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1. • CommentRowNumber1.
   • CommentAuthorFosco
   • CommentTimeNov 4th 2017
   • PermaLink
   Author: Fosco
   Format: MarkdownItexIt seems to me that the category of marked simplicial sets exhibits cohesion over sSet, since the sharp marking functor has an additional left adjoint defined by the property that $$ msSet((X,S),(Y,Y^\sharp))\cong sSet(X[S^{-1}], Y) $$ I chose the notation to suggest that the action of this functor is a "localization" procedure. Is this cohesion actually true? Is this a different look at localization of a quasicategory with respect to a class of edges?

   It seems to me that the category of marked simplicial sets exhibits cohesion over sSet, since the sharp marking functor has an additional left adjoint defined by the property that

   $$msSet((X,S),(Y,Y^\sharp))\cong sSet(X[S^{-1}], Y)$$

   I chose the notation to suggest that the action of this functor is a “localization” procedure.

   Is this cohesion actually true? Is this a different look at localization of a quasicategory with respect to a class of edges?