I want to be straight!

Dominic Verity – 13 May 2020

One of the key constructions in Jacob Lurie's development of quasi-category theory is the (un)straightening equivalence which, for our purposes here, we might decompose into two stages. The first of these is, in essence, the Grothendieck construction, which straightens functors f: A^{op}\to Q , with codomain the quasi-category of small infinity-categories Q, into cartesian (nee Grothendieck) fibrations p:E \to A with small fibres. The second relates a functor quasi-category [A^{op},Q] to the homotopy coherent nerve of a corresponding enriched model category [CA^{op}, sSet_J] of simplicial presheaves under the Joyal model structure. This step straightens homotopy coherent natural transformations into (strict) simplicially natural ones. In this talk we propose to implement the second of these steps in the complicial model of (\infty,\infty)-categories and thereby provide a foundation for developing the category theory of those structures.

Specifically, let msSet_C denote the model category of marked (nee stratified) simplicial sets equipped with a complicial model structure and let M denote a model category enriched in there. Assume further that M is well behaved enough to ensure that for each (small) msSet -enriched category D the enriched functor category [D,M] admits the projective model structure, thereby making it into a msSet_C -enriched model category (M is an admissible model category say). Then given any (small) marked simplicial set X \in msSet we shall construct a straightening equivalence

N([CX,M]_{cf}) \to [X, N(M_{cf})] of complicial sets, wherein N denotes the homotopy coherent nerve construction for complicially enriched categories and C denotes its left adjoint, homotopy coherent realisation.