The main focus of this 2-part special is to show how two important constructions of higher category theory, namely the homotopy coherent nerve and the (un)straightening adjunction, are most naturally seen as cubical-to-simplicial. To this end, I will define the homotopy coherent nerve functor to be taking a *cubical* category to a simplicial set and will rephrase the (un)straightening adjunction accordingly. I will then show that the straightening-over-the-point functor is a co-reflective embedding of the category of simplicial sets into the category of cubical sets.
This talk is based on:
K, Voevodsky, "A cubical approach to straightening", preprint, 2018.
K, Lindsey, Wong, "A co-reflection of cubical sets into simplicial
sets with applications to model structures", New York Journal of
Mathematics 25 (2019).