Sebastian Öberg: Rigidifying homotopy commutative diagrams

In this talk we investigate functors indexed by simplex categories that send certain face maps to weak equivalences. We explain why such functors can be regarded as homotopy commutative diagrams. The key question we consider is related to rigidifications of such functors: under what circumstances is such a functor weakly equivalent to a functor that send these face maps to isomorphisms? We show that if the simplicial set is the nerve of a small category then such an homotopy commutative diagram can indeed be rigidified. We conjecture that this is also true whenever the simplicial set is a quasi-category. If there is time we will also look at connection between these homotopy commutative diagrams and mapping spaces of model categories via hammock localization.