Let be the category of meet-semilattices (i.e., posets admitting finite meets); equivalently, the category of idempotent commutative monoids. Regarding Sierpinski space ambimorphically either as a topological space or as a meet-semilattice, i.e., an internal idempotent commutative monoid in seen as a dualizing object, there is a pair of adjoint functors given by homming into ,
In more detail: for a space , the set of continuous maps is naturally identified with the set of open subsets, i.e., the topology (where the pointwise meet-semilattice structure on coincides with the meets defined by intersection on ). For a meet-semilattice , the set is topologized as a subspace of the product space , a product of copies of Sierpinski space. The adjunction says that there is a natural bijection
between semilattice maps above and continuous maps below.
The filter monad is the monad on induced by this adjunction. (Compare ultrafilter monad.)
For a space , is the set of meet-preserving maps . This is in natural bijection with subsets of that are upward-closed and closed under finite intersections, i.e., filters in . The unit of the monad evaluated at is , taking to the filter of open sets containing .
We can read off the multiplication on the filter monad directly from this description. For , we have
However, there is another description often encountered in the literature:
To see , suppose , i.e., suppose there is with and . Then for all , and since already , we see by upward closure of . Hence .
To see , suppose , and then put . Then , and by definition, so .
Last revised on November 17, 2024 at 23:47:11. See the history of this page for a list of all contributions to it.