# Algebraic characterization of convergence spaces

Is there an algebraic characterization of convergence spaces similar to Barr's characterization of topological spaces as lax algebras for the ultrafilter monad? I'm also curious about the same question for uniform and Cauchy convergence spaces.

• Nice question! This has been bugging me for a long time :) – goblin Feb 11 '17 at 15:41

Note that the axioms for a convergence space are all phrased in terms of convergence to a single point $x$. So in a Barr-like axiomatization of convergence spaces, one expects there will be no analog of Barr's lax associativity axiom, which relates convergence at many different points to convergence at a single other point. In fact, if you simply drop the lax associativity axiom from Barr's characterization, (so you just have a relation from ultrafilters to the original set which is lax-unital) you get pseudotopoogical spaces, which can be identified as a certain subcategory of convergence spaces.
To get convergence spaces you would presumably have to work with the filter monad rather than the ultrafilter monad. Be aware that there is more than one reasonable way to extend the filter monad from $\mathsf{Set}$ to $\mathsf{Rel}$ -- see Seal, which discusses two important ones, and Schubert and Seal, which discusses them more generally.