2-classifiers for 2-algebras

Abstract

In this talk I will report on work in progress, joint with David Jaz Myers, about lifting discrete opfibration classifiers (2-classifiers, i.e. a ‘Set’-like object) from a 2-category K to the 2-category of algebras of a 2-monad T. In the setting of DOTS, we often construct behaviour functors as ‘representables’, but without a 2-classifier one can’t really call these ‘representables’. Moreover, there is a strong connection between compositionality of such functors, the properties of the algebra they map out of, and the properties of the object(s) that represents them. These phenomena are in fact completely general, so we set out to better understand the situation and found some frankly interesting notions and results, chiefly a tight result on the existence of 2-classifiers for 2-algebras.