nLab Moore closure

Idea

The concept of Moore closure is a very general idea of what it can mean for a set to be closed under some condition. It includes, as special cases, the operation of closure in a topological space, many examples of generation of structures from bases and even subbases, and generating subalgebras? from subsets of an algebra.

Secretly, it is the same thing as the collection of subsets preserved by some monad on a power set (the subset of “modal types”). In fact it is a special case of the notion of closure operator or modality in logic/type theory, namely the special case where the ambient category/hyperdoctrine is the topos Set.

Definitions

We give two equivalent definitions. The first one

• In terms of closure condition

gives the explicit condition for a subset of a power set to qualify as a Moore closure, the second

• In terms of closure operators

characterizes Moore closures as the collections of modal types of suitable closure operators. More abstractly, this characterizes Moore closures

• In terms of monads

on the subobject lattice of the given set.

In terms of closure condition

Definition

Let $X$ be a set, and let $\mathcal{C} \subset \mathcal{P}X$ be a collection of subsets of $X$. Then $\mathcal{C}$ is a Moore collection if every intersection of members of $\mathcal{C}$ belongs to $\mathcal{C}$.

That is, given a family $(A_i)_i$ of sets in $X$,

$$\forall i,\; A_i \in \mathcal{C} \;\Rightarrow\; \bigcap_i A_i \in \mathcal{C} .$$

Definition

Given any collection $\mathcal{B}$ whatsoever of subsets of $X$, the Moore collection generated by $\mathcal{B}$ is the collection of all intersections of members of $\mathcal{B}$.

Remark

This is indeed a Moore collection, and it equals $\mathcal{B}$ if and only if $\mathcal{B}$ is a Moore collection.

In terms of closure operators

Definition

Again let $X$ be a set, and now let $Cl$ be an operation on subsets of $X$. Then $Cl$ is a closure operation if $Cl$ is isotone, extensive, and idempotent. That is,

1. $A \subseteq B \;\Rightarrow\; Cl(A) \subseteq Cl(B)$,
2. $A \subseteq Cl(A)$, and
3. $Cl(Cl(A)) \subseteq Cl(A)$ (the reverse inclusion follows from the previous two properties).

Proposition

If $Cl$ is a closure operation, then let $\mathcal{C}$ be the collection of sets that equal their own closures (the “modal types” or “local objects”). Then $\mathcal{C}$ is a Moore collection.

Conversely, if $\mathcal{C}$ is a Moore collection, then let $Cl(A)$ be the intersection of all closed sets that contain $A$. Then $Cl$ is a closure operator.

Furthermore, the two maps above, from closure operators to Moore collections and vice versa, are inverses.

In terms of monads

Moore closures on $X$ are precisely monads on the subobject lattice $\mathcal{P}X$. The property (1) of a closure operator, def. , corresponds to the action of the monad on morphisms, while (2,3) are the unit and multiplication of the monad. (The rest of the requirements of a monad are trivial in a poset, since they state the equality of various morphisms with common source and target.)

Examples

The closed subsets in a topological space form a Moore collection; here the closure of a set $A$ is its closure in the usual sense. In fact, a topological space can be defined as a set equipped with a Moore closure with either of these additional properties (which are equivalent):

• $Cl(\empty) = \empty$ and $Cl(A \cup B) = Cl(A) \cup Cl(B)$.
• $\empty$ is closed, and so is $A \cup B$ if $A$ and $B$ are closed.

(However, these properties may fail in constructive mathematics; in fact, a topology cannot be constructively recovered from its closure operation.)

The first pair of properties is equivalent to the following weaker ones given that $A \subseteq Cl(A)$:

• $Cl(\empty) \subseteq \empty$ and $Cl(A \cup B) \subseteq Cl(A) \cup Cl(B)$.

The closure operator of a topological space with underlying set $X$, and thus in classical mathematics, a topology on a set $X$, can thus be described as an oplax monoidal monad on the join semilattice $\mathcal{P}X$ considered as a cocartesian monoidal category.

Notice that a preclosure in a pretopological space, is not a closure in the above sense, even though some authors do call this ‘closure’.

Here are some algebraic examples:

• The subgroups of a group $G$ form a Moore collection; the closure of a subset $B$ of $G$ is the subgroup generated by $B$.

• The subrings of a ring $R$ form a Moore collection; the closure of a subset $B$ of $R$ is the subring generated by $B$.

• The subspaces of a vector space $V$ form a Moore collection; the closure of a subset $B$ of $V$ is the subspace spanned by $B$.

• OK, you get the idea. This applies to any algebraic theory. For a finitary algebraic theory, the lattice of closed elements is an algebraic lattice.

But also:

• The normal subgroups of $G$ form a Moore collection; the closure of $B$ is the normal subgroup generated by $B$.

• The ideals of a ring $R$ form a Moore collection; the closure of $B$ is the ideal generated by $B$.

• The (topologically) closed subspaces of a Hilbert space $H$ form a Moore collection; the closure of $B$ is the closed subspace generated by $B$.

• And many further examples.

Here are some examples on power sets:

• The topologies on $X$ form a Moore collection on $\mathcal{P}X$; the closure of a subset $\mathcal{B}$ of $\mathcal{P}X$ is the topology generated by $\mathcal{B}$ as a subbase.

• The filters on $X$ form a Moore collection on $\mathcal{P}X$; the closure of $\mathcal{B}$ is the filter generated by $\mathcal{B}$ as a subbase. (The proper filters on $X$ do not form a Moore collection; not every $\mathcal{B}$ generates a proper filter.)

• The $\sigma$-algebras on $X$ form a Moore collection on $\mathcal{P}X$; the closure of $\mathcal{B}$ is the $\sigma$-algebra generated by $\mathcal{B}$. (This is the ‘abstract nonsense’ way to generate a $\sigma$-algebra; else you have to do transfinite induction on countable ordinals.)

• And so on.

Topping off these, the Moore collections on $X$ form a Moore collection on $\mathcal{P}X$; the closure of $\mathcal{B}$ is the Moore collection generated by $\mathcal{B}$ as described in the definitions.

See also at matroid.

Generalisations

The definition of Moore collection really makes sense in any inflattice; even better, the definition of closure operator makes sense in any poset. This context is the generic meaning of closure operator; here are some examples:

• Instead of $\mathcal{P}X$, work in the opposite poset $\mathcal{P}^{op}X$. Then the open sets in a topological space $X$ form a Moore collection whose closure operator is the usual interior operation. Now we can define a topological space as a set equipped with a Moore closure operator on $\mathcal{P}^{op}X$ that preserves joins (which here are intersections); this definition is even valid constructively.

• Let $f \dashv g$ be a Galois connection between posets $A$ and $B$. Call an element of $A$ normal if $g(f(a)) \leq a$ (the reverse is always true). Then $g \circ f$ is a closure operator. This generalises the case of the normal subgroups of $G$ when $G$ is the Galois group of an extension of fields.

Since Galois connections are simply adjunctions between posets, the concept of Moore closure cries out for categorification. And in fact, the answer is well known in category theory: it is a monad.

Related concepts

• matroid

• modal operator, modality

• modal logic, modal type theory, modal homotopy type theory

References

• Erich Schechter, Section 4.1-4.12 in: Handbook of Analysis and its Foundations

See also:

• Wikipedia, Closure operator