Draft:Ultrafilter monad

In category theory and set theory, the ultrafilter monad, also known as the compact Hausdorff monad, is a monad that maps a set $S$ to the set of ultrafilters on $S.$

Formal definition

Given a set $S$, define $US$ to be the set $$\{{\mathcal {D}}\in {\mathcal {P}}({\mathcal {P}}(S))|{\mathcal {D}}{\text{ is an ultrafilter on }}S\}.$$ Since $U$ is a functor, it must also map functions to functions. Given a function $f:S\to T,$ and given some ultrafilter ${\mathcal {D}}\in US,$ $$f({\mathcal {D}})=\{A\in {\mathcal {P}}(T)|f^{-1}A\in {\mathcal {D}}\}.$$

Since $U$ is a monad, it also has a unit map and a multiplication map. The unit map ${\displaystyle \eta$ :\mathrm {id} _{\mathrm {Set} }\to U} maps an element $s\in S$ to the principal ultrafilter on $S.$ The multiplication map $\mu :UU\to U$ maps ultrafilters of ultrafilters to ultrafilters. For a set $S$, and for an ultrafilter of ultrafilters ${\mathcal {E}}\in UUS,$ $$\mu _{S}({\mathcal {E}})=\{A\subseteq S|\{{\mathcal {D}}\in US|A\in {\mathcal {D}}\}\in {\mathcal {E}}\}.$$

Alternative characterizations

The ultrafilter monad is the codensity monad of the map $\mathrm {FinSet} \to \mathrm {Set}$ from the category of finite sets to the category of sets.

The functor $U$, without its monad structure, is the terminal object in the category of endofunctors on $\mathrm {Set}$ preserving finite coproducts. The universal morphisms $\mathrm {id_{Set}} \to U$ and $UU\to U$ are the monad transformations.

$U$ can also be defined using Boolean algebras, which form the category $\mathrm {BoolAlg}$. If $F$ is the free functor from $\mathrm {Set}$ to $\mathrm {BoolAlg}$ and $2$ is the two-element Boolean algebra, the functor $\operatorname {Hom} (-,2)\circ F$ is naturally isomorphic to $U.$

Relation with compact Hausdorff spaces

The Eilenberg–Moore category over the ultrafilter monad is equivalent to the category of compact Hausdorff topological spaces.

This equivalence is related to convergence of filters. In a Hausdorff space, ultrafilters converge to at most one point, while in a compact space, they converge to at least one point.

The Eilenberg–Moore category of a monad on $\mathrm {Set}$ is equipped with an adjoint pair of functors: a forgetful functor from the category to $\mathrm {Set}$ and a free functor from $\mathrm {Set}$ to the category. The forgetful functor $\mathrm {CHaus} \to \mathrm {Set}$ is the standard forgetful functor (it forgets the topology), while the free functor $\mathrm {Set} \to \mathrm {CHaus}$ corresponds to the Stone–Čech compactification of a set.

In the more general context of topological spaces, which need not be compact or Hausdorff, ultrafilters need not converge to a unique point; instead, they converge to a set of points, and convergence is a relation rather than a function. For example, in a discrete topology, any non-principal ultrafilter does not converge to any points, and in the indiscrete topology, every ultrafilter converges to every point. However, ultrafilter convergence is still sufficient to recover the topology. In this Barr has shown that the category of topological spaces can be seen as the category of relational algebras over the ultrafilter monad.

Relation to measure theory

Ultrafilters on a set can be seen as finitely additive measures taking values in $\{0,1\}$.

Analogues

The ultrafilter monad has several generalizations and analogues. Many variants of the ultrafilter monad resemble a functor of the form $\operatorname {Hom} (\operatorname {Hom} (-,X))$, where $X$ is an appropriate object.

Double dualization

Given any $K$-vector space $V$, one can define the dual vector space $V^{*}=\operatorname {Hom} (V,K).$ The functor $(-)^{*}$ is contravariant. The functor $(-)^{**}=\operatorname {Hom} (\operatorname {Hom} (-,K),K)$ is covariant and can be given a monad structure. With this monad stucture, it is called the double dualization monad.

While the Eilenberg-Moore category of the ultrafilter monad is equivalent to the category of compact Hausdorff spaces, the Eilenberg Moore category of the double dualization monad is equivalent to the category of linearly compact vector spaces. A linearly compact vector space is a Hausdorff topological vector space such that:

• There is a basis of open affine sets.
• If the intersection of an infinite set of closed affine spaces is empty, the intersection of a finite subset must be empty.

The double dualization monad's Eilenberg-Moore category is also equivalent to $K{\textrm {-}}\mathrm {Vect} ^{op}$.

The double dualization monad is the codensity monad of the injection $K{\textrm {-}}\mathrm {FinVect} \to K{\textrm {-}}\mathrm {Vect} ,$ where $K{\textrm {-}}\mathrm {FinVect}$ is the category of finite-dimensional $K$-vector spaces.

The Giry monad

Main article: Giry monad

The Giry monad can be realized as the codensity monad of a functor related to convex sets.

The ultrafilter monad, like the Giry monad, can be seen as a monad of measures.

The filter monad

A monad on filters can be defined. Its Eilenberg-Moore category is equivalent to the category of continuous lattices as defined by Scott.

Prime filters

The category of partially ordered sets can be equipped with a monad based on prime filters. The prime filter monad is related to Priestley duality, whereas the ultrafilter monad is related to Stone duality.

The Eilenberg-Moore category of this monad is the category of compact Hausdorff spaces with closed partial orders. (A partial order on a Hausdorff space $H$ is a binary relation, so it can be represented as a subset of $H\times H$. The partial order is said to be closed if this set is closed.)

D-ultrafilters

Adámek and Sousa have described a generalization of ultrafilters, which they call D-ultrafilters, defined over an arbitrary locally finitely presentable category $C.$ In the special case where $C=\mathrm {Set} ,$ these coincide with standard ultrafilters. There is a D-ultrafilter monad that generalizes the standard ultrafilter monad. This monad is the codensity monad on the embedding $C_{fp}\to C$ from the category $C_{fp}$ of finitely presented objects to $C.$

Notes

Manes, Ernest (1969). "A triple theoretic construction of compact algebras". Seminar on Triples and Categorical Homology Theory. Lecture Notes in Mathematics. Vol. 80. pp. 91–118. doi:10.1007/BFB0083083. ISBN 978-3-540-04601-1. Comfort, W. Wistar; Negrepontis, Stylianos (1974). The Theory of Ultrafilters. doi:10.1007/978-3-642-65780-1. ISBN 978-3-642-65782-5. Halmos, Paul. Finite-Dimensional Vector Spaces. Kennison, J.F.; Gildenhuys, Dion (1971). "Equational completion, model induced triples and pro-objects". Journal of Pure and Applied Algebra. 1 (4): 317–346. doi:10.1016/0022-4049(71)90001-6.

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