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5.5.1 The $\infty $-Category of Spaces

We begin by introducing a refinement of Construction 3.1.4.10.

Construction 5.5.1.1 (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 construction

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

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.

Proof. By virtue of Theorem 2.4.5.1, it suffices to show that the simplicial category $\operatorname{Kan}$ is locally Kan: that is, for every pair of Kan complexes $X$ and $Y$, the simplicial set $\operatorname{Fun}(X,Y)$ is also a Kan complex. This is a special case of Corollary 3.1.3.4. $\square$

Remark 5.5.1.3. Let $\operatorname{N}_{\bullet }(\operatorname{Kan})$ denote the nerve of the category of Kan complexes, where we view $\operatorname{Kan}$ as an ordinary category. There is an evident monomorphism of simplicial sets

\[ \iota : \operatorname{N}_{\bullet }(\operatorname{Kan}) \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Kan}) = \operatorname{\mathcal{S}}, \]

which is bijective on simplices of dimension $\leq 1$ (Example 2.4.3.9). In other words:

  • The objects of the $\infty $-category $\operatorname{\mathcal{S}}$ are Kan complexes.

  • If $X$ and $Y$ are Kan complexes, then morphisms $f: X \rightarrow Y$ in the $\infty $-category $\operatorname{\mathcal{S}}$ can be identified with morphisms of simplicial sets from $X$ to $Y$.

However, $\iota $ is not bijective on simplices of dimension $\geq 2$. For example, $2$-simplices of $\operatorname{\mathcal{S}}$ can be identified with diagrams of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} \ar@ {=>}[]+<0pt,-15pt>;+<0pt,-60pt>^-{\mu } & \\ X \ar [ur]^{f} \ar [rr]_{h} & & Z } \]

which commute up to a specified homotopy $\mu : (g \circ f) \rightarrow h$.

Remark 5.5.1.4. 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 5.5.1.5. Let $f: X \rightarrow Y$ be a morphism of Kan complexes. Then $f$ is a homotopy equivalence (in the sense of Definition 3.1.5.1) it and only if it is an isomorphism when viewed as a morphism of the $\infty $-category $\operatorname{\mathcal{S}}$.

Remark 5.5.1.6. Let $X$ and $Y$ be Kan complexes. Then Remark 4.6.7.6 supplies a canonical homotopy equivalence of Kan complexes $\operatorname{Fun}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}}( X,Y)$. Beware that this homotopy equivalence is generally not an isomorphism.

Remark 5.5.1.7 (Comparison with Sets). For every set $S$, let $\underline{S}$ denote the associated constant simplicial set (Construction 1.1.4.2). The construction $S \mapsto \underline{S}$ determines a fully faithful embedding from the category of sets to the category of Kan complexes. Passing to homotopy coherent nerve, we obtain a functor of $\infty $-categories $\operatorname{N}_{\bullet }( \operatorname{Set}) \rightarrow \operatorname{\mathcal{S}}$. This functor is fully faithful: in fact, it is an isomorphism from $\operatorname{N}_{\bullet }(\operatorname{Set})$ to the full subcategory of $\operatorname{\mathcal{S}}$ spanned by Kan complexes of the form $\underline{S}$. We will generally abuse notation by identifying (the nerve of) the category $\operatorname{Set}$ with its image in $\operatorname{\mathcal{S}}$: in particular, we will not distinguish between a set $S$ and the associated constant simplicial set $\underline{S}$, viewed as an object of $\operatorname{\mathcal{S}}$. We can summarize the situation informally by saying that the $\infty $-category $\operatorname{\mathcal{S}}$ is an enlargement of the ordinary category $\operatorname{Set}$.

Remark 5.5.1.8 (Comparison with Groupoids). Let $\mathbf{Cat}$ denote the (strict) $2$-category of small categories, let $\mathbf{Gpd} \subseteq \mathbf{Cat}$ denote the full subcategory spanned by the groupoids, and let $\mathbf{Gpd}_{\bullet }$ denote the associated simplicial category (Example 2.4.2.7), which we can describe concretely as follows:

  • The objects of the simplicial category $\mathbf{Gpd}_{\bullet }$ are small groupoids.

  • If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are groupoids, then the simplicial set $\operatorname{Hom}_{ \mathbf{Gpd} }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{\bullet }$ is the nerve of the functor category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.

Note that if $\operatorname{\mathcal{C}}$ is a groupoid, then the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a Kan complex (Proposition 1.2.4.2). By virtue of Proposition 1.4.3.3, the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ determines a fully faithful embedding of simplicial categories $\mathbf{Gpd}_{\bullet } \hookrightarrow \operatorname{Kan}$. Passing to homotopy coherent nerves and invoking Example 2.4.3.11, we obtain a functor of $\infty $-categories

\[ \operatorname{N}^{\operatorname{D}}_{\bullet }( \mathbf{Gpd} ) \simeq \operatorname{N}^{\operatorname{hc}}_{\bullet }( \mathbf{Gpd}_{\bullet } ) \hookrightarrow \operatorname{N}^{\operatorname{hc}}_{\bullet }( \operatorname{Kan}) = \operatorname{\mathcal{S}}, \]

where $\operatorname{N}^{\operatorname{D}}_{\bullet }( \mathbf{Gpd} )$ is the Duskin nerve of the $2$-category $\mathbf{Gpd}$ (Construction 2.3.1.1). This functor restricts to an isomorphism of $\operatorname{N}^{\operatorname{D}}_{\bullet }( \mathbf{Gpd} )$ with the full subcategory of $\operatorname{\mathcal{S}}$ spanned by those Kan complexes of the form $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, where $\operatorname{\mathcal{C}}$ is a small groupoid. We can informally summarize the situation informally by saying that the $\infty $-category $\operatorname{\mathcal{S}}$ is an enlargement of the $2$-category of groupoids $\mathbf{Gpd}$.

Remark 5.5.1.9 (Comparison with Topological Spaces). Let $\operatorname{Top}$ denote the category of topological spaces and continuous functions, endowed with the simplicial enrichment described in Example 2.4.1.5. The geometric realization construction $X \mapsto |X|$ determines a functor of simplicial categories $| \bullet |: \operatorname{Kan}\rightarrow \operatorname{Top}$ (see Construction 3.5.5.1). Moreover, if $X$ and $Y$ are Kan complexes, then Proposition 3.5.5.2 guarantees that the induced map

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

is a homotopy equivalence of Kan complexes. Applying Corollary 4.6.7.8, we deduce that the induced map

\[ \operatorname{\mathcal{S}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}) \xrightarrow { | \bullet | } \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Top}) \]

is a fully faithful functor of $\infty $-categories. The essential image of this embedding is spanned by those topological spaces which have the homotopy type of a CW complex (Proposition 3.5.5.3).