Triangles and globes

When explaining higher categories to other mathematicians, one is sometimes tempted to make them believe that these are natural generalizations of categories, where one would allow cells of higher dimension. Obviously, this oversimplification elides the many choices and difficulties necessary to a working definition of higher categories, but one of them is particularly critical: the choice of the shape of these higher cells. Two candidates, however, stand out from the rest. The first, which we shall call globular, is arguably the closest to intuition, and the second, simplicial, allows one to connect higher categories to simplicial sets, these "lovely objects about which algebraic topologists know a lot", to quote Street. Identifying the right simplicial definition of higher categories, and the link it would have with their globular sibling, has been a rich program which has kept many mathematicians busy since the 70s and to which Dominic Verity has brought major contributions.

We will begin by recalling the definition of (globular) $\omega$-categories, as well as the Roberts-Street Nerve $N:\omega \text{-cat}\to sSet$. We will use it to justify the definition of "complicial sets", and state the Roberts-Street Conjecture, which stipulates an equivalence between the globular and simplicial versions of higher categories, and which was resolved positively by Dominic Verity. From this, we will discuss the homotopical version of complicial sets, introduced by Street, and studied in depth by Verity. We will then state the homotopical version of the Roberts-Street Conjecture, resolved positively by the speaker, building on the work of Ozornova-Rovelli.