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\begin{document}

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\section*{atom}

\begin{quote}%
This article is on the mathematical concept of atom as used in the theory of [[preorder|preorders]], and related mathematical notions. For small projective objects in categories see at [[atomic object]]. For still other uses, see [[atom (disambiguation)]].


\end{quote}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{compact_objects}{}\paragraph*{{Compact objects}}\label{compact_objects}

[[!include compact object - contents]]

\hypertarget{atoms}{}\section*{{Atoms}}\label{atoms}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{remarks_on_terminology}{Remarks on terminology}\dotfill \pageref*{remarks_on_terminology} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{atoms_and_tiny_elements}{Atoms and tiny elements}\dotfill \pageref*{atoms_and_tiny_elements} \linebreak
\noindent\hyperlink{generalization_and_categorification}{Generalization and categorification}\dotfill \pageref*{generalization_and_categorification} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

An atom in a [[poset]] is a [[minimal element]] among those which are not actually the [[bottom element|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.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

Let $S$ be a [[poset]] (or [[proset]]) with a [[bottom element]] $\bot$. Recall that an element of $S$ is \textbf{[[positive element|positive]]} if it is not a bottom element. An element $a$ of $S$ is \textbf{atomic} if, given any element $p \leq a$, $p$ is positive iff $a \leq p$. An \textbf{atom} of $S$ is simply an atomic element of $S$. Note that every atom must be positive (since $a \leq a$).

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

The p(r)oset $S$ is \textbf{atomic} (or more commonly in the literature, \textbf{atomistic}; see remarks below) if every element is a [[supremum]] of atoms. In this case, every element $x$ is a supremum of those atoms $a \leq x$. 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 mathematics|predicative]] constructive mathematics, positivity cannot be defined at all, and $S$ must come equipped with a [[positivity predicate]] before we may consider its atoms.

\hypertarget{remarks_on_terminology}{}\subsubsection*{{Remarks on terminology}}\label{remarks_on_terminology}

There is some terminological variance in the literature to the notion of atomic poset as defined here. In particular, \href{http://en.wikipedia.org/wiki/Atom_%28order_theory%29}{Wikipedia} defines an \emph{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 $n$. The atoms in this lattice are prime numbers while it may also contain [[semi-atom|semi-atoms]] which are powers of primes. This lattice is atomic because any object not the [[bottom]], $1$, 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 \emph{unique factorization theorem}.

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

\begin{displaymath}
c = b \wedge \neg (\bigvee \{atoms\, a: a \leq b\})
\end{displaymath}
To show $B$ is atomistic, it suffices to show $c = 0$. If not, then there is an atom $a'$ such that $a' \leq c$, which means both $a' \leq b$ and

\begin{displaymath}
a' = a' \wedge a' \leq a' \wedge \bigvee \{atoms\, a: a \leq b\} = 0
\end{displaymath}
since $a' \leq \neg(\bigvee \{atoms\, a: a \leq b\})$. This is a contradiction.

Our (\emph{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.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{}\hypertarget{}{}
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.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
In a [[lattice of subtoposes]] the atoms are the 2-valued [[Boolean toposes]]. See \href{subtopos#BooleantoposesAreAtoms}{this proposition}.

\end{example}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

If $a$ is an atom in a [[lattice]] or more generally a [[meet]] [[semilattice]] and $b$ any other element then (using classical logic)

\begin{displaymath}
a \wedge b \in \{a, \bot\} .
\end{displaymath}
This is simply because $a \wedge b \leq a$, so equals either $a$ or $\bot$.

\hypertarget{atoms_and_tiny_elements}{}\subsection*{{Atoms and tiny elements}}\label{atoms_and_tiny_elements}

If $E$ is a poset or preorder, in other words a $\mathbf{2}$-[[enriched category]], an element $e \in E$ is \emph{[[tiny object|tiny]]} if the hom $E(e, -)\colon E \to \mathbf{2}$ preserves all [[supremum|sups]] that exist in $E$. 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.

\begin{prop}
\label{}\hypertarget{}{}
A tiny element in a Boolean algebra is precisely an atom.

\end{prop}
\begin{proof}
Let $a$ be an atom. Let $\{x_i\}$ be a collection of elements that admits a supremum such that $a \leq \bigvee_i x_i$. Then

\begin{displaymath}
a = a \wedge \bigvee_i x_i = \bigvee_i a \wedge x_i
\end{displaymath}
(where the second equation holds since $a \wedge -$ is a left [[adjoint functor|adjoint]], because $B$ is a [[Heyting algebra]]). Since $a$ is positive, for some $i$ the element $a \wedge x_i$ is positive as well. Trivially it holds that $a \wedge x_i \leq a$; since $a$ is an atom, the inequality is an equality. Thus $a \leq x_i$ for some $i$, which is what we want.

If $a$ is not an atom, i.e., if $0 \lt b \lt a$ for some $b$, then

\begin{displaymath}
a = b \vee (a \wedge \neg b)
\end{displaymath}
If $B(a, -)$ preserved the join on the right, then either $a \leq b$ which is evidently false, or $a \leq a \wedge \neg b$, i.e., $a \leq \neg b$, i.e., $b = a \wedge b \leq 0$, also evidently false. Thus $B(a, -)$ does not preserve suprema.

\end{proof}
Only one half of this proposition holds (an atom is a tiny element) if we replace the Boolean algebra $B$ by a general [[frame]]. (In fact, this direction even holds in impredicative constructive mathematics, if the frame is equipped with a [[positive element|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, \href{http://www.acsu.buffalo.edu/~wlawvere/ToposMotion.pdf#page=6}{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 $(-)^D$, particularly in the case of [[synthetic differential geometry]] where such adjoints exist for [[infinitesimal object|infinitesimal objects]] $D$.

\hypertarget{generalization_and_categorification}{}\subsection*{{Generalization and categorification}}\label{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]] $\mathbf{2} = (-1)Grpd$ of [[(-1)-groupoid|(-1)-groupoids]] -- may be generalized and categorified as follows.

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

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

\begin{displaymath}
E \to V^{Tiny(E)^{op}}
\end{displaymath}
sending an object $e$ to $E(i-, e)$. We say that $E$ is \textbf{[[atomic category|atomic]]} if $i \colon Tiny(E) \hookrightarrow E$ is $V$-[[dense functor|dense]], in other words if every object $e$ of $E$ is a canonical colimit of atomic objects below it, in the precise sense that the following enriched [[coend]] exists, and its canonical map to $e$,

\begin{displaymath}
\int^{a \in Tiny(E)} E(i a, e) \cdot i a \to e,
\end{displaymath}
is an isomorphism.

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

\begin{displaymath}
\sup \{i a: i a \leq p\}
\end{displaymath}
so that $E$ is atomic precisely if every element is the sup of the tiny elements below it.

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

\end{thm}
N.B. ``Free sup-lattice'' refers to a left adjoint of the forgetful functor $U \colon SupLat \to Preord$ from sup-lattices to preorders.

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

\begin{displaymath}
L \colon \mathbf{2}^{Tiny(E)^{op}} \to E
\end{displaymath}
which sends $X \colon Tiny(E)^{op} \to \mathbf{2}$ to

\begin{displaymath}
\int^{a \in Tiny(E)} X(a) \cdot i a = \sup \{i a: X(a) = 1\}.
\end{displaymath}
This $L$ is left adjoint to the restricted Yoneda embedding $R \colon E \to \mathbf{2}^{Tiny(E)^{op}}$. The condition that $E$ is atomic says that for each $e \in E$, the value of the counit of $L \dashv R$ at $e$ is an isomorphism

\begin{displaymath}
\int^a E(i a, e) \cdot i a \cong e.
\end{displaymath}
On the other hand, the value of the unit of $L \dashv R$ at an object $X$ is given by a string of isomorphisms

\begin{displaymath}
\itexarray{
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)
}
\end{displaymath}
where the last isomorphism obtains from the fact that $E(i a, -)$ preserves colimits if $a$ is tiny. Thus the unit is also an isomorphism.

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

\end{proof}
\begin{cor}
\label{}\hypertarget{}{}
A complete atomic Boolean algebra $B$ is isomorphic to $2^T$, where $T$ is the discrete preorder of atoms of $B$.

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

\begin{thm}
\label{}\hypertarget{}{}
A category $E$ 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 $Set$-category. Representables $C(-, c)$ are (among the) atomic objects of $Set^{C^{op}}$, and generate the presheaf topos by closing under all small colimits.

\end{thm}
(The literal statement in Bunge's thesis is that a category is equivalent to a presheaf category $Set^{C^{op}}$ if and only if it is cocomplete, regular, and has a generating set of atomic objects, but this is trivially the same since presheaf toposes are of course [[regular category|regular]].)

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[compact object]]


\item [[connected object]]


\item [[Cauchy completion]]



\end{itemize}
[[!redirects atom]] [[!redirects atoms]] [[!redirects atomic element]] [[!redirects atomic elements]]



\end{document}