nLab
nucleus

Idea

Recall that frames are dual to locales, and locales are kinds of spaces. So, if you adopt locales as models for spaces, then your models for subspaces are quotient frames. However, much as a quotient set can be described by an equivalence relation on the original set, so a quotient frame may be described by an appropriate structure on the original frame. That structure is a nucleus.

Thus, nuclei correspond to sublocales.

Definitions

Let $L$ be a frame, that is a suplattice satisfying the infinite distributivity law.

A nucleus on $L$ is a function $j:L\to L$ that satisfies the following identities:

1. $j(a\wedge b)=j(a)\wedge j(b)$,
2. $a\le j(a)$,
3. $j(j(a))\le j(a)$.

In other words, a nucleus on $L$ is a meet-preserving monad on $L$.

Note that the following properties of a nucleus might be included in the definition, but they follow from the above:

1. $j(\top )=\top$,
2. $j(a)\le j(b)$ if $a\le b$,
3. $j(j(a))=j(a)$.

The subset of closed elements

Let $L$ be a frame.

As a nucleus $j$ on $L$ (being a monad on a poset) is a kind of Moore closure, we say that an element $a$ of $L$ is $j$-closed if $j(a)=a$. (But note that this has nothing to do with the closed subspaces of the locale $L$.)

We may equivalently define a nucleus on $L$ to be a subset $J$ of $L$ that satisfies certain conditions, namely these identities:

1. $\bigwedge A\in J$ whenever $A\subseteq J$ (using that $L$ is a complete lattice),
2. $a\Rightarrow b\in J$ whenever $b\in J$ (using that $L$ is a Heyting algebra).

Then we recover $j:L\to L$ by

$$j(a)≔\bigwedge \{b:L\mid b\in J,a\le b\}$$j(a) \coloneqq \bigwedge \{ b\colon L \;|\; b \in J,\; a \leq b \}

and we have

$$J=\{a:L\mid j(a)=a\}.$$J = \{ a \colon L \;|\; j(a) = a \} .

Check all this, and expand on it if necessary.

This approach to nuclei is not appropriate in a predicative approach to topology, where we want to use large (but accessible) frames, which may not be meet-complete.

Quotient frames and sublocales

Let $L$ be a frame, and let $j$ be a nucleus on $L$.

Let $L/j$ be the subset of $L$ consisting of the $j$-closed elements of $L$ (those elements $a$ such that $j(a)=a$). Note that, by property (3) above, we may interpret $j$ as a function $j^{*}:L\to L/j$, which is a surjective frame homomorphism. Since Frm is an algebraic category, this means that $L/j$ is a regular quotient of $L$.

Conversely, suppose that $M$ is any regular quotient of $L$; that is, we have a surjective frame homomorphism $k:L\to M$. Since $k$ is a frame homomorphism, it has (by the adjoint functor theorem) a right adjoint $k_{*}:M\to L$. Let $j:L\to L$ be the composite of $k$ followed by $k_{*}$. Then $j$ is a nucleus, and $k_{*}$ is an embedding (in Pos, not $\mathrm{Frm}$) whose image is $L/j$.

In short, given a nucleus $j$, we have an adjunction $j^{*}:L\rightleftarrows L/j:j_{*}$, where $j^{*}$ is a surjective homomorphism and $j_{*}$ is the inclusion function; while, given a surjective homomorphism $k$, we have an adjunction $k:L\rightleftarrows M:k_{*}$, where $k_{*}\circ k$ is a nucleus and $k_{*}$ is an embedding.

If we think of $L$ as a locale, then we define a sublocale of $L$ to be a quotient frame of $L$, which corresponds to a nucleus on $L$ as above.

Categorification

If you categorify from locales to toposes, then nuclei become Lawvere–Tierney topologies, and the operation of the nucleus becomes sheafification.

References

• Stone Spaces, II.2