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5.5.3 The $\infty $-Category of Pointed Spaces

We now study a variant of Construction 5.5.1.1.

Construction 5.5.3.1 (The $\infty $-Category of Pointed Spaces). Let $\operatorname{\mathcal{S}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan})$ denote the $\infty $-category of spaces, and regard the Kan complex $\Delta ^0$ as an object of $\operatorname{\mathcal{S}}$. We let $\operatorname{\mathcal{S}}_{\ast }$ denote the coslice $\infty $-category $\operatorname{\mathcal{S}}_{ \Delta ^0 / }$. We will refer to $\operatorname{\mathcal{S}}_{\ast }$ as the $\infty $-category of pointed spaces.

Proposition 5.5.3.2. The simplicial set $\operatorname{\mathcal{S}}_{\ast }$ is an $\infty $-category, and the projection map $\operatorname{\mathcal{S}}_{\ast } \rightarrow \operatorname{\mathcal{S}}$ is a left fibration of $\infty $-categories.

Proof. By virtue of Proposition 5.5.1.2, the simplicial set $\operatorname{\mathcal{S}}$ is an $\infty $-category. It follows that for every object $X \in \operatorname{\mathcal{S}}$, the projection map $\operatorname{\mathcal{S}}_{X/} \rightarrow \operatorname{\mathcal{S}}$ is a left fibration (Corollary 4.3.6.8). Taking $X = \Delta ^{0}$, we conclude that the projection map $\operatorname{\mathcal{S}}_{\ast } \rightarrow \operatorname{\mathcal{S}}$ is a left fibration, so that $\operatorname{\mathcal{S}}_{\ast }$ is an $\infty $-category (Remark 4.2.1.5). $\square$

Example 5.5.3.3 (Objects of $\operatorname{\mathcal{S}}_{\ast }$). By definition, an object of the $\infty $-category $\operatorname{\mathcal{S}}_{\ast }$ is an edge $e: \Delta ^{0} \rightarrow X$ of the simplicial set $\operatorname{\mathcal{S}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Kan})$ whose source is the Kan complex $\Delta ^{0}$. By virtue of Remark 5.5.1.3, this is the same data as a morphism $e: \Delta ^{0} \rightarrow X$ in the ordinary category of Kan complexes: that is, the data of a pointed Kan complex $(X,x)$ (Definition 3.2.1.1).

Example 5.5.3.4 (Morphisms of $\operatorname{\mathcal{S}}_{\ast }$). Let $(X,x)$ and $(Y,y)$ be pointed Kan complexes, regarded as objects of the $\infty $-category $\operatorname{\mathcal{S}}_{\ast }$. By definition, a morphism from $(X,x)$ to $(Y,y)$ in the $\infty $-category $\operatorname{\mathcal{S}}_{\ast }$ can be identified with a $2$-simplex $\sigma $ of the simplicial set $\operatorname{\mathcal{S}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan})$, which we can identify with a diagram of simplicial sets

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

which commutes up to a specified homotopy $h$. In other words, a morphism from $(X,x)$ to $(Y,y)$ in the $\infty $-category $\operatorname{\mathcal{S}}_{\ast }$ can be identified with a pair $(f,h)$, where $f: X \rightarrow Y$ is a morphism of Kan complexes and $h: f(x) \rightarrow y$ is an edge of the simplicial set $Y$.

Let $\operatorname{Kan}_{\ast }$ denote the category of pointed Kan complexes (Definition 3.2.1.1). For every pair of pointed Kan complexes $(X,x)$ and $(Y,y)$, we let

\[ \operatorname{Hom}_{\operatorname{Kan}_{\ast }}( (X,x), (Y,y) )_{\bullet } = \operatorname{Fun}( X, Y) \times _{ \operatorname{Fun}( \{ x\} , Y) } \{ y\} \]

be the simplicial set parametrizing pointed morphisms from $X$ to $Y$. If $(Z,z)$ is another pointed Kan complex, we have an evident composition law

\[ \circ : \operatorname{Hom}_{\operatorname{Kan}_{\ast }}( (Y,y), (Z,z) )_{\bullet } \times \operatorname{Hom}_{\operatorname{Kan}_{\ast }}( (X,x), (Y,y) )_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{Kan}_{\ast } }( (X,x), (Z,z) ), \]

which endows $\operatorname{Kan}_{\ast }$ with the structure of a simplicial category. Note that this construction is a special case of Variant 5.5.2.3, since $\operatorname{Kan}_{\ast }$ can be identified with the coslice category $\operatorname{Kan}_{\Delta ^{0} / }$. Applying Construction 5.5.2.17, we obtain a coslice comparison functor

\[ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}_{\ast } ) = \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}_{ \Delta ^0 / } ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Kan})_{ \Delta ^{0} / } = \operatorname{\mathcal{S}}_{\ast }. \]

Proposition 5.5.3.5. The coslice comparison functor $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}_{\ast } ) \rightarrow \operatorname{\mathcal{S}}_{\ast }$ is an equivalence of $\infty $-categories.

Proof. Note that, for every pair of pointed Kan complexes $(X,x)$ and $(Y,y)$, the evaluation map $\operatorname{Fun}(X,Y) \rightarrow \operatorname{Fun}( \{ x\} , Y)$ is a Kan fibration (Corollary 3.1.3.3). Proposition 5.5.3.5 is therefore a special case of Theorem 5.5.2.21. $\square$

Warning 5.5.3.6. The coslice comparison functor $F: \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}_{\ast } ) \rightarrow \operatorname{\mathcal{S}}_{\ast }$ of Proposition 5.5.3.5 is bijective on vertices: objects of either $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}_{\ast } )$ and $\operatorname{\mathcal{S}}_{\ast }$ can be identified with pointed Kan complexes $(X,x)$. However, it is not bijective on edges (and is therefore not an isomorphism of simplicial sets). If $(X,x)$ and $(Y,y)$ are pointed Kan complexes, then a morphism from $(X,x)$ to $(Y,y)$ in the $\infty $-category $\operatorname{\mathcal{S}}_{\ast }$ can be identified with a pair $(f,h)$, where $f: X \rightarrow Y$ is a morphism of Kan complexes and $h: f(x) \rightarrow y$ is an edge of the Kan complex $Y$. The pair $(f,h)$ belongs to the image of $F$ if and only if the edge $h$ is degenerate (which guarantees in particular that $f(x) = y$, so that $f$ is a morphism of pointed Kan complexes).

Corollary 5.5.3.7. The coslice comparison functor $F: \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}_{\ast } ) \rightarrow \operatorname{\mathcal{S}}_{\ast }$ induces an isomorphism of homotopy categories $\mathrm{h} \mathit{F}: \mathrm{h} \mathit{\operatorname{Kan}}_{\ast } \xrightarrow {\sim } \mathrm{h} \mathit{\operatorname{\mathcal{S}}}_{\ast }$, where $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$ denotes the homotopy category of pointed Kan complexes (Construction 3.2.1.10).

Proof. It follows from Propositions 2.4.6.8 and 5.5.3.5 that the functor $\mathrm{h} \mathit{F}$ is an equivalence of categories. Since it is bijective on objects, it is an isomorphism of categories. $\square$

Note that the coslice comparison functor $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}_{\ast } ) \rightarrow \operatorname{\mathcal{S}}_{\ast }$ is a monomorphism of simplicial sets (Exercise 5.5.2.20). Heuristically, we can think of $\operatorname{\mathcal{S}}_{\ast }$ as an enlargement of the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}_{\ast } )$ which is obtained by allowing morphisms between pointed Kan complexes which preserve base points only up to (specified) homotopy. By virtue of Proposition 5.5.3.5, this enlargement gives rise to an equivalent $\infty $-category. However, the $\infty $-category $\operatorname{\mathcal{S}}_{\ast }$ is in some respects more convenient to work with, because the forgetful functor $\operatorname{\mathcal{S}}_{\ast } \rightarrow \operatorname{\mathcal{S}}$ is a left fibration of $\infty $-categories (this observation will play an important role in ยง, when we discuss an $\infty $-categorical version of Construction 5.1.1.1). The composite functor $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}_{\ast } ) \rightarrow \operatorname{\mathcal{S}}_{\ast } \rightarrow \operatorname{\mathcal{S}}$ does not share this property:

Warning 5.5.3.8. There is an evident simplicial functor from the category $\operatorname{Kan}_{\ast }$ of pointed Kan complexes to the category $\operatorname{Kan}$ of Kan complexes, given on objects by the construction $(X,x) \mapsto X$. Passing to homotopy coherent nerves, we obtain a functor of $\infty $-categories $U: \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}_{\ast } ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}) = \operatorname{\mathcal{S}}$. Beware that the functor $U$ is not a left fibration of simplicial sets. For example, suppose we are given a $2$-simplex $\sigma $ of $\operatorname{\mathcal{S}}$, corresponding to a diagram 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 commutes up to a homotopy $\mu : (g \circ f) \rightarrow h$ (see Remark 5.5.1.3). Pick a vertex $x \in X$ and set $y = f(x)$ and $z = h(x)$, so that we have morphisms of pointed Kan complexes Kan complexes $f: (X,x) \rightarrow (Y,y)$ and $h: (X,x) \rightarrow (Z,z)$. This data determines a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{2}_{0} \ar [r]^-{ (\bullet , h, f) } \ar [d] & \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}_{\ast } ) \ar [d]^{U} \\ \Delta ^{2} \ar [r]^-{\sigma } \ar@ {-->}[ur] & \operatorname{\mathcal{S}}, } \]

which admits a solution if and only if $\mu (x): g(y) \rightarrow z$ is a degenerate edge of the Kan complex $Z$ (in which case $g(y) = z$, so that $g: (Y,y) \rightarrow (Z,z)$ is also a morphism of pointed Kan complexes).

Example 5.5.3.9 (Pointed Sets as Pointed Spaces). Let $\operatorname{Set}_{\ast }$ denote the category of pointed sets (see Remark 5.1.1.6. Every pointed set $(X,x)$ can be regarded as a pointed Kan complex by identifying $X$ with the corresponding constant simplicial set. This construction determines a fully faithful embedding $\operatorname{Set}_{\ast } \hookrightarrow \operatorname{Kan}_{\ast }$. Composing with the equivalence of Proposition 5.5.3.5, we obtain a functor of $\infty $-categories

\[ \operatorname{N}_{\bullet }( \operatorname{Set}_{\ast } ) \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{Kan}_{\ast } ) \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}_{\ast } ) \hookrightarrow \operatorname{\mathcal{S}}_{\ast }. \]

It follows from Remark 5.5.1.7 that this functor is fully faithful: in fact, it is an isomorphism from $\operatorname{N}_{\bullet }( \operatorname{Set}_{\ast } )$ to the full subcategory of $\operatorname{\mathcal{S}}_{\ast }$ spanned by those pointed Kan complexes $(X,x)$ where the simplicial set $X$ is constant.

For every group $G$, let $B_{\bullet }G$ denote the Milnor construction on $G$ (Example 1.2.4.3), which we regard as a Kan complex (Proposition 1.1.9.9) having a unique vertex. The construction $G \mapsto B_{\bullet }G$ determines a functor from the category $\mathbf{Group}$ of groups to the category $\operatorname{Kan}_{\ast }$ of pointed Kan complexes. Passing to nerves, we obtain a functor of $\infty $-categories

\[ \operatorname{N}_{\bullet }( \mathbf{Group} ) \rightarrow \operatorname{N}_{\bullet }( \operatorname{Kan}_{\ast } ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}_{\ast } ) \rightarrow \operatorname{\mathcal{S}}_{\ast }. \]

Proposition 5.5.3.10. The functor

\[ \operatorname{N}_{\bullet }( \mathbf{Group} ) \rightarrow \operatorname{\mathcal{S}}_{\ast } \quad \quad G \mapsto B_{\bullet } G \]

is fully faithful.

Proof. By virtue of Proposition 5.5.3.5 and Corollary 4.6.7.8, it will suffice to show that the construction $G \mapsto B_{\bullet }G$ determines a weakly fully faithful functor from $\mathbf{Group}$ (regarded as a constant simplicial category) to the simplicial category $\operatorname{Kan}_{\ast }$. In other words, we must show that for every pair of groups $G$ and $H$, the canonical map

\[ \theta : \{ \text{Group homomorphisms from $G$ to $H$} \} \rightarrow \operatorname{Hom}_{\operatorname{Kan}_{\ast }}( B_{\bullet } G, B_{\bullet } H)_{\bullet } \]

is a homotopy equivalence of Kan complexes. In fact, we claim that $\theta $ is an isomorphism of simplicial sets. Let $BG$ denote the category having a single object $X$ with automorphism group $G$, and let $BH$ denote the category having a single object $Y$ with automorphism group $H$. Proposition 1.4.3.3 then supplies an isomorphism

\begin{eqnarray*} \operatorname{Hom}_{\operatorname{Kan}_{\ast }}( B_{\bullet } G, B_{\bullet } H)_{\bullet } & = & \operatorname{Fun}( \operatorname{N}_{\bullet }(BG), \operatorname{N}_{\bullet }(BH) ) \times _{ \operatorname{N}_{\bullet }(BH) } \operatorname{N}_{\bullet }( \{ Y\} ) \\ & \simeq & \operatorname{N}_{\bullet }( \operatorname{Fun}(BG,BH) ) \times _{ \operatorname{N}_{\bullet }( BH ) } \operatorname{N}_{\bullet }( \{ Y \} ) \\ & \simeq & \operatorname{N}_{\bullet }( \operatorname{Fun}( BG, BH ) \times _{BH} \{ Y \} ). \end{eqnarray*}

Note that if $F,F': BG \rightarrow BH$ are functors and $\alpha : F \rightarrow F'$ is a natural transformation with the property that $\alpha _{X}: F(X) \rightarrow F'(X)$ is the identity morphism $\operatorname{id}_{Y}$, then the functors $F$ and $F'$ are equal and $\alpha $ is the identity transformation (since $X$ is the only object of the category $BG$). It follows that the fiber product category $\operatorname{Fun}(BG, BH) \times _{BH} \{ Y\} $ is discrete: that is, it has only identity morphisms. We conclude by observing that the set of objects of the category $\operatorname{Fun}(BG, BH) \times _{BH} \{ Y\} $ can be identified with the set of group homomorphisms from $G$ to $H$. $\square$

Remark 5.5.3.11 (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 3.5.5.2). 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 4.6.7.7), 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 4.6.7.8). 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 5.5.3.5, we obtain a fully faithful functor $\operatorname{\mathcal{S}}_{\ast } \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Top}_{\ast } )$.

Exercise 5.5.3.12. Let $(X,x)$ be a pointed topological space. Show that $(X,x)$ belongs to the essential image of the functor $\operatorname{\mathcal{S}}_{\ast } \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Top}_{\ast } )$ if and only if the topological space $X$ has the homotopy type of a CW complex and the inclusion map $\{ x\} \hookrightarrow X$ is a Hurewicz fibration (that is, the union $(\{ 0\} \times X) \cup ( [0,1] \times \{ x\} )$ is a retract of the product space $[0,1] \times X$).