Definition 3.1.5.1. Let $X_{}$ and $Y_{}$ be simplicial sets, and suppose we are given a pair of maps $f,g: X_{} \rightarrow Y_{}$, which we identify with vertices of the simplicial set $\operatorname{Fun}( X_{}, Y_{} )$. We will say that $f$ and $g$ are homotopic if they belong to the same connected component of the simplicial set $\operatorname{Fun}(X_{}, Y_{} )$ (Definition 1.2.1.8).
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3.1.5 The Homotopy Category of Kan Complexes
The category of simplicial sets is equipped with a good notion of homotopy.
Let us now make Definition 3.1.5.1 more concrete.
Definition 3.1.5.2. Let $X_{}$ and $Y_{}$ be simplicial sets, and suppose we are given a pair of morphisms $f_0, f_1: X_{} \rightarrow Y_{}$. A homotopy from $f_0$ to $f_1$ is a morphism $h: \Delta ^1 \times X_{} \rightarrow Y_{}$ satisfying $f_0 = h|_{ \{ 0\} \times X_{}}$ and $f_1 = h|_{ \{ 1\} \times X_{} }$.
Remark 3.1.5.3 (Homotopy Extension Lifting Property). Let $f: X \rightarrow S$ be a Kan fibration of simplicial sets. Suppose we are given a morphism of simplicial sets $u: B \rightarrow X$ and a homotopy $\overline{h}$ from $f \circ u$ to another map $\overline{v}: B \rightarrow S$. Then we can choose a map of simplicial sets $h: \Delta ^1 \times B_{} \rightarrow X_{}$ satisfying $f \circ h = \overline{h}$ and $h|_{ \{ 0\} \times B_{} } = u$: in other words, $\overline{h}$ can be lifted to a homotopy $h$ from $u$ to another map $v = h|_{ \{ 1\} \times B_{} }$. Moreover, given any simplicial subset $A_{} \subseteq B_{}$ and any map $h_0: \Delta ^1 \times A_{} \rightarrow X_{}$ satisfying $f \circ h_0 = \overline{h}|_{ \Delta ^1 \times A_{}}$ and $h_0|_{ \{ 0\} \times A_{} } = u|_{ A_{}}$, we can arrange that $h$ is an extension of $h_0$. This follows from Theorem 3.1.3.1, which guarantees that the restriction map is a Kan fibration (and therefore weakly right orthogonal to the inclusion $\{ 0\} \hookrightarrow \Delta ^1$). For a partial converse, see Corollary 4.2.6.2.
\[ \operatorname{Fun}( B_{}, X_{} ) \rightarrow \operatorname{Fun}( B_{}, S_{} ) \times _{ \operatorname{Fun}( A_{}, S_{} )} \operatorname{Fun}( A_{}, X_{} ) \]
Proposition 3.1.5.4. Let $X_{}$ and $Y_{}$ be simplicial sets, and suppose we are given a pair of morphisms $f,g: X_{} \rightarrow Y_{}$. Then:
The morphisms $f$ and $g$ are homotopic 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 $1 \leq i \leq n$, either there exists a homotopy from $f_{i-1}$ to $f_{i}$ or a homotopy from $f_{i}$ to $f_{i-1}$.
Suppose that $Y_{}$ is a Kan complex. Then $f$ and $g$ are homotopic if and only if there exists a homotopy from $f$ to $g$.
Proof. The first assertion follows by applying Remark 1.2.1.23 to the simplicial set $\operatorname{Fun}( X_{}, Y_{} )$. If $Y_{}$ is a Kan complex, then $\operatorname{Fun}( X_{}, Y_{} )$ is also a Kan complex (Corollary 3.1.3.4), so the second assertion follows from Proposition 1.2.5.10. $\square$
Example 3.1.5.5. Let $X_{}$ be a simplicial set and let $Y$ be a topological space. Suppose we are given a pair of continuous functions $f_0, f_1: | X_{} | \rightarrow Y$, corresponding to morphisms of simplicial sets $f'_0, f'_1: X_{} \rightarrow \operatorname{Sing}_{\bullet }(Y)$. Let $h: [0,1] \times | X_{} | \rightarrow Y$ be a continuous function satisfying $f_0 = h|_{ \{ 0\} \times |X_{}|}$ and $f_1 = h|_{ \{ 1\} \times | X_{} |}$ (that is, a 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 homotopy from $f'_0$ to $f'_1$ (in the sense of Definition 3.1.5.2). We will show later that $\theta $ is a homeomorphism of topological spaces (Corollary 3.6.2.2), so every homotopy from $f_0$ to $f_1$ arises in this way. In other words, the construction $h \mapsto h'$ induces a bijection
\[ | \Delta ^1 \times X_{} | \xrightarrow {\theta } | \Delta ^1 | \times | X_{} | = [0,1] \times | X_{} | \xrightarrow {h} Y \]
\[ \{ \text{(Continuous) homotopies from $f_0$ to $f_1$} \} \simeq \{ \text{(Simplicial) homotopies from $f'_0$ to $f'_1$} \} . \]
Example 3.1.5.6. Let $X$ and $Y$ be topological spaces, and let $h: [0,1] \times X \rightarrow Y$ be a continuous function, which we regard as a 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 3.1.5.5 to the composite map $[0,1] \times | \operatorname{Sing}_{\bullet }(X) | \rightarrow [0,1] \times X \xrightarrow {h} Y$.
Example 3.1.5.7. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories and suppose we are given a pair of functors $F,G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, which we identify with morphisms of simplicial sets $\operatorname{N}_{\bullet }(F), \operatorname{N}_{\bullet }(G): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$. By definition, a homotopy from $\operatorname{N}_{\bullet }(F)$ to $\operatorname{N}_{\bullet }(G)$ is a map of simplicial sets satisfying $h|_{ \{ 0\} \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } = \operatorname{N}_{\bullet }(F)$ and $h|_{ \{ 1\} \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} = \operatorname{N}_{\bullet }(G)$. By virtue of Proposition 1.3.3.1, this is equivalent to the datum of a functor $H: [1] \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ satisfying $H|_{\{ 0\} \times \operatorname{\mathcal{C}}} = F$ and $H|_{ \{ 1\} \times \operatorname{\mathcal{C}}} = G$. In other words, we have a canonical bijection In particular, if there exists a natural transformation from $F$ to $G$, then $\operatorname{N}_{\bullet }(F)$ and $\operatorname{N}_{\bullet }(G)$ are homotopic.
\[ h: \Delta ^1 \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \simeq \operatorname{N}_{\bullet }( [1] \times \operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) \]
\[ \xymatrix@R =50pt@C=50pt{ \{ \text{Natural transformations from $F$ to $G$} \} \ar [d]^{\sim } \\ \{ \text{Homotopies from $\operatorname{N}_{\bullet }(F)$ to $\operatorname{N}_{\bullet }(G)$} \} }. \]
Example 3.1.5.8. Let $X$ be a simplicial set, let $M_{\ast }$ be a chain complex of abelian groups, and let $\mathrm{K}(M_{\ast })$ denote the associated Eilenberg-MacLane space (Construction 2.5.6.3). Suppose we are given a pair of morphisms $f,g: X \rightarrow \mathrm{K}(M_{\ast })$ in the category of simplicial sets, which we can identify with morphisms $f',g': \mathrm{N}_{\ast }(X; \operatorname{\mathbf{Z}}) \rightarrow M_{\ast }$ in the category of chain complexes (Corollary 2.5.6.12); here $\mathrm{N}_{\ast }(X; \operatorname{\mathbf{Z}})$ denotes the normalized Moore complex of $X$ (Construction 2.5.5.9). The following conditions are equivalent:
To prove this, we note that $(1)$ is equivalent to the assertion that there is a homotopy from $f$ to $g$ (since $\mathrm{K}(M_{\ast })$ is a Kan complex; see Remark 2.5.6.4): that is, a map of simplicial sets $h: \Delta ^{1} \times X \rightarrow \mathrm{K}(M_{\ast })$ satisfying $h|_{ \{ 0\} \times X} = f$ and $h|_{ \{ 1\} \times X} = g$. By virtue of Corollary 2.5.6.12, this is equivalent to the existence of a chain map $h': \mathrm{N}_{\ast }( \Delta ^1 \times X; \operatorname{\mathbf{Z}}) \rightarrow M_{\ast }$ which is compatible with $f'$ and $g'$. For any such chain map $h'$, the composition
\[ \mathrm{N}_{\ast }( \Delta ^1 ) \boxtimes \mathrm{N}_{\ast }( X; \operatorname{\mathbf{Z}}) \xrightarrow { \mathrm{EZ} } \mathrm{N}_{\ast }( \Delta ^1 \times X) \xrightarrow {h'} M_{\ast } \]
determines a chain homotopy from $f'$ to $g'$ (where $\mathrm{EZ}$ denotes the Eilenberg-Zilber homomorphism of Example 2.5.7.12). More explicitly, this chain homotopy is given by the map of graded abelian groups
\[ \mathrm{N}_{\ast }(X; \operatorname{\mathbf{Z}}) \rightarrow M_{\ast +1} \quad \quad \sigma \mapsto h'( \tau \triangledown \sigma ), \]
where $\tau $ is the generator of $\mathrm{N}_{1}( \Delta ^1 ) \simeq \operatorname{\mathbf{Z}}$ and $\triangledown $ is the shuffle product of Construction 2.5.7.9. This proves that $(1)$ implies $(2)$. Conversely, if $(2)$ is satisfied, then there exists a chain map $u: \mathrm{N}_{\ast }( \Delta ^1 ) \boxtimes \mathrm{N}_{\ast }( X; \operatorname{\mathbf{Z}}) \rightarrow M_{\ast }$ compatible with $f'$ and $g'$, and we can verify $(1)$ by taking $h'$ to be the composite map
\[ \mathrm{N}_{\ast }( \Delta ^1 \times X; \operatorname{\mathbf{Z}}) \xrightarrow { \mathrm{AW} } \mathrm{N}_{\ast }( \Delta ^1 ) \boxtimes \mathrm{N}_{\ast }( X; \operatorname{\mathbf{Z}}) \xrightarrow {u} M_{\ast } \]
where $\mathrm{AW}$ is the Alexander-Whitney homomorphism of Construction 2.5.8.6.
Notation 3.1.5.9. Let $f: X_{} \rightarrow Y_{}$ be a morphism of simplicial sets. We let $[f]$ denote the homotopy class of $f$: that is, the image of $f$ in the set $\pi _0 \operatorname{Fun}( X_{}, Y_{} )$ of homotopy classes of maps from $X_{}$ to $Y_{}$.
Construction 3.1.5.10 (The Homotopy Category of Kan Complexes). We define a category $\mathrm{h} \mathit{\operatorname{Kan}}$ as follows:
The objects of $\mathrm{h} \mathit{\operatorname{Kan}}$ are Kan complexes.
If $X_{}$ and $Y_{}$ are Kan complexes, then $\operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}}}( X_{}, Y_{} ) = [ X_{}, Y_{} ] = \pi _0( \operatorname{Fun}( X_{}, Y_{} ) )$ is the set of homotopy classes of morphisms from $X_{}$ to $Y_{}$.
If $X_{}$, $Y_{}$, and $Z_{}$ are Kan complexes, then the composition law
is characterized by the formula $[g] \circ [f] = [g \circ f]$.
\[ \circ : \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}}}( Y_{}, Z_{} ) \times \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}}}( X_{}, Y_{} ) \rightarrow \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}}}( X_{}, Z_{} ) \]
We will refer to $\mathrm{h} \mathit{\operatorname{Kan}}$ as the homotopy category of Kan complexes.
Remark 3.1.5.11. Let $\operatorname{Kan}$ denote the full subcategory of $\operatorname{Set_{\Delta }}$ spanned by the Kan complexes, and let $\operatorname{\mathcal{C}}$ be any category. Then precomposition with the quotient map $\operatorname{Kan}\rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ induces an isomorphism from the functor category $\operatorname{Fun}( \mathrm{h} \mathit{\operatorname{Kan}}, \operatorname{\mathcal{C}})$ to the full subcategory of $\operatorname{Fun}( \operatorname{Kan}, \operatorname{\mathcal{C}})$ spanned by those functors $F: \operatorname{Kan}\rightarrow \operatorname{\mathcal{C}}$ which satisfy the following condition:
If $X$ and $Y$ are Kan complexes and $u_0, u_1: X \rightarrow Y$ are homotopic morphisms, then $F(u_0) = F(u_1)$ in $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(X), F(Y) )$.
Remark 3.1.5.12. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category (Definition 2.4.1.8). Then the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ of Construction 2.4.6.1 inherits the structure of an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched category, which can be described concretely as follows:
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the mapping object $\underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}} }( X,Y )$ is the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$, regarded as an object of $\mathrm{h} \mathit{\operatorname{Kan}}$.
For every pair of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, the composition law
is the homotopy class of the composition map $\circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet }$.
\[ \underline{\operatorname{Hom}}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }( Y,Z ) \times \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}} }( X,Y ) \rightarrow \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}} }( X,Z ) \]
Note that the passage from the category $\operatorname{Kan}$ to its homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ can be viewed as a special case of Construction 2.4.6.1, where we view $\operatorname{Kan}$ as a simplicial category with morphism spaces given by $\operatorname{Hom}_{\operatorname{Kan}}(X,Y)_{\bullet } = \operatorname{Fun}(X,Y)$. Applying Construction 2.4.6.16 to this simplicial category, we obtain the following variant:
Construction 3.1.5.13 (The Homotopy $2$-Category of Kan Complexes). We define a strict $2$-category $\mathrm{h}_{2} \mathit{\operatorname{Kan}}$ as follows:
The objects of $\mathrm{h}_{2} \mathit{\operatorname{Kan}}$ are Kan complexes.
If $X$ and $Y$ are Kan complexes, then $\underline{\operatorname{Hom}}_{\mathrm{h}_{2} \mathit{\operatorname{Kan}} }(X,Y)$ is the fundamental groupoid of the Kan complex $\operatorname{Fun}(X,Y)$.
If $X$, $Y$, and $Z$ are Kan complexes, then the composition law on $\mathrm{h}_{2} \mathit{\operatorname{Kan}}$ is given by
\begin{eqnarray*} \underline{\operatorname{Hom}}_{\mathrm{h}_{2} \mathit{\operatorname{Kan}}}( Y,Z) \times \underline{\operatorname{Hom}}_{\mathrm{h}_{2} \mathit{\operatorname{Kan}}}( X,Y) & = & \pi _{\leq 1}( \operatorname{Fun}(Y,Z) ) \times \pi _{\leq 1}( \operatorname{Fun}(X,Y) ) \\ & \simeq & \pi _{\leq 1}( \operatorname{Fun}(Y,Z) \times \operatorname{Fun}(X,Y) ) \\ & \xrightarrow {\circ } & \pi _{\leq 1}( \operatorname{Fun}(X,Z) ) \\ & = & \underline{\operatorname{Hom}}_{\mathrm{h}_{2} \mathit{\operatorname{Kan}}}( X, Z ). \end{eqnarray*}
We will refer to $\mathrm{h}_{2} \mathit{\operatorname{Kan}}$ as the homotopy $2$-category of Kan complexes.
Remark 3.1.5.14. We can describe the strict $2$-category $\mathrm{h}_{2} \mathit{\operatorname{Kan}}$ more informally as follows:
The objects of $\mathrm{h}_{2} \mathit{\operatorname{Kan}}$ are Kan complexes.
The morphisms of $\mathrm{h}_{2} \mathit{\operatorname{Kan}}$ are morphisms of Kan complexes $f: X \rightarrow Y$.
If $f_0, f_1: X \rightarrow Y$ are morphisms of Kan complexes, then a $2$-morphism $f_0 \Rightarrow f_1$ in $\mathrm{h}_{2} \mathit{\operatorname{Kan}}$ is an equivalence class of homotopies $h: \Delta ^1 \times X \rightarrow Y$ from $f_0 = h|_{\{ 0\} \times X}$ to $f_1 = h|_{\{ 1\} \times X}$, where we regard $h$ and $h'$ as equivalent if they are homotopic relative to $\operatorname{\partial \Delta }^1 \times X$.
Remark 3.1.5.15. Every $2$-morphism in the $2$-category $\mathrm{h}_{2} \mathit{\operatorname{Kan}}$ is invertible: that is, $\mathrm{h}_{2} \mathit{\operatorname{Kan}}$ is a $(2,1)$-category in the sense of Definition 2.2.8.5. Moreover, the homotopy category of $\mathrm{h}_{2} \mathit{\operatorname{Kan}}$ (in the sense of Construction 2.2.8.12) can be identified with the category $\mathrm{h} \mathit{\operatorname{Kan}}$ of Construction 3.1.5.10 (see Remark 2.4.6.18).