A study of Kock's fat delta

The simplicial category \Delta plays an important role in category theory. One of the reasons is that there is a fully faithful nerve functor from the category Cat to the category of simplicial sets (that is functors from the opposite of \Delta to Set). Its essential image consists of simplicial sets satisfying additional conditions that the Segal maps are isomorphisms. This allows to think of a small category as a type of simplicial set, and this idea has been carried on in higher dimensions in defining appropriate notions of higher categories. This talk is about a modification of \Delta, introduced by J. Kock, called the fat delta. After explaining the motivation for our interest in fat delta, both from higher category theory and from type theory, we present a study of fat delta in terms of monad with arities. This leads to a nerve theorem for relative semicategories, as well as a description of fat delta as a hypermoment categories in the sense of Berger. This is joint work with Tom de Jong, Nicolai Kraus and Stiephen Pradal, arXiv.2503.10963v1.