This is a subentry of [[a reading guide to HTT]]. # Contents * Automatic table of contents {: toc} ## 1.1.1 Goals and obstacles ## 1.1.2 $\infty$-categories ## 1.1.3 Equivalences of topological categories ## 1.1.4 Simplicial categories ## 1.1.5 Comparing $\infty$-categories with simplicial categories The adjunction $$(||\dashv N):sSet Cat\stackrel{N}{\to} sSet$$ shall be described. The functor $||$ is constructed by the general technic of [[nLab:nerve and realization]] via the cosimplicial object $\mathfrak{C}:\Delta\to sSet Cat$. To be precise we define $||:=Lan_y \mathfrak{C}: sSet\to sSet Cat$ as the Kan extension of the simplicial-thickening functor $\mathfrak{C}$ along the Yoneda embedding $y:\Delta\to sSet$. ### The simplicial-thickening functor [ncafe](http://golem.ph.utexas.edu/category/2010/04/understanding_the_homotopy_coh.html) +-- {: .un_defn} ###### Definition 1.1.5.1 (the simplicial category assigned to a linearly ordered set) =-- +-- {: .un_remark} ###### Remark 1.1.5.2 We can consider a linearly ordered set $[n]$ 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 $\mathfrak{C}[n]$ such that it is a cofibrant replacement of $[n]$ with respect to a [[model structure on sSet Cat in HTT|suitable model category]]. =-- +-- {: .un_defn} ###### Definition 1.1.5.4 =--