This article is on the mathematical concept of atom as used in the theory of preorders, and related mathematical notions. For small projective objects in categories see at atomic object. For still other uses, see atom (disambiguation).
objects such that commutes with certain colimits
An atom in a poset is a minimal element among those which are not actually the minimum. Thus an atom is as small as possible without being nothing. In an atomic poset, every element may be broken down (typically not uniquely) into atoms.
A related but slightly weaker concept is that of “tiny element”, which has important generalizations in the context of enriched category theory.
Let be a poset (or proset) with a bottom element . Recall that an element of is positive if it is not a bottom element. An element of is atomic if, given any element , is positive iff . An atom of is simply an atomic element of . Note that every atom must be positive (since ).
The p(r)oset is atomic (or more commonly in the literature, atomistic; see remarks below) if every element is a supremum of atoms. In this case, every element is a supremum of those atoms . Note that is a supremum of no atoms, and every atom is a supremum of itself, so the condition is really about the nontrivial nonatomic elements.
In constructive mathematics, we require a more complicated definition of a positive element, but the other definitions above remain correct (under the stated conditions), once we have that. In predicative constructive mathematics, positivity cannot be defined at all, and must come equipped with a positivity predicate before we may consider its atoms.
There is some terminological variance in the literature to the notion of atomic poset as defined here. In particular, Wikipedia defines an atomic poset to be a poset in which every positive element has an atom below it, and refers to our stronger notion of atomic poset by the term “atomistic poset”. Note well that the Wikipedia conventions seem to be the ones observed in most lattice-theoretic texts.
“Atomic” and “Atomistic” differ for the simple example of the divisor lattice? for some number . The atoms in this lattice are prime numbers while it may also contain semi-atoms which are powers of primes. This lattice is atomic because any object not the bottom, , is divisible by a prime in the lattice. However it is not generally atomistic, but is instead uniquely semi-atomistic (every object is the product of a unique set of semi-atoms with bottom corresponding to the empty set), which is one way of stating the fundamental theorem of arithmetic?, also known as the unique factorization theorem.
The two notions coincide in the case of complete Boolean algebras . Indeed, suppose is atomic in the Wikipedia sense, and for any element , consider the relative complement
To show is atomistic, it suffices to show . If not, then there is an atom such that , which means both and
since . This is a contradiction.
Our (pro tem) decision to define the word “atomic” in the idiosyncratic nLab sense above is consistent with its use elsewhere in category theory; see the sections below on atomic objects and on categorification.
This is simply because , so equals either or .
If is a poset or preorder, in other words a -enriched category, an element is tiny if the hom preserves all sups that exist in . It is arguable (from an nPOV) that the weaker concept of tiny element is more fundamental than the notion of atom; for example, as we will see below, replacing atoms by tiny elements permits one to generalize the characterization of power sets as complete atomic Boolean algebras.
A tiny element in a Boolean algebra is precisely an atom.
Let be an atom. Let be a collection of elements that admits a supremum such that . Then
(where the second equation holds since is a left adjoint, because is a Heyting algebra). Since is positive, for some the element is positive as well. Trivially it holds that ; since is an atom, the inequality is an equality. Thus for some , which is what we want.
If is not an atom, i.e., if for some , then
If preserved the join on the right, then either which is evidently false, or , i.e., , i.e., , also evidently false. Thus does not preserve suprema.
Only one half of this proposition holds (an atom is a tiny element) if we replace the Boolean algebra by a general frame. (In fact, this direction even holds in impredicative constructive mathematics, if the frame is equipped with a positivity predicate.) On the other hand, tiny elements need not be atoms (an easy example is the frame of down-sets of a poset, where principal down-sets are atomic objects, but generally not atoms in the underlying poset of the frame).
Be this as it may, Lawvere has written, “In order to settle once and for all the various terminological differences, perhaps we can use a.t.o.m. as an abbreviation for ‘amazing tiny object model’.” This is Lawvere’s ‘objective’ way of abbreviating “atomic object”; the word ‘amazing’ here is presumably chosen to evoke what Lawvere has called the “amazing right adjoint” to an exponential functor , particularly in the case of synthetic differential geometry where such adjoints exist for infinitesimal objects .
Let be a -category, where is a cosmos (a complete, cocomplete, symmetric monoidal closed category). We define an object of to be tiny or atomic if preserves any -colimit that exists in . (As usual, the appropriate notion of colimit in the enriched setting is weighted colimit.)
In what follows, we suppose the full -subcategory of atomic objects in is essentially small. The inclusion induces a restricted Yoneda embedding
sending an object to . We say that is atomic if is -dense, in other words if every object of is a canonical colimit of atomic objects below it, in the precise sense that the following enriched coend exists, and its canonical map to ,
is an isomorphism.
If is a preorder, i.e., is -enriched where is the category of -categories, the coend amounts to the supremum
so that is atomic precisely if every element is the sup of the tiny elements below it.
A small-cocomplete atomic preorder is equivalent to the free sup-lattice generated by the preorder of tiny elements. Conversely, every free sup-lattice is small-cocomplete and atomic, where is the poset of tiny elements.
N.B. “Free sup-lattice” refers to a left adjoint of the forgetful functor from sup-lattices to preorders.
Since is cocomplete, and since is the free sup-lattice or cocomplete preorder generated from , the inclusion extends uniquely to a sup-preserving map
which sends to
This is left adjoint to the restricted Yoneda embedding . The condition that is atomic says that for each , the value of the counit of at is an isomorphism
On the other hand, the value of the unit of at an object is given by a string of isomorphisms
where the last isomorphism obtains from the fact that preserves colimits if is tiny. Thus the unit is also an isomorphism.
For the converse: each representable object of is tiny, because the covariant functor , being the same as evaluation at by the Yoneda lemma, preserves colimits. Furthermore, every functor is a canonical colimit of representables, so that is atomic in addition to being cocomplete.
A complete atomic Boolean algebra is isomorphic to , where is the discrete preorder of atoms of .
The argument given for the theorem above carries over without obstruction to the general enriched setting. In particular, replacing - by its categorification -, we get the following result, first enunciated in Bunge’s thesis.
A category is equivalent to a presheaf topos (functors with values in the 1-category Set of 0-groupoids) if and only if it is cocomplete and atomic as a -category. Representables are (among the) atomic objects of , and generate the presheaf topos by closing under all small colimits.
Bunge proves in fact a slight strengthening as follows: a category is equivalent to a presheaf category if and only if it is cocomplete, regular, and has a generating set of atomic objects.