An exactness property of T-Alg

John Bourke – 10 June 2009

Given a monad T on Set, many properties of the category of Sets are also true for Set^T, the Eilenberg-Moore category of algebras for the monad. Some such properties are completeness, the existence of coequalizers of equivalence relations, and the fact that equivalence relations are effective. This talk is about 2-dimensional analogues to the above situation. T is now a 2-monad on Cat, and we are interested in T-Alg, the 2-category of strict algebras, pseudomorphisms, and the usual 2-cells. Dealing with pseudomorphisms makes T-Alg more bicategorical and less like Cat, yet T-Alg still inherits the pie limits of Cat and has all limits involving only strict morphisms. These limits are enough to define the analogues of kernel pair (higher kernel) and equivalence relation (congruence) in T-Alg, and the corresponding notion to coequalizer becomes codescent object. We will see that if a 2-monad T is strongly finitary then T-Alg has codescent objects of congruences, and congruences are effective.