is locally small, cocomplete, and co-well-powered (because for any object $X$, equivalence classes of epimorphisms with domain $X$ are in natural bijection with internal equivalence relations on $X$, and there is a small set of these because they are contained in a set isomorphic to $\hom(X \times X, \Omega)$), and finally $E$ has a generating set (namely, the set of associated sheaves of representables coming from a small site presentation for $E$). Under these conditions, the SAFT guarantees that any cocontinuous functor $E \to C$ has a right adjoint, provided that $C$ is locally small; then apply this to the case $C = Set$.
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 $R$, the tiny objects in the category of left $R$-modules $Ab^R$, considered as an Ab-enriched category, are the finitely generated projective modules. Certainly f.g. projective modules are tiny because $R$ is tiny (the forgetful functor $\hom(R, -): Ab^R \to Ab$ preserves $Ab$-colimits) and the closure of $R$ under finite direct sums and retracts, which are absolute $Ab$-colimits, comprise finitely generated projective modules. See also Cauchy completion.
On the other hand, when the category $Ab^R$ 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 $X$ be a tiny object. The morphism $X \to 0$ induces a map $Hom(X,X) \to Hom(X,0)$. This map has empty codomain (since $Hom(X,-)$ preserves the zero object, as an empty colimit). Thus $Hom(X,X) = \emptyset$ in contradiction to $id_X \in Hom(X,X)$.
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 $F : J \to PSh$ 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).
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 $C$ with finite products, the notions of externally tiny object and internally tiny object coincide.
Without loss of generality, we may assume $C$ 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 $C(-, c)$.
Let $E$ denote the presheaf category. Given that the empty product $1$ is tiny, if $e \in Ob(E)$ is internally tiny, then the composite
is cocontinuous, hence $e$ is externally tiny.
In the other direction, recall how exponentials $G^F$ in $E = PSh(C)$ are constructed: we have the formula
In particular, if $F$ is externally tiny, hence a representable $C(-, c')$, we have
where the last isomorphism is by the Yoneda lemma. Since colimits in $PSh(C)$ are computed pointwise, whereby evaluation functors $ev_c: PSh(C) \to Set$ preserve colimits, we see that $(-)^{C(-, c')}: G \mapsto G(- \times c')$ preserves colimits, so that $F = C(-, c')$ is internally tiny. The amazing right adjoint $R$ in this case takes a presheaf $H$ to the presheaf $R H$ that takes an object $d$ to the set $(R H)(d) = E(C(- \times c', d), H)$.
(Compare the result here.)
In the context of topos theory we say, for $C$ small category, that an adjoint triple of functors
is an essential geometric morphism of toposes $f : Set \to [C,Set]$; or an essential point of $[C,Set]$.
By the adjoint functor theorem this is equivalently simply a single functor $f^* : [C, Set] \to Set$ 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 $C$ a small category there is an equivalence of categories
of the tiny objects of $[C,Set]$ with the category of essential points of $[C,Set]$.
We first exhibit a full inclusion $Topos_{ess}(Set,[C,Set])^{op} \hookrightarrow \overline{C}$.
So let $Set \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}} [C,Set]$ be an essential geometric morphism. Then because $f_!$ is left adjoint and thus preserves all small colimits and because every set $S \in Set$ is the colimit over itself of the singleton set we have that
is fixed by a choice of copresheaf
The $(f_! \dashv f^*)$-adjunction isomorphism then implies that for all $H \in [C,Set]$ we have
naturally in $H$, and hence that
By assumption this has a further right adjoint $f_!$ and hence preserves all colimits. By the discussion at tiny object it follows that $F \in [C,Set]$ is a tiny object. By prop. this means that $F$ belongs to $\overline{C} \subset [C,Set]$.
A morphism $f \Rightarrow g$ between geometric morphisms $f,g : Set \to [C,Set]$ is a geometric transformation, which is a natural transformation $f^* \Rightarrow g^*$, hence by the above a natural transformation $[C,Set](F,-) \Rightarrow [C,Set](G,-)$. By the Yoneda lemma these are in bijection with morphisms $G \to H$ in $[C,Set]$. This gives the full inclusion $Topos_{ess}(Set,[C,Set])^{op} \subset \overline{C}$.
The converse inclusion is now immediate by the same arguments: since the objects in $\overline{C}$ are precisely the tiny objects $F \in [C,Set]$ each of them corresponds to a functor $[C,Set](F,-) : [C,Set] \to Set$ that has a right adjoint. Since this generally also has a left adjoint, it is the inverse image of an essential geometric morphism $f : Set \to [C,Set]$.
The terminal object in any local topos is atomic.
In particular for $\mathbf{H}$ a topos and $X \in \mathbf{H}$ an object, the slice topos $\mathbf{H}_{/X}$ is local precisely if $X$ is atomic.
This is discuss at local geometric morphism – Local over-toposes.
Let $\mathbf{H}$ 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 $X \in \mathbf{H}$ is called geometrically contractible if its shape is contractible, in that $\int X \simeq \ast$.
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 $S \in \infty Grpd \hookrightarrow \mathbf{H}$ 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 $(\int \dashv \flat)$-adjunction, the $\infty$-tensoring, the fact that $\flat$ is also left adjoint (hence the existence of the sharp modality), the assumption that $X$ is atomic, then again the fact that $\flat$ is right adjoint, that $\ast$ is the terminal object and finally again the $\infty$-tensoring.
Let $\mathbf{H}$ be a cohesive (∞,1)-topos over ∞Grpd and let $X \in \mathbf{H}$ be an atomic object. Then the slice (∞,1)-topos $\mathbf{H}_{/X}$ 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
The bottom composite $\Gamma\circ \prod_X$ has an extra right adjoint by prop . The extra left adjoint $\Pi \circ \sum_X$ preserves the terminal object by prop. .
The term small projective object is used in section 5.5. of
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
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