## 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.

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