Remark 5.5.3.14 (0206): Comparison with Pointed Topological Spaces

Remark 5.5.3.14 (Comparison with Pointed Topological Spaces). Let $\operatorname{Top}_{\ast }$ denote the category whose objects are pointed topological spaces $(X,x)$ and whose morphisms $f: (X,x) \rightarrow (Y,y)$ are continuous functions $f: X \rightarrow Y$ satisfying $f(x) = y$. We regard $\operatorname{Top}_{\ast }$ as a simplicial category, where the $n$-simplices of $\operatorname{Hom}_{\operatorname{Top}_{\ast }}( (X,x), (Y,y) )_{\bullet }$ are continuous maps $f: | \Delta ^{n} | \times X \rightarrow Y$ satisfying $f(t,x) = y$ for every point $t \in | \Delta ^ n |$.

The construction $(X,x) \mapsto ( |X|, x)$ determines a simplicial functor from the category $\operatorname{Kan}_{\ast }$ of pointed Kan complexes to the category $\operatorname{Top}_{\ast }$ of pointed topological spaces. Moreover, if $(X,x)$ and $(Y,y)$ are pointed Kan complexes, then we have a commutative diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{Kan}}(X,Y)_{\bullet } \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{Top}}( |X|, |Y| )_{\bullet } \ar [d] \\ Y \ar [r] & \operatorname{Sing}_{\bullet }( |Y| ), } \]

where the vertical maps are Kan fibrations given by evaluation at $x$ and the horizontal maps are homotopy equivalences (Proposition 3.6.5.2). Passing to the fiber over the vertex $y \in Y$, we deduce that the induced map

\[ \operatorname{Hom}_{\operatorname{Kan}_{\ast }}( (X,x), (Y,y) )_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{Top}_{\ast }}( ( |X|, x), (|Y|, y) )_{\bullet } \]

is also a homotopy equivalence of Kan complexes. Allowing $(X,x)$ and $(Y,y)$ to vary, we deduce that geometric realization $| \bullet |: \operatorname{Kan}_{\ast } \rightarrow \operatorname{Top}_{\ast }$ is a weakly fully faithful functor of simplicial categories (Definition 4.6.8.7), and therefore induces a fully faithful functor of $\infty $-categories $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}_{\ast } ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Top}_{\ast } )$ (Corollary 4.6.8.8). Composing this functor with a homotopy inverse to the equivalence $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}_{\ast } ) \rightarrow \operatorname{\mathcal{S}}_{\ast }$ of Proposition 5.5.3.8, we obtain a fully faithful functor $\operatorname{\mathcal{S}}_{\ast } \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Top}_{\ast } )$.