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. If $(X,x)$ and $(Y,y)$ are pointed Kan complexes, then a pointed map from $(X,x)$ to $(Y,y)$ is a morphism of Kan complexes $f: X \rightarrow Y$ satisfying $f(x) = y$. We let $\operatorname{Kan}_{\ast }$ denote the category whose objects are pointed Kan complexes and whose morphisms are pointed maps.
3.2.1 Pointed Kan Complexes
In ยง3.1.5, 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 3.1.5.10). 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
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 3.1.1.9). 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 Corollary 3.5.2.2, 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
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: in general, 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 $[g] \circ [f] = [g \circ f]$.
We will refer to $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$ as the homotopy category of pointed Kan complexes.