Stefan Milius - Demystifying Codensity Monads through Duality

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. 

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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 Schützenberger 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 distributive laws by Richard Garner. Also weak distributive law in nlab, and Stone Duality for Monads by Richard Garner, Alyssa Renata and Nicolas Wu. 

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