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This is a subentry of a reading guide to HTT.
The adjunction
shall be described. The functor is constructed by the general technic of nerve and realization via the cosimplicial object . To be precise we define as the Kan extension of the simplicial-thickening functor along the Yoneda embedding .
(the simplicial category assigned to a linearly ordered set)
We can consider a linearly ordered set as a category, and as a simplicially enriched category in obvious trivial ways. The idea behind the definition of the simplicial thickening is to construct the category such that it is a cofibrant replacement of with respect to a suitable model category.