nLab
nucleus

Nuclei

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 LL be a frame, that is a suplattice satisfying the infinite distributivity law.

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

  1. j(ab)=j(a)j(b)j(a \wedge b) = j(a) \wedge j(b),
  2. aj(a)a \leq j(a),
  3. j(j(a))j(a)j(j(a)) \leq j(a).

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

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

  1. j()=j(\top) = \top,
  2. j(a)j(b)j(a) \leq j(b) if aba \leq b,
  3. j(j(a))=j(a)j(j(a)) = j(a).

The subset of closed elements

Let LL be a frame.

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

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

  1. AJ\bigwedge A \in J whenever AJA \subseteq J (using that LL is a complete lattice),
  2. abJa \Rightarrow b \in J whenever bJb \in J (using that LL is a Heyting algebra).

Then we recover j:LLj\colon L \to L by

j(a){b:LbJ,ab} j(a) \coloneqq \bigwedge \{ b\colon L \;|\; b \in J,\; a \leq b \}

and we have

J={a:Lj(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 LL be a frame, and let jj be a nucleus on LL.

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

Conversely, suppose that MM is any regular quotient of LL; that is, we have a surjective frame homomorphism k:LMk\colon L \to M. Since kk is a frame homomorphism, it has (by the adjoint functor theorem) a right adjoint k *:MLk_*\colon M \to L. Let j:LLj\colon L \to L be the composite of kk followed by k *k_*. Then jj is a nucleus, and k *k_* is an embedding (in Pos, not FrmFrm) whose image is L/jL/j.

In short, given a nucleus jj, we have an adjunction j *:LL/j:j *j^*\colon L \rightleftarrows L/j: j_*, where j *j^* is a surjective homomorphism and j *j_* is the inclusion function; while, given a surjective homomorphism kk, we have an adjunction k:LM:k *k\colon L \rightleftarrows M: k_*, where k *kk_* \circ k is a nucleus and k *k_* is an embedding.

If we think of LL as a locale, then we define a sublocale of LL to be a quotient frame of LL, which corresponds to a nucleus on LL 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

Revised on July 2, 2010 23:25:06 by Toby Bartels (98.19.48.177)