The decalage comonads and their restrictions

Adrian Miranda – 14 September 2020

I will begin by reminding everyone of the Decalage construction on simplicial sets, and describe how it inherits a comonad structure from the 'free monoid'. I will then describe its category of coalgebras in a few different ways, and mention a conjectural description which I am currently checking. We will then consider its restrictions to three distinguished subcategories of sSet which will be of interest:

1 Cat, 2 2-Cat_nlax; 2-categories and normal lax functors, (this is new work, joint with Steve) 3 qCat; quasi-categories.

In each of these cases, I will provide a more familiar description of the coalgebra structures. I will also describe some examples of decalages and Dec-coalgebras along the way.