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. We begin by introducing a slight generalization of Definition 3.1.5.1.
Definition 3.2.1.1. Let $X$ and $Y$ be simplicial sets, and let $K \subseteq X$ be a simplicial subset. We say that morphisms $f_0, f_1: X \rightarrow Y$ are homotopic relative to $K$ if the following conditions are satisfied:
The morphisms $f_0$ and $f_1$ have the same restriction to $K$: that is, there is a morphism $\overline{f}: K \rightarrow Y$ satisfying $f_0 |_{K} = \overline{f} = f_{1}|_{K}$.
The morphisms $f_0$ and $f_1$ belong to the same connected component of the simplicial set $\{ \overline{f} \} \times _{ \operatorname{Fun}(K,Y) } \operatorname{Fun}(X, Y )$.
Example 3.2.1.2. Let $f_0, f_1: X \rightarrow Y$ be morphisms of simplicial sets. Then $f_0$ and $f_1$ are homotopic (in the sense of Definition 3.1.5.1) if and only if they are homotopic relative to the empty subset $\emptyset \subset X$ (in the sense of Definition 3.2.1.1).
Definition 3.2.1.3. Let $f_0, f_1: X \rightarrow Y$ be a pair of morphisms of simplicial sets and let $h: \Delta ^1 \times X \rightarrow Y$ be a homotopy from $f_0$ to $f_1$. If $K \subseteq X$ is a simplicial subset, we say that $h$ is constant along $K$ if the restriction $h|_{ \Delta ^1 \times K}$ factors through the projection map $\Delta ^1 \times K \twoheadrightarrow K$.
Proposition 3.2.1.4. Let $f,g: X \rightarrow Y$ be morphisms of simplicial sets and let $K \subseteq X$ be a simplicial subset. Then:
The morphisms $f$ and $g$ are homotopic relative to $K$ if and only if there exists a sequence of morphisms $f = f_0, f_1, \ldots , f_ n = g$ from $X_{}$ to $Y_{}$ having the property that, for each $1 \leq i \leq n$, there exists either a homotopy from $f_{i-1}$ to $f_{i}$ which is constant along $K$, or a homotopy from $f_{i}$ to $f_{i-1}$ which is constant along $K$.
Suppose that $Y_{}$ is a Kan complex. Then $f$ and $g$ are homotopic relative to $K$ if and only if there exists a homotopy from $f$ to $g$ which is constant along $K$.
Proof.
Set $\overline{f} = f|_{K}$. Without loss of generality, we may assume that $\overline{f}$ is also equal to $g|_{K}$. The first assertion follows by applying Remark 1.2.1.23 to the simplicial set $Z = \{ \overline{f} \} \times _{ \operatorname{Fun}(K,Y) } \operatorname{Fun}(X,Y)$. If $Y_{}$ is a Kan complex, then the restriction map $\operatorname{Fun}(X, Y) \rightarrow \operatorname{Fun}(K,Y)$ is a Kan fibration (Corollary 3.1.3.3), so that $Z$ is a Kan complex (Remark 3.1.1.9). The second assertion now follows from Proposition 1.2.5.10.
$\square$
We will be primarily interested in applying Definition 3.2.1.3 in the special case where $K = \{ x\} $ is a vertex of $X$.
Definition 3.2.1.5. 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.
Definition 3.2.1.7. Let $(X,x)$ and $(Y,y)$ be simplicial sets. We say that pointed maps $f_0, f_1: (X,x) \rightarrow (Y,y)$ are pointed homotopic if they are homotopic relative to the simplicial subset $\{ x\} \subseteq X$, in the sense of Definition 3.2.1.3. A pointed homotopy from $f_0$ to $f_1$ is a homotopy $h: \Delta ^1 \times X \rightarrow Y$ which is constant along $\{ x\} $ (Definition 3.2.1.3): that is, which carries $\Delta ^1 \times \{ x\} $ to the degenerate edge $\operatorname{id}_{y}$.
Example 3.2.1.8. 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
\[ | \Delta ^1 \times X_{} | \xrightarrow {\theta } | \Delta ^1 | \times | X_{} | = [0,1] \times | X_{} | \xrightarrow {h} Y \]
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 3.2.1.7). By virtue of Corollary 3.6.2.2, 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
\[ \xymatrix@R =50pt@C=50pt{ \{ \text{(Continuous) pointed homotopies from $f_0$ to $f_1$} \} \ar [d]^{\sim } \\ \{ \text{(Simplicial) pointed homotopies from $f'_0$ to $f'_1$} \} . } \]
Example 3.2.1.9. 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 3.2.1.8 to the composite map $[0,1] \times | \operatorname{Sing}_{\bullet }(X) | \rightarrow [0,1] \times X \xrightarrow {h} Y$.
Notation 3.2.1.10. 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 3.2.1.11. Notation 3.2.1.10 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 3.2.1.12 (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
\[ \circ : \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}}}( (Y,y) , (Z,z) ) \times \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}}}( (X,x), (Y,y) ) \rightarrow \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}}}( (X,x), (Z,z)) \]
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.
Note that there is a forgetful functor $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast } \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$, given on objects by the construction $(X,x) \mapsto X$. This forgetful functor is conservative:
Proposition 3.2.1.13. Let $f: (X,x) \rightarrow (Y,y)$ be a morphism of pointed Kan complexes. The following conditions are equivalent:
- $(1)$
The underlying morphism of simplicial sets $f: X \rightarrow Y$ is a homotopy equivalence (Definition 3.1.6.1): that is, there exists a morphism of simplicial sets $g: Y \rightarrow X$ such that $g \circ f$ and $f \circ g$ are homotopic to the identity maps $\operatorname{id}_{X}$ and $\operatorname{id}_{Y}$, respectively.
- $(2)$
The map $f$ is a pointed homotopy equivalence: that is, there exists a morphism of pointed simplicial sets $g: (Y,y) \rightarrow (X,x)$ such that $g \circ f$ and $f \circ g$ are pointed homotopic to the identity maps $\operatorname{id}_{X}$ and $\operatorname{id}_{Y}$, respectively.
We will deduce Proposition 3.2.1.13 from the following slightly more precise result:
Lemma 3.2.1.14. Let $f: (X,x) \rightarrow (Y,y)$ be a morphism of pointed Kan complexes, and suppose that the homotopy class $[f]$ admits a left inverse in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. Then $[f]$ also admits a left homotopy inverse in the pointed homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$.
Proof.
Let $g: Y \rightarrow X$ be a left homotopy inverse of $f$. Then there exists a homotopy $\alpha : \Delta ^1 \times X \rightarrow X$ from the identity morphism $\operatorname{id}_{X} = \alpha |_{ \{ 0\} \times X }$ to $g \circ f = \alpha |_{ \{ 1\} \times X}$. Then the restriction $\alpha |_{ \Delta ^1 \times \{ x\} }$ determines an edge $e: x \rightarrow g(y)$ of $X$. Since $X$ is a Kan complex, we can use Remark 3.1.5.3 to construct another map $g': Y \rightarrow X$ and a homotopy $\beta : \Delta ^1 \times Y \rightarrow X$ from $g' = \beta |_{ \{ 0\} \times Y }$ to $g = \beta |_{ \{ 1\} \times Y}$, such that $\beta |_{ \{ y\} \times \Delta ^1 }$ is the edge $e$. Precomposing $\beta $ with $\operatorname{id}_{ \Delta ^1 } \times f$, we obtain a homotopy $\beta _{f}$ from $g' \circ f$ to $g \circ f$. Let $\sigma = s^{1}_{0}(e)$ denote the degenerate $2$-simplex of $X$ depicted in the diagram
\[ \xymatrix@R =50pt@C=50pt{ & x \ar [dr]^{ e } & \\ x \ar [ur]^{\operatorname{id}_ x} \ar [rr]^{e} & & g(y). } \]
Corollary 3.1.3.3 guarantees that the evaluation map $\operatorname{ev}_{x}: \operatorname{Fun}( X, X ) \rightarrow \operatorname{Fun}( \{ x\} , X ) \simeq X$ is a Kan fibration, so we can lift $\sigma $ to a $2$-simplex of $\operatorname{Fun}(X,X)$ depicted in the diagram
\[ \xymatrix@R =50pt@C=50pt{ & g' \circ f \ar [dr]^{ \beta _{f} } & \\ \operatorname{id}_{X} \ar [ur]^{\gamma } \ar [rr]^{\alpha } & & g \circ f. } \]
By construction, $\gamma $ is a pointed homotopy from $\operatorname{id}_{X}$ to the composition $g' \circ f$, so that the homotopy class $[g']$ is a left inverse to $[f]$ in the pointed homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$.
$\square$
Proof of Proposition 3.2.1.13.
Let $f: (X,x) \rightarrow (Y,y)$ be a morphism of pointed Kan complexes which is a homotopy equivalence; we wish to show that $f$ is a pointed homotopy equivalence (the reverse implication follows immediately from the definitions). Using Lemma 3.2.1.14, we deduce that there is a morphism of pointed Kan complexes $g: (Y,y) \rightarrow (X,x)$ such that the homotopy class $[g]$ is a left inverse of $[f]$ in the pointed homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$. Since $f$ is a homotopy equivalence, it follows that $g$ is also a homotopy equivalence. Applying Lemma 3.2.1.14 again, we conclude that $[g]$ admits a left inverse in the pointed homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$. In particular, $[g]$ is an isomorphism in $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$, so its right inverse $[f]$ is also an isomorphism.
$\square$
Proposition 3.2.1.15. Let $f: (X,x) \rightarrow (Y,y)$ be a morphism of pointed simplicial sets, where $Y$ is a Kan complex. The following conditions are equivalent:
- $(1)$
The morphism $f$ is nullhomotopic as an unpointed map. That is, there exists a vertex $z \in Y$ and a homotopy from $f$ to the constant map $\underline{z}: X \rightarrow Y$ taking the value $z$ (see Definition 3.2.4.5).
- $(2)$
The morphism $f$ is nullhomotopic as a pointed map: that is, there exists a vertex $y \in Y$ and a pointed homotopy from $f$ to the constant map $\underline{y}: X \rightarrow Y$.
Proof.
The implication $(2) \Rightarrow (1)$ is immediate from the definition. To prove the converse, suppose that there exists a a homotopy $h: \Delta ^1 \times X \rightarrow Y$ satisfying $h|_{ \{ 0\} \times X} = f$ and $h |_{ \{ 1\} \times X} = \underline{z}$ for some vertex $z \in Y$. Let $e: y \rightarrow z$ be the edge of $Y$ given by the restriction $h|_{ \Delta ^1 \times \{ x\} }$ and let $\sigma = s^{1}_0(e)$ denote the degenerate $2$-simplex of $Y$ depicted in the diagram
\[ \xymatrix@R =50pt@C=50pt{ & y \ar [dr]^{e} & \\ y \ar [ur]^{\operatorname{id}_ y} \ar [rr]^{e} & & z. } \]
Let $\underline{e}: \underline{y} \rightarrow \underline{z}$ denote the image of $e$ in $\operatorname{Fun}(X,Y)$. Since $Y$ is a Kan complex, the restriction map $q: \operatorname{Fun}(X,Y) \rightarrow \operatorname{Fun}(\{ x\} ,Y) \simeq Y$ is a Kan fibration (Corollary 3.1.3.3). It follows that the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{2}_{2} \ar [r]^-{ (\bullet , h, \underline{e}) } \ar [d] & \operatorname{Fun}(X,Y) \ar [d]^{q} \\ \Delta ^2 \ar [r]^-{\sigma } \ar@ {-->}[ur] & Y, } \]
admits a solution which carries the edge $\operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \subseteq \Delta ^2$ to a pointed homotopy from $f$ to $\underline{y}$.
$\square$