Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

6.7 Temporarily Removed Tags

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Remark 6.7.0.1. Let $X$ and $Y$ be Kan complexes, and let $f,g: X \rightarrow Y$ be morphisms. Then $f$ and $g$ are homotopic as morphisms of simplicial sets (that is, they belong to the same connected component of the Kan complex $\operatorname{Fun}(X,Y)$) if and only if they are homotopic as morphisms in the $\infty $-category $\operatorname{\mathcal{S}}$ (Definition 1.3.3.1). Consequently, the category $\mathrm{h} \mathit{\operatorname{Kan}}$ of Construction 3.1.4.10 can be identified with the homotopy category of the $\infty $-category $\operatorname{\mathcal{S}}$ (this is a special case of Proposition 2.4.6.8).

Remark 6.7.0.2. Let $\mathrm{h} \mathit{\operatorname{Kan}}$ denote the homotopy category of Kan complexes (Construction 3.1.4.10). Then $\mathrm{h} \mathit{\operatorname{Kan}}$ can be identified with the homotopy category of the $\infty $-category $\operatorname{\mathcal{S}}$ of Construction . If $q: X \rightarrow S$ is a left fibration of simplicial sets, then Construction 5.3.3.7 determines a covariant transport functor

\[ T: \mathrm{h} \mathit{S} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}. \]

In ยง, we show that (up to equivalence) $T$ can be promoted to a map of simplicial sets $S \rightarrow \operatorname{\mathcal{S}}$. In other words, the formation of covariant transport morphisms $(e: s \rightarrow s') \mapsto (e_{!}: X_{s} \rightarrow X_{s'})$ is compatible with composition not only up to homotopy, but up to coherent homotopy.

Construction 6.7.0.3 (The $\infty $-Category of Spaces). Let $\operatorname{Kan}$ denote the category of Kan complexes. We view $\operatorname{Kan}$ as a simplicial category, with simplicial morphism sets given by the constructoin

\[ \operatorname{Hom}_{\operatorname{Kan}}(X,Y)_{\bullet } = \operatorname{Fun}(X,Y). \]

We let $\operatorname{\mathcal{S}}$ denote the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan})$ (Definition 2.4.3.5). We will refer to $\operatorname{\mathcal{S}}$ as the $\infty $-category of spaces.