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A cubical approach to straightening
Journal of Topology  (IF1.582),  Pub Date : 2020-10-23, DOI: 10.1112/topo.12173
Krzysztof Kapulkin, Vladimir Voevodsky

For a suitable choice of the cube category, we construct a Grothendieck topology on it such that sheaves with respect to this topology are exactly simplicial sets (thus establishing simplicial sets as a reflective subcategory of cubical sets). We then extend the construction of the homotopy coherent nerve to cubical categories and establish an analogue of Lurie's straightening–unstraightening construction.