Algebras of classified restriction categories

JS Lemay – 20 September 2023

For a classified restriction category, there is a monad on its subcategory of total maps, called the classifying monad. The Kleisli category of the classifying monad is isomorphic to the classified restriction category. Is there anything interesting we can say about the algebras of the classifying monad? In general it doesn’t appear so… However for many important examples of classified restriction categories, such as the category of sets and partial functions, the Eilenberg-Moore category of the classifying monad is equivalent to the Kleisli category. I will give a restriction category explanation for why this is the case by using restriction idempotents and relating them to B. Jacobs’ notion of bases for algebras in “Bases as Coalgebras”.