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.
Let be a frame, that is a suplattice satisfying the infinite distributivity law.
A nucleus on is a function that satisfies the following identities:
In other words, a nucleus on is a meet-preserving monad on .
Note that the following properties of a nucleus might be included in the definition, but they follow from the above:
- if ,
The subset of closed elements
Let be a frame.
As a nucleus on (being a monad on a poset) is a kind of Moore closure, we say that an element of is -closed if . (But note that this has nothing to do with the closed subspaces of the locale .)
We may equivalently define a nucleus on to be a subset of that satisfies certain conditions, namely these identities:
- whenever (using that is a complete lattice),
- whenever (using that is a Heyting algebra).
Then we recover by
and we have
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 be a frame, and let be a nucleus on .
Let be the subset of consisting of the -closed elements of (those elements such that ). Note that, by property (3) above, we may interpret as a function , which is a surjective frame homomorphism. Since Frm is an algebraic category, this means that is a regular quotient of .
Conversely, suppose that is any regular quotient of ; that is, we have a surjective frame homomorphism . Since is a frame homomorphism, it has (by the adjoint functor theorem) a right adjoint . Let be the composite of followed by . Then is a nucleus, and is an embedding (in Pos, not ) whose image is .
In short, given a nucleus , we have an adjunction , where is a surjective homomorphism and is the inclusion function; while, given a surjective homomorphism , we have an adjunction , where is a nucleus and is an embedding.
If we think of as a locale, then we define a sublocale of to be a quotient frame of , which corresponds to a nucleus on as above.
If you categorify from locales to toposes, then nuclei become Lawvere–Tierney topologies, and the operation of the nucleus becomes sheafification.