Some facts about sound classes of weights

Giacomo Tendas – 2 March 2022

In this talk I will consider the (enriched) notions of sound and weakly-sound classes of weights, first introduced in the ordinary setting by Adamek, Borceux, Lack, and Rosicky. We’ll see (1) that these two notions are actually equivalent in many cases and (2) that a sound class \Phi is all you need to obtain a theory of locally \Phi-presentable V-categories, even if V is not locally presentable itself but just locally bounded. The class Q of the Cauchy weights will play an important role in (2), in particular we’ll obtain a new proof of the fact that Q is a small class whenever V is locally presentable (this was first shown by Scott Johnson in 1989).