Tininess and right adjoints to exponentials

Abstract: Objects $T$ whose exponential functor $(-)<sup>T$ admits a right adjoint $(-)_T$ are known under different names. The fact that they exist, yet that the only set that satisfies this in the category of sets is the singleton made Lawvere suggest they ought to be amazingly tiny'' -- hence Lawvere's acronymA.T.O.M.'' This report explores how intuitively tiny any such object is. Evidences both in favor and to the contrary are produced by looking at their categorical behavior (subobjects, quotients, retracts, etc) when the ambient category is a topos. The topological behavior (connectedness, contractibility, connected components, etc) of both $T$ and $(-)_T$ is further analyzed in toposes that satisfy certain precohesive conditions over their decidable objects, where this tininess is tested against parts of Lawvere's foundational proposal for Synthetic Differential Geometry.

• The paper develops the basic theory of atomic objects, proving closure under finite products and retracts, faithfulness for pointed objects, and generalized singleton embeddings.
• In McLarty toposes, atomic decidable objects are terminal, while right adjoints preserve global points and connected components; atomicity is also equivalent to contractibility when tested on the decidable object 2.
• The paper proves atomic objects are contractible in broad presheaf and Grothendieck toposes, but shows subobjects and quotients may fail to remain atomic and leaves noncontractible precohesive examples as an open problem.

Overview

The paper studies objects $T$ in a cartesian closed category whose exponential functor $(-)^T$ admits a right adjoint $(-)_T$. Such objects are called atomic (also tiny or infinitesimal in the literature), and their existence is central to Synthetic Differential Geometry (SDG), where the tangent structure is represented by $(-)^T$ with an "amazing" right adjoint. Lawvere coined the acronym A.T.O.M. ("amazingly tiny object") because the only set with this property is the singleton. The paper's stated aim is to assess how "tiny" such objects actually are: it examines their categorical behavior (subobjects, quotients, retracts) in toposes, and their topological behavior (connectedness, contractibility, points) in toposes satisfying precohesion conditions over their decidable objects.

The ambient setting is the string of adjunctions of a precohesive geometric morphism $F:E\to\mathcal S$, with $F^*$ fully faithful, monic counit for $F^*\dashv F_*$, and $F_!$ preserving finite products. A McLarty topos is defined as a 2-valued topos where supports split and which is precohesive over a Boolean base; examples include models of ETCS and any Grothendieck topos whose canonical geometric morphism is precohesive over $\mathrm{Set}$. In these toposes, $\Pi$ computes connected components, $(-)^T$0 the set of global elements, and $(-)^T$1 corresponds to the inclusion of $(-)^T$2-sheaves.

The subcategory of atomic objects

The paper first develops general facts valid in any cartesian closed category. Finite products of atomic objects are atomic; initial objects are never atomic unless the category is trivial; terminal objects always are. For any arrow $(-)^T$3 between atomic objects, a natural transformation $(-)^T$4 is constructed analogously to $(-)^T$5, making $(-)^T$6 a covariant functor on the full subcategory of atomic objects. If $(-)^T$7 has a point, both $(-)^T$8 and $(-)^T$9 are faithful, since any pointed atomic $(-)_T$0 makes every $(-)_T$1 a retract of $(-)_T$2.

A structural theorem establishes a bijective correspondence between natural transformations $(-)_T$3 and natural transformations $(-)_T$4, via transposition through the two adjunctions; the analogous correspondence between $(-)_T$5 and $(-)_T$6 is also recorded. Combined with a characterization of when a natural transformation $(-)_T$7 is of the form $(-)_T$8, this yields a criterion: $(-)_T$9 for some $(-)^T$0 exactly when $(-)^T$1 commutes with the canonical maps $(-)^T$2 and $(-)^T$3 down to $(-)^T$4.

Retracts of atomic objects

In any regular cartesian closed category, every retract of an atomic object is atomic. The proof explicitly constructs the right adjoint for the retract $(-)^T$5 of $(-)^T$6: given an epi–mono factorization of the idempotent-induced map $(-)^T$7, one defines mutually inverse bijections between hom-sets out of $(-)^T$8 and into the factorization object $(-)^T$9, and verifies naturality in both variables. The paper notes this parallels known results on small-projective objects, where preservation of colimits by exponentiation is inherited by retracts. The converse fails at the level of subobjects and quotients: explicit examples show that neither subobjects nor quotients of atomic objects need be atomic (see below).

Generalized singletons

For a topos $F:E\to\mathcal S$0 with a pointed atomic object $F:E\to\mathcal S$1, the paper constructs, for each object $F:E\to\mathcal S$2, a monomorphism

$F:E\to\mathcal S$3

given internally by $F:E\to\mathcal S$4, where $F:E\to\mathcal S$5 is a point. This map is called a generalized singleton: for $F:E\to\mathcal S$6 it recovers the usual singleton map $F:E\to\mathcal S$7. Three commutativity properties are established: compatibility with the singleton map along $F:E\to\mathcal S$8, compatibility of the counit $F:E\to\mathcal S$9 with $F^*$0 (so $F^*$1 factors through $F^*$2), and naturality in the point—i.e., for $F^*$3 between atomic objects, $F^*$4 maps to $F^*$5 under $F^*$6. The proofs use the internal language together with careful diagrammatic manipulations involving exponent swapping and product distribution isomorphisms. Since $F^*$7 is faithful when $F^*$8 is pointed, these generalized singletons provide a uniform way of embedding the values of the amazing right adjoint into power-type objects.

Atomicity under precohesion

For a precohesive morphism $F^*$9, two preservation/reflection results hold:

• If $F^*\dashv F_*$0 is a reflective and coreflective exponential ideal in $F^*\dashv F_*$1, then the inclusion reflects atomic objects.
• $F^*\dashv F_*$2 preserves atomic objects: using the isomorphism $F^*\dashv F_*$3 (valid because $F^*\dashv F_*$4 is an exponential ideal), a chain of natural bijections exhibits a right adjoint to $F^*\dashv F_*$5 in $F^*\dashv F_*$6.

Counterexamples are given showing that $F^*\dashv F_*$7 need not reflect atomicity and that $F^*\dashv F_*$8 may neither preserve nor reflect it. Regarding $F^*\dashv F_*$9, the paper cites evidence that reflection should not be expected in general but concedes that no explicit construction within the precohesive context is currently available.

Atomic objects in McLarty toposes

The strongest rigidity results concern McLarty toposes, which satisfy a Nullstellensatz: every object is either initial or has a global element (equivalently, the local set theory is strongly witnessed). The main findings are:

• Atomic decidable objects are terminal. The proof is a counting argument: if a decidable atomic $F_!$0 had two distinct points, then $F_!$1 would force at least sixteen subobjects of $F_!$2 in $F_!$3, contradicting that $F_!$4 has only four. Consequently, in Boolean McLarty toposes all atomic objects are terminal, and in any McLarty topos atomic objects are connected.
• Points and components are preserved by $F_!$5: for any atomic $F_!$6 and arbitrary $F_!$7,

$F_!$8

The first follows from Yoneda applied to the chain relating maps $F_!$9 and $\mathrm{Set}$0 via evaluation isomorphisms $\mathrm{Set}$1 on decidable $\mathrm{Set}$2; the second from a splitting argument showing that the complement of $\mathrm{Set}$3 inside $\mathrm{Set}$4 must be initial, using completeness and consistency of the local theory.

• Contractibility criterion: an atomic $\mathrm{Set}$5 in a McLarty topos is contractible if and only if $\mathrm{Set}$6 is an isomorphism for every decidable $\mathrm{Set}$7; moreover, it suffices to check this for $\mathrm{Set}$8, i.e., $\mathrm{Set}$9 is contractible iff $\Pi$0 is an isomorphism. The proof of the converse direction uses unique extension of maps to 2 along $\Pi$1 and a complementation argument in the Boolean topos $\Pi$2.

These results bear directly on Lawvere's foundational proposal for SDG, in which connectedness of the pullback $\Pi$3 of the evaluation $\Pi$4 along $\Pi$5 is argued equivalent to $\Pi$6, assuming $\Pi$7 is a retract of $\Pi$8. Prior work of the authors showed that in McLarty toposes this condition holds iff $\Pi$9 is contractible; the present paper argues that requiring atomicity is reasonable, since atomicity implies contractibility in the contexts studied.

Presheaf and Grothendieck toposes

Two contractibility theorems are proven here. First, if $(-)^T$00 is a small category with finite products such that $(-)^T$01 is a McLarty topos, then every atomic object of $(-)^T$02 is contractible. The argument combines Yetter's observation that representables are atomic, Madanshekaf's theorem that atomic objects are retracts of representables, and the direct computation that each representable $(-)^T$03 satisfies $(-)^T$04, hence is contractible; retract-closure of contractibility completes the proof.

Second, if $(-)^T$05 is a Grothendieck topos whose canonical geometric morphism is precohesive and $(-)^T$06 preserves colimits, then every atomic sheaf is contractible. Here the proof embeds $(-)^T$07 into a sheaf topos on its molecular site, uses that molecules have $(-)^T$08, expresses an atomic sheaf as a colimit of representables over its category of elements, and extracts a splitting of $(-)^T$09 through some representable, again reducing to the retract case. A useful observation records that in this setting $(-)^T$10 is connected iff its category of elements is connected.

Examples and counterexamples

Three constructions calibrate the theory:

• No nontrivial atomic objects: for $(-)^T$11 the monoid with unit and one idempotent, the topos $(-)^T$12 (a quality type) has no atomic objects other than $(-)^T$13. The finite case is ruled out by a cardinality contradiction—the putative cardinality of $(-)^T$14 would depend on $(-)^T$15—and the infinite case by exhibiting $(-)^T$16 as a retract of any infinite atomic object, contradicting finiteness.
• Atomicity does not bound the number of points: a McLarty topos is built from a presented category $(-)^T$17 (obtained from $(-)^T$18 minus its terminal object by adjoining copairing arrows into $(-)^T$19 subject to explicit relations) for which $(-)^T$20 is precohesive and sufficiently cohesive. The representable $(-)^T$21 is atomic and has countably infinitely many points, since the copoints $(-)^T$22 yield distinct maps from the terminal functor. This shows that "atomic" does not force infinitesimal-like point behavior.
• Quotients need not be atomic: on the same category $(-)^T$23, a quotient $(-)^T$24 of $(-)^T$25 is constructed (constant value 2, acting trivially except on new arrows factoring through $(-)^T$26). One computes $(-)^T$27 while $(-)^T$28 has cardinality $(-)^T$29 for every $(-)^T$30, so $(-)^T$31 cannot preserve colimits and $(-)^T$32 is not atomic.
• Trivial action on 2: for any small cartesian closed category $(-)^T$33, the representable $(-)^T$34 is atomic in $(-)^T$35 and $(-)^T$36; hence the contractibility criterion of the McLarty section is satisfied in these presheaf examples.

Taken together, the examples support the paper's critical assessment of terminology: there exist disconnected objects with amazing right adjoints (e.g., the terminal object in $(-)^T$37, which has two complemented subobjects), not every connected object is atomic, and SDG infinitesimals are not obtained as geometrical limits. Neither "atomic," "tiny," nor "infinitesimal" is therefore fully adequate, and the authors state that further knowledge of the behavior of these objects is needed before proposing a better name.

Limitations and open questions

Several concessions are made explicitly. The reflection result for $(-)^T$38 on atomic objects is conjectural: prior work suggests it should fail in general, but no counterexample within the precohesive context is constructed. In Boolean McLarty toposes, atomic objects being terminal makes the theory degenerate there, so the interesting cases are necessarily non-Boolean. Most significantly, the authors state they are unaware of any noncontractible atomic object in any precohesive context, and pose the open question of whether such objects exist in McLarty toposes. They note that under the weaker assumption of a mere connectedness structure on a cartesian closed category there ought to be more room for counterexamples, but this lies outside the scope of the manuscript.

Conclusion

The paper systematizes the elementary theory of atomic objects—closure under products and retracts, faithfulness of $(-)^T$39 and $(-)^T$40, generalized singleton monomorphisms—and then tests the intuitive notion of tininess against cohesion axioms. Its principal findings are the rigidity of $(-)^T$41 with respect to points and connected components ($(-)^T$42, $(-)^T$43), the terminality of atomic decidable objects in McLarty toposes, and the contractibility of all atomic objects in precohesive Grothendieck and presheaf toposes, alongside counterexamples showing that quotients and subobjects of atomic objects can fail to be atomic and that atomic objects may carry infinitely many points. These results sharpen the picture of what the existence of an amazing right adjoint does and does not entail, and leave the existence of noncontractible atomic objects in precohesive settings as the central open problem.