Following on from last week's talk, I will introduce some examples of augmentation categories. The first will be the class of crossed simplicial groups, of which Connes' cyclic category is an example. We will show that using the cyclic category we can retrieve SO(2)-equivariant cohomology. Moreover, using the cyclic nerve construction, we can define an equivariant version of the derived stack of local systems, which is a canonical example of 1-Artin derived stack.
Next, we will look at the dendroidal category, and see what the Kan model structure on dendroidal sets should be thought of geometrically. Finally, we combine these two examples with a categorical amalgamation construction forming a new augmentation category.
If time allows, I will present some conjectural links of augmentation categories and test categories, along with some other model structures that we can equip augmented sets with.