D-Ultrafilters and their Monads

TL;DR

This paper characterizes the codensity monad in various categories using D-ultrafilters, providing a unified framework for understanding these monads across different mathematical structures.

Contribution

It introduces D-ultrafilters and demonstrates that the codensity monad corresponds to objects representing all D-ultrafilters, extending to multiple categories.

Findings

The codensity monad can be described via D-ultrafilters.

Applicable to categories like sets, vector spaces, posets, and graphs.

Provides a unified approach across diverse mathematical structures.

Abstract

For a number of locally finitely presentable categories K we describe the codensity monad of the full embedding of all finitely presentable objects into K. We introduce the concept of D-ultrafilter on an object, where D is a "nice" cogenerator of K. We prove that the codensity monad assigns to every object an object representing all D-ultrafilters on it. Our result covers e.g. categories of sets, vector spaces, posets, semilattices, graphs and M-sets for finite commutative monoids M.