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The ultrafilter endofunctor with principal unit and flattening multiplication
Definition
The ultrafilter endofunctor sends to the set of ultrafilters on and sends to ultrafilter pushforward . Its principal unit and flattening multiplication are
The preceding pushforward and flattening lemmas establish that these assignments are well-defined and natural.
Depends on
Used by
- The open-set family induced by an ultrafilter algebra Definition
- On finite sets the ultrafilter monad is naturally isomorphic to the identity; assuming the ultrafilter lemma, its unit is not invertible on the natural numbers Example
- The ultrafilter algebra on a finite discrete space Example
- βℕ as the free ultrafilter algebra Example
- The ultrafilter-limit map of a compact Hausdorff space is an algebra for the ultrafilter monad Lemma
- Under the ultrafilter lemma, an ultrafilter algebra maps each ultrafilter to its unique limit Lemma
- Under the ultrafilter lemma, closure in an ultrafilter-algebra topology is the image of ultrafilters containing the set Lemma
- The codensity monad of the small skeleton of finite sets is the ultrafilter monad Theorem
- The ultrafilter endofunctor with principal unit and flattening multiplication is a monad Theorem
Dependency tree · two levels
5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- E. Riehl, Category Theory in Context, 2nd ed., Example 5.1.4(v) and Exercise 5.1.ii (standard reference, not scraped)