For the lax-algebraic presentations of Top as (F,2)-Cat, via the power-enriched filter monad 08b2eb28d678e7e6f9a2328ff3" title="Click to view the MathML source">F and of App as (I,2)-Cat, via the power-enriched functional ideal monad I, we present weaker conditions in terms of convergence of filters and functional ideals respectively, equivalent to the usual regularity in Top and App.
For the lax-algebraic presentation of App as (B,2)-Cat, via the prime functional ideal monad B, a submonad of I with the initial extension to Rel, restricting to proper elements already gives more interesting results. We prove that B-regularity (restricted to proper prime functional ideals) is equivalent to the approach space being topological and regular. However it requires further weakening of the concept to obtain a characterization of the usual regularity in App in terms of convergence of prime functional ideals.