nLab
atom

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).

Atoms

Idea

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.

Definitions

Let SS be a poset (or proset) with a bottom element \bot. Recall that an element of SS is positive if it is not a bottom element. An element aa of SS is atomic if, given any element pap \leq a, pp is positive iff apa \leq p. An atom of SS is simply an atomic element of SS. Note that every atom must be positive (since aaa \leq a).

The atoms are precisely the minimal elements of the set of positive elements. For a poset, aa is atomic iff every pap \leq a is positive iff p=ap = a. Using classical logic too, aa is atomic iff every pap \leq a satisfies p=p = \bot xor p=ap = a.

The p(r)oset SS is atomic if every element is a supremum of atoms. In this case, every element xx is a supremum of those atoms axa \leq x. Note that \bot 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 SS must come equipped with a positivity predicate before we may consider its atoms.

Remarks on terminology

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”. The Wikipedia conventions are observed in (many? most?) lattice-theoretic texts such as the book by Grätzer (citation needed).

The two notions coincide in the case of complete Boolean algebras BB. Indeed, suppose BB is atomic in the Wikipedia sense, and for any element bBb \in B, consider the relative complement

c=b¬({atomsa:ab})c = b \wedge \neg (\bigvee \{atoms\, a: a \leq b\})

To show BB is atomistic, it suffices to show c=0c = 0. If not, then there is an atom aa' such that aca' \leq c, which means both aba' \leq b and

a=aaa{atomsa:ab}=0a' = a' \wedge a' \leq a' \wedge \bigvee \{atoms\, a: a \leq b\} = 0

since a¬({atomsa:ab})a' \leq \neg(\bigvee \{atoms\, a: a \leq b\}). This is a contradiction.

Our pro tem decision to define the word “atomic” in the manner above is consistent with its use elsewhere in category theory; see the sections below on atomic objects and on categorification.

Examples

Example

In a power set the atoms are the singleton subsets. Every power set is atomic, and in fact every atomic complete boolean algebra is (up to isomorphism) a power set.

Example

In a lattice of subtoposes the atoms are the 2-valued Boolean toposes. See this proposition.

Properties

If aa is an atom in a lattice or more generally a meet semilattice and bb any other element then (using classical logic)

ab{a,}. a \wedge b \in \{a, \bot\} .

This is simply because abaa \wedge b \leq a, so equals either aa or \bot.

Atoms and tiny elements

If EE is a poset or preorder, in other words a 2\mathbf{2}-enriched category, an element eEe \in E is tiny if the hom E(e,):E2E(e, -)\colon E \to \mathbf{2} preserves all sups that exist in EE. 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.

Proposition

A tiny element in a Boolean algebra is precisely an atom.

Proof

Let aa be an atom. Let {x i}\{x_i\} be a collection of elements that admits a supremum such that a ix ia \leq \bigvee_i x_i. Then

a=a ix i= iax ia = a \wedge \bigvee_i x_i = \bigvee_i a \wedge x_i

(where the second equation holds since aa \wedge - is a left adjoint, because BB is a Heyting algebra). Since aa is positive, for some ii the element ax ia \wedge x_i is positive as well. Trivially it holds that ax iaa \wedge x_i \leq a; since aa is an atom, the inequality is an equality. Thus ax ia \leq x_i for some ii, which is what we want.

If aa is not an atom, i.e., if 0<b<a0 \lt b \lt a for some bb, then

a=b(a¬b)a = b \vee (a \wedge \neg b)

If B(a,)B(a, -) preserved the join on the right, then either aba \leq b which is evidently false, or aa¬ba \leq a \wedge \neg b, i.e., a¬ba \leq \neg b, i.e., b=ab0b = a \wedge b \leq 0, also evidently false. Thus B(a,)B(a, -) does not preserve suprema.

Only one half of this proposition holds (an atom is a tiny element) if we replace the Boolean algebra BB 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 () D(-)^D, particularly in the case of synthetic differential geometry where such adjoints exist for infinitesimal objects DD.

Generalization and categorification

The result that an atomic complete Boolean algebra is isomorphic to a power set – hence to a presheaf with values in the 0-category 2=(1)Grpd\mathbf{2} = (-1)Grpd of (-1)-groupoids – may be generalized and categorified as follows.

Let EE be a VV-category, where VV is a cosmos (a complete, cocomplete, symmetric monoidal closed category). We define an object ee of EE to be tiny or atomic if E(e,):EVE(e, -) \colon E \to V preserves any VV-colimit that exists in EE. (As usual, the appropriate notion of colimit in the enriched setting is weighted colimit.)

In what follows, we suppose the full VV-subcategory Tiny(E)Tiny(E) of atomic objects in EE is essentially small. The inclusion i:Tiny(E)Ei \colon Tiny(E) \hookrightarrow E induces a restricted Yoneda embedding

EV Tiny(E) opE \to V^{Tiny(E)^{op}}

sending an object ee to E(i,e)E(i-, e). We say that EE is atomic if i:Tiny(E)Ei \colon Tiny(E) \hookrightarrow E is VV-dense, in other words if every object ee of EE is a canonical colimit of atomic objects below it, in the precise sense that the following enriched coend exists, and its canonical map to ee,

aTiny(E)E(ia,e)iae,\int^{a \in Tiny(E)} E(i a, e) \cdot i a \to e,

is an isomorphism.

If EE is a preorder, i.e., is 2\mathbf{2}-enriched where 2\mathbf{2} is the category of (1)(-1)-categories, the coend amounts to the supremum

sup{ia:iap}\sup \{i a: i a \leq p\}

so that EE is atomic precisely if every element is the sup of the tiny elements below it.

Theorem

A small-cocomplete atomic preorder EE is equivalent to the free sup-lattice 2 T op2^{T^{op}} generated by the preorder T=Tiny(E)T = Tiny(E) of tiny elements. Conversely, every free sup-lattice 2 T op2^{T^{op}} is small-cocomplete and atomic, where TT is the poset of tiny elements.

N.B. “Free sup-lattice” refers to a left adjoint of the forgetful functor U:SupLatPreordU \colon SupLat \to Preord from sup-lattices to preorders.

Proof

Since EE is cocomplete, and since 2 Tiny(E) op\mathbf{2}^{Tiny(E)^{op}} is the free sup-lattice or cocomplete preorder generated from Tiny(E)Tiny(E), the inclusion i:Tiny(E)Ei \colon Tiny(E) \to E extends uniquely to a sup-preserving map

L:2 Tiny(E) opEL \colon \mathbf{2}^{Tiny(E)^{op}} \to E

which sends X:Tiny(E) op2X \colon Tiny(E)^{op} \to \mathbf{2} to

aTiny(E)X(a)ia=sup{ia:X(a)=1}.\int^{a \in Tiny(E)} X(a) \cdot i a = \sup \{i a: X(a) = 1\}.

This LL is left adjoint to the restricted Yoneda embedding R:E2 Tiny(E) opR \colon E \to \mathbf{2}^{Tiny(E)^{op}}. The condition that EE is atomic says that for each eEe \in E, the value of the counit of LRL \dashv R at ee is an isomorphism

aE(ia,e)iae.\int^a E(i a, e) \cdot i a \cong e.

On the other hand, the value of the unit of LRL \dashv R at an object XX is given by a string of isomorphisms

X Yoneda aX(a)Tiny(E)(,a) aX(a)E(i,ia) E(i, aX(a)ia)\array{ X & \stackrel{Yoneda}{\cong} & \int^a X(a) \cdot Tiny(E)(-, a) \\ & \cong & \int^a X(a) \cdot E(i-, i a) \\ & \cong & E(i-, \int^a X(a) \cdot i a) }

where the last isomorphism obtains from the fact that E(ia,)E(i a, -) preserves colimits if aa is tiny. Thus the unit is also an isomorphism.

For the converse: each representable object T(,t)T(-, t) of 2 T op\mathbf{2}^{T^{op}} is tiny, because the covariant functor 2 T op(T(,t),)2^{T^{op}}(T(-, t), -), being the same as evaluation at tt by the Yoneda lemma, preserves colimits. Furthermore, every functor X:T op2X: T^{op} \to \mathbf{2} is a canonical colimit of representables, so that 2 T op2^{T^{op}} is atomic in addition to being cocomplete.

Corollary

A complete atomic Boolean algebra BB is isomorphic to 2 T2^T, where TT is the discrete preorder of atoms of BB.

The argument given for the theorem above carries over without obstruction to the general enriched setting. In particular, replacing 2=(1)\mathbf{2} = (-1)-CatCat by its categorification Set=0Set = 0-CatCat, we get the following result, first enunciated in Bunge’s thesis.

Theorem (Bunge)

A category EE 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 SetSet-category. Representables C(,c)C(-, c) are (among the) atomic objects of Set C opSet^{C^{op}}, 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 Set C opSet^{C^{op}} if and only if it is cocomplete, regular, and has a generating set of atomic objects.

Revised on November 26, 2014 19:42:34 by Urs Schreiber (82.224.164.72)