Definition 3.2.1.1. A pointed simplicial set is a pair $(X,x)$, where $X$ is a simplicial set and $x$ is a vertex of $X$. If $X$ is a Kan complex, then we refer to the pair $(X,x)$ as a pointed Kan complex.
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3.2.1 Pointed Kan Complexes
In §3.1.4, we showed that the collection of Kan complexes can be organized into a category $\mathrm{h} \mathit{\operatorname{Kan}}$ whose morphisms are given by homotopy classes of maps (Construction 3.1.4.9). In this section, we describe a variant of this construction for Kan complexes which are equipped with a specified base point.
Remark 3.2.1.2. We will often abuse terminology by identifying a pointed simplicial set $(X,x)$ with the underlying simplicial set $X$. In this case, we will refer to $x$ as the base point of $X$.
Definition 3.2.1.3. Let $(X,x)$ and $(Y,y)$ be simplicial sets, and suppose we are given a pair of pointed maps $f,g: X \rightarrow Y$, which we identify with vertices of the simplicial set $\operatorname{Fun}( X, Y ) \times _{ \operatorname{Fun}( \{ x\} , Y) } \{ y\} $. We will say that $f$ and $g$ are pointed homotopic if they belong to the same connected component of $\operatorname{Fun}( X, Y ) \times _{ \operatorname{Fun}( \{ x\} , Y) } \{ y\} $ (Definition 1.1.6.8).
Definition 3.2.1.4. Let $(X,x)$ and $(Y,y)$ be pointed simplicial sets, and suppose we are given a pair of pointed maps $f_0, f_1: X \rightarrow Y$. A pointed homotopy from $f_0$ to $f_1$ is a morphism $h: \Delta ^1 \times X_{} \rightarrow Y_{}$ for which $f_0 = h|_{ \{ 0\} \times X_{}}$, $f_1 = h|_{ \{ 1\} \times X_{} }$, and $h|_{ \Delta ^1 \times \{ x\} }$ is the degenerate edge associated to the vertex $y \in Y$.
Proposition 3.2.1.5. Let $(X,x)$ and $(Y,y)$ be pointed simplicial sets, and suppose we are given a pair of pointed morphisms $f,g: X_{} \rightarrow Y_{}$. Then:
The morphisms $f$ and $g$ are pointed homotopic if and only if there exists a sequence of pointed morphisms $f = f_0, f_1, \ldots , f_ n = g$ from $X_{}$ to $Y_{}$ having the property that, for each $1 \leq i \leq n$, either there exists a pointed homotopy from $f_{i-1}$ to $f_{i}$ or a pointed homotopy from $f_{i}$ to $f_{i-1}$.
Suppose that $Y_{}$ is a Kan complex. Then $f$ and $g$ are pointed homotopic if and only if there exists a pointed homotopy from $f$ to $g$.
Proof. The first assertion follows by applying Remark 1.1.6.23 to the simplicial set
\[ \operatorname{Fun}( X, Y ) \times _{ \operatorname{Fun}( \{ x\} , Y) } \{ y\} . \]
If $Y_{}$ is a Kan complex, then the evaluation map $\operatorname{Fun}(X,Y) \rightarrow \operatorname{Fun}( \{ x\} , Y)$ is a Kan fibration (Corollary 3.1.3.3), so the fiber $\operatorname{Fun}(X,Y) \times _{ \operatorname{Fun}( \{ x\} , Y) } \{ y\} $ is a Kan complex (Remark 3.1.1.8). The second assertion now follows from Proposition 1.1.9.10. $\square$
Example 3.2.1.6. Let $(X,x)$ be a pointed simplicial set and let $(Y,y)$ be a pointed topological space. Suppose we are given a pair of continuous functions $f_0, f_1: | X_{} | \rightarrow Y$ carrying $x$ to $y$, which we can identify with pointed morphisms $f'_0, f'_1: X_{} \rightarrow \operatorname{Sing}_{\bullet }(Y)$. Let $h: [0,1] \times | X_{} | \rightarrow Y$ be a continuous function satisfying $f_0 = h|_{ \{ 0\} \times |X_{}|}$, $f_1 = h|_{ \{ 1\} \times | X_{} |}$, and $h(t,x) = y$ for $0 \leq t \leq 1$ (that is, $h$ is a pointed homotopy from $f_0$ to $f_1$ in the category of topological spaces). Then the composite map classifies a morphism of simplicial sets $h': \Delta ^1 \times X_{} \rightarrow \operatorname{Sing}_{\bullet }(Y)$, which is a pointed homotopy from $f'_0$ to $f'_1$ (in the sense of Definition 3.2.1.4). By virtue of Theorem
\[ | \Delta ^1 \times X_{} | \xrightarrow {\theta } | \Delta ^1 | \times | X_{} | = [0,1] \times | X_{} | \xrightarrow {h} Y \]
, the map $\theta $ is a homeomorphism, so every pointed homotopy from $f_0$ to $f_1$ arises in this way. In other words, the construction $h \mapsto h'$ induces a bijection
\[ \xymatrix { \{ \text{(Continuous) pointed homotopies from $f_0$ to $f_1$} \} \ar [d]^{\sim } \\ \{ \text{(Simplicial) pointed homotopies from $f'_0$ to $f'_1$} \} . } \]
Example 3.2.1.7. Let $(X,x)$ and $(Y,y)$ be pointed topological spaces, and let $h: [0,1] \times X \rightarrow Y$ be a continuous function satisfying $h(t,x) = y$ for $0 \leq t \leq 1$, which we regard as a pointed homotopy from $f_0 = h|_{ \{ 0\} \times X}$ to $f_1 = h|_{ \{ 1\} \times X}$. Then $h$ determines a homotopy between the induced map of simplicial sets $\operatorname{Sing}_{\bullet }(f_0), \operatorname{Sing}_{\bullet }(f_1): \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{Sing}_{\bullet }(Y)$: this follows by applying Example 3.2.1.6 to the composite map $[0,1] \times | \operatorname{Sing}_{\bullet }(X) | \rightarrow [0,1] \times X \xrightarrow {h} Y$.
Notation 3.2.1.8. Let $(X,x)$ and $(Y,y)$ be pointed simplicial sets. We let $[X, Y]_{\ast }$ denote the set $\pi _{0}( \operatorname{Fun}(X,Y) \times _{ \operatorname{Fun}( \{ x\} , Y) } \{ y\} )$ of pointed homotopy classes of morphisms from $(X,x)$ to $(Y,y)$. If $f: X \rightarrow Y$ is a morphism of pointed simplicial sets, we denote its pointed homotopy class by $[f] \in [X,Y]_{\ast }$.
Warning 3.2.1.9. Notation 3.2.1.8 has the potential to create confusion. If $(X,x)$ and $(Y,y)$ are pointed simplicial sets and $f: X \rightarrow Y$ is a morphism satisfying $f(x) = y$, then we use the notation $[f]$ to represent both the homotopy class of $f$ as a map of simplicial sets (that is, the image of $f$ in the set $\pi _0( \operatorname{Fun}(X,Y) )$), and the pointed homotopy class of $f$ as a map of pointed simplicial sets (that is, the image of $f$ in the set $[X,Y]_{\ast } = \pi _{0}( \operatorname{Fun}(X,Y) \times _{ \operatorname{Fun}( \{ x\} , Y) } \{ y\} )$). Beware that these usages are not the same: it is possible for a pair of pointed morphisms $f,g: X \rightarrow Y$ to be homotopic without being pointed homotopic.
Construction 3.2.1.10 (The Homotopy Category of Pointed Kan Complexes). We define a category $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$ as follows:
The objects of $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$ are pointed Kan complexes $(X,x)$.
If $(X_{},x)$ and $(Y_{},y)$ are Kan complexes, then $\operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}}}( (X,x), (Y,y) ) = [ X_{}, Y_{} ]_{\ast }$ is the set of pointed homotopy classes of morphisms from $(X,x)$ to $(Y,y)$.
If $(X_{},x)$, $(Y_{},y)$, and $(Z_{},z)$ are Kan complexes, then the composition law
is characterized by the formula $[g] \circ [f] = [g \circ f]$.
\[ \circ : \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}}}( (Y,y) , (Z,z) ) \times \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}}}( (X,x), (Y,y) ) \rightarrow \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}}}( (X,x), (Z,z)) \]
We will refer to $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$ as the homotopy category of pointed Kan complexes.
Like the unpointed homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$, the pointed homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$ can be realized as the homotopy category of an $\infty $-category.
Construction 3.2.1.11 (The $\infty $-Category of Pointed Kan Complexes). Let $(\operatorname{Set_{\Delta }})_{\ast }$ denote the category whose objects are pointed simplicial sets $(X,x)$ and pointed morphisms between them. We regard $(\operatorname{Set_{\Delta }})_{\ast }$ as a simplicial category, where the simplicial set of morphisms from $(X,x)$ to $(Y,y)$ is given by the fiber product $\operatorname{Fun}(X,Y) \times _{ \operatorname{Fun}( \{ x\} , Y) } \{ y\} $. Let $(\operatorname{Set}_{\Delta }^{\circ })_{\ast }$ denote the full subcategory of $(\operatorname{Set_{\Delta }})_{\ast }$ spanned by the pointed Kan complexes, so that $(\operatorname{Set}_{\Delta }^{\circ })_{\ast }$ inherits the structure of a simplicial category. We let $\operatorname{Kan}_{\ast }$ denote the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( (\operatorname{Set}_{\Delta }^{\circ })_{\ast } )$. Corollary 3.1.3.3 (and Remark 3.1.1.8) guarantee that the simplicial category $(\operatorname{Set}_{\Delta }^{\circ })_{\ast }$ is locally Kan, so the simplicial set $\operatorname{Kan}_{\ast } = \operatorname{N}_{\bullet }^{\operatorname{hc}}( (\operatorname{Set}_{\Delta }^{\circ })_{\ast } )$ is an $\infty $-category (Theorem 2.4.5.1). We will refer to $\operatorname{Kan}_{\ast }$ as the $\infty $-category of pointed Kan complexes.
Remark 3.2.1.12. The definition of the pointed homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$ can be viewed as a special case of Construction 2.4.6.1, applied to the simplicial category $(\operatorname{Set}_{\Delta }^{\circ })_{\ast }$ of Construction 3.2.1.11. Invoking Proposition 2.4.6.8, we see that the category $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$ of Construction 3.2.1.10 can be identified with the homotopy category of the $\infty $-category $\operatorname{Kan}_{\ast }$ (as suggested by the notation).
Remark 3.2.1.13. In §, we will give a different description of the $\infty $-category $\operatorname{Kan}_{\ast }$ (at least up to equivalence): it can be realized as the $\infty $-category of pointed objects of the $\infty $-category $\operatorname{Kan}$ (that is, the full subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{Kan})$ spanned by those diagrams $f: X \rightarrow Y$ where Kan complex $X$ is contractible; see Example ). Beware that the analogous statement does not hold at the level of homotopy categories: there is no formal mechanism to extract the pointed homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$ from the unpointed homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ (see Warning 3.2.1.9).