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3.1.4 The $\infty $-Category of Kan Complexes

The category of simplicial sets is equipped with a good notion of homotopy.

Definition 3.1.4.1. 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 1.1.6.8).

Let us now make Definition 3.1.4.1 more concrete.

Definition 3.1.4.2. 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 $f_1 = h|_{ \{ 1\} \times X_{} }$.

Proposition 3.1.4.3. 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 1.1.6.23 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 1.1.9.10. $\square$

Example 3.1.4.4. 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

\[ | \Delta ^1 \times X_{} | \xrightarrow {\theta } | \Delta ^1 | \times | X_{} | = [0,1] \times | X_{} | \xrightarrow {h} Y \]

classifies a morphism of simplicial sets $h': \Delta ^1 \times X_{} \rightarrow \operatorname{Sing}_{\bullet }(Y)$, which is a homotopy from $f'_0$ to $f'_1$ (in the sense of Definition 3.1.4.2). We will show later that $\theta $ is a homeomorphism of topological spaces (Theorem ), so every homotopy from $f_0$ to $f_1$ arises in this way. In other words, the construction $h \mapsto h'$ induces a bijection

\[ \{ \text{(Continuous) homotopies from $f_0$ to $f_1$} \} \simeq \{ \text{(Simplicial) homotopies from $f'_0$ to $f'_1$} \} . \]

Example 3.1.4.5. 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 3.1.4.4 to the composite map $[0,1] \times | \operatorname{Sing}_{\bullet }(X) | \rightarrow [0,1] \times X \xrightarrow {h} Y$.

Notation 3.1.4.6. 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 3.1.4.7 (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

    \[ \circ : \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}}}( Y_{}, Z_{} ) \times \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}}}( X_{}, Y_{} ) \rightarrow \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}}}( X_{}, Z_{} ) \]

    is characterized by the formula $[g] \circ [f] = [g \circ f]$.

We will refer to $\mathrm{h} \mathit{\operatorname{Kan}}$ as the homotopy category of Kan complexes.

For many purposes, it is useful to consider a more refined version of Construction 3.1.4.7.

Construction 3.1.4.8 (The $\infty $-Category of Kan Complexes). Let $\operatorname{Set_{\Delta }}$ denote the category of simplicial sets, which we view as a simplicial category (where the simplicial set of morphisms from $X_{}$ to $Y_{}$ is given by $\operatorname{Fun}( X_{}, Y_{} )$). Let $\operatorname{Set}_{\Delta }^{\circ }$ denote the full subcategory of $\operatorname{Set}_{\Delta }$ spanned by the Kan complexes, so that $\operatorname{Set}_{\Delta }^{\circ }$ inherits the structure of a simplicial category. We let $\operatorname{Kan}$ denote the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set}_{\Delta }^{\circ } )$. Corollary 3.1.3.4 implies that the simplicial category $\operatorname{Set}_{\Delta }^{\circ }$ is locally Kan, so the simplicial set $\operatorname{Kan}= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set}_{\Delta }^{\circ } )$ is an $\infty $-category (Theorem 2.4.5.1). We will refer to $\operatorname{Kan}$ as the $\infty $-category of Kan complexes.

Remark 3.1.4.9. The definition of the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ can be viewed as a special case of Construction 2.4.6.1, applied to the simplicial category $\operatorname{Set}_{\Delta }^{\circ }$ of Construction 3.1.4.8. Invoking Proposition 2.4.6.8, we see that the category $\mathrm{h} \mathit{\operatorname{Kan}}$ of Construction 3.1.4.7 can be identified with the homotopy category of the $\infty $-category $\operatorname{Kan}$ (as suggested by the notation).