Duality is not just good, it's the good! Codensity monads though,... See William Lawvere's remarks on the semi-lattice construction of Set in About Logic Interview with Urs Schreiber . 52:21 In case you're like me and don't remember what an ultrafilter on a set is: Filters are dual to ideals . Subscribe to Topos Institute . Daniela Petrisan - Operations on languages and Codensity monads Talk given at Simons Institute in November 2016: Given some operation on languages, a natural question to ask is what is the corresponding operation at the level of recognisers. For example, the concatenation of two languages recognized by monoids M and N, respectively, is recognized by the Schutzenberger product of M and N.... See The SchĂĽtzenberger product for syntactic spaces by Mai Gehrke, Daniela Petrisan and Luca Reggio. See also https://www.irif.fr/~petrisan/ and The Vietoris monad and weak distrib...