filter monad

Let SemiLatSemiLat be the category of meet-semilattices (i.e., posets admitting finite meets); equivalently, the category of idempotent commutative monoids. Regarding Sierpinski space 2={01}\mathbf{2} = \{0 \leq 1\} ambimorphically either as a topological space or as a meet-semilattice, i.e., an internal idempotent commutative monoid in TopTop seen as a dualizing object, there is a pair of adjoint functors given by homming into 2\mathbf{2},

TopTop(,2) opSemiLat opSemiLat opSemiLat(,2)Top.Top \stackrel{Top(-, \mathbf{2})^{op}}{\to} SemiLat^{op} \;\; \dashv \;\; SemiLat^{op} \stackrel{SemiLat(-, \mathbf{2})}{\to} Top.

In more detail: for a space XX, the set of continuous maps X2X \to \mathbf{2} is naturally identified with the set of open subsets, i.e., the topology 𝒪(X)\mathcal{O}(X) (where the pointwise meet-semilattice structure on hom(X,2)\hom(X, \mathbf{2}) coincides with the meets defined by intersection on 𝒪(X)\mathcal{O}(X)). For a meet-semilattice LL, the set SemiLat(L,2)SemiLat(L, \mathbf{2}) is topologized as a subspace of the product space 2 |L|\mathbf{2}^{{|L|}}, a product of |L|{|L|} copies of Sierpinski space. The adjunction says that there is a natural bijection

LTop(X,2)XSemiLat(L,2)\frac{L \to Top(X, \mathbf{2})}{X \to SemiLat(L, \mathbf{2})}

between semilattice maps above and continuous maps below.

The filter monad is the monad FiltFilt on TopTop induced by this adjunction. (Compare ultrafilter monad.)

For a space XX, Filt(X)Filt(X) is the set of meet-preserving maps 𝒪(X)2\mathcal{O}(X) \to \mathbf{2}. This is in natural bijection with subsets of 𝒪(X)\mathcal{O}(X) that are upward-closed and closed under finite intersections, i.e., filters in 𝒪(X)\mathcal{O}(X). The unit of the monad evaluated at XX is u X:XFilt(X)u_X: X \to Filt(X), taking xXx \in X to the filter u X(x)u_X(x) of open sets containing xx.

We can read off the multiplication mm on the filter monad directly from this description. For ΦFilt(Filt(X))\Phi \in Filt(Filt(X)), we have

μ X(Φ){U:𝒪(X)|{F:Filt(X)|UF}Φ}.\mu_X(\Phi) \coloneqq \{U:\mathcal{O}(X)\; |\; \{F: Filt(X)\; |\; U \in F\} \in \Phi\}.

However, there is another description often encountered in the literature:

μ X(Φ)={𝒰:𝒰Φ}.\mu_X(\Phi) = \bigcup \{\bigcap \mathcal{U}: \mathcal{U} \in \Phi\}.

To see {𝒰:𝒰Φ}μ X(Φ)\bigcup \{\bigcap \mathcal{U}: \mathcal{U} \in \Phi\} \subseteq \mu_X(\Phi), suppose U{𝒰:𝒰Φ}U \in \bigcup \{\bigcap \mathcal{U}: \mathcal{U} \in \Phi\}, i.e., suppose there is 𝒰\mathcal{U} with 𝒰Φ\mathcal{U} \in \Phi and U𝒰U \in \bigcap \mathcal{U}. Then UFU \in F for all F𝒰F \in \mathcal{U}, and since already 𝒰Φ\mathcal{U} \in \Phi, we see {FFilt(X):UF}Φ\{F \in Filt(X): U \in F\} \in \Phi by upward closure of Φ\Phi. Hence Uμ X(Φ)U \in \mu_X(\Phi).

To see μ X(Φ){𝒰:𝒰Φ}\mu_X(\Phi) \subseteq \bigcup \{\bigcap \mathcal{U}: \mathcal{U} \in \Phi\}, suppose Uμ X(Φ)U \in \mu_X(\Phi), and then put 𝒰={F:Filt(X)|UF}\mathcal{U}' = \{F: Filt(X)\; |\; U \in F\}. Then 𝒰Φ\mathcal{U}' \in \Phi, and U𝒰U \in \bigcap \mathcal{U}' by definition, so U{𝒰:𝒰Φ}U \in \bigcup \{\bigcap \mathcal{U}: \mathcal{U} \in \Phi\}.

Last revised on July 17, 2014 at 03:34:57. See the history of this page for a list of all contributions to it.