Spahn HTT, 1.1, fondations for higher category theory

This is a subentry of a reading guide to HTT.

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 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

Definition 1.1.5.1

(the simplicial category assigned to a linearly ordered set)

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 suitable model category.

Definition 1.1.5.4