1.4.5 The Homotopy Category of an $\infty $-Category
To any topological space $X$, one can associate a category $\pi _{\leq 1}(X)$, called the fundamental groupoid of $X$. This category can be described informally as follows:
The objects of $\pi _{\leq 1}(X)$ are the points of $X$.
Given a pair of points $x,y \in X$, we can identify $\operatorname{Hom}_{\pi _{\leq 1}(X) }(x,y)$ with the set of homotopy classes of continuous paths $p: [0,1] \rightarrow X$ satisfying $p(0) = x$ and $p(1) = y$.
Composition in $\pi _{\leq 1}(X)$ is given by concatenation of paths (see Example 1.4.4.5).
All of the concepts needed to define the fundamental groupoid $\pi _{\leq 1}(X)$ (such as points, paths, homotopies, and concatenation) can be formulated in terms of singular $n$-simplices of $X$ (for $n \leq 2$). Consequently, one can view the fundamental groupoid $\pi _{\leq 1}(X)$ as an invariant of the simplicial set $\operatorname{Sing}_{\bullet }(X)$, rather than the topological space $X$. In this section, we describe an extension of this invariant, where the simplicial set $\operatorname{Sing}_{\bullet }(X)$ is replaced by an arbitrary $\infty $-category $\operatorname{\mathcal{C}}$. In this case, the fundamental groupoid $\pi _{\leq 1}(X)$ is replaced by a category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ which we call the homotopy category of $\operatorname{\mathcal{C}}$ (beware that the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is generally not a groupoid: in fact, we will later see that it is a groupoid if and only if $\operatorname{\mathcal{C}}$ is a Kan complex (Proposition 4.4.2.1).
Construction 1.4.5.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, we let $\operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y)$ denote the set of homotopy classes of morphisms from $X$ to $Y$ in $\operatorname{\mathcal{C}}$. For every morphism $f: X \rightarrow Y$, we let $[f]$ denote its equivalence class in $\operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }(X,Y)$.
It follows from Propositions 1.4.4.2 and 1.4.4.7 that, for every triple of objects $X,Y, Z \in \operatorname{\mathcal{C}}$, there is a unique composition law
\[ \circ : \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y, Z) \times \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }(X, Y) \rightarrow \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X, Z) \]
satisfying the identity $[g] \circ [f] = [h]$ whenever $h: X \rightarrow Z$ is a composition of $f$ and $g$ in the $\infty $-category $\operatorname{\mathcal{C}}$.
Proposition 1.4.5.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then:
- $(1)$
The composition law of Construction 1.4.5.1 is associative. That is, for every triple of composable morphisms $f: W \rightarrow X$, $g: X \rightarrow Y$, and $h: Y \rightarrow Z$ in $\operatorname{\mathcal{C}}$, we have an equality $( [h] \circ [g]) \circ [f] = [h] \circ ([g] \circ [f])$ in $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( W, Z)$.
- $(2)$
For every object $X \in \operatorname{\mathcal{C}}$, the homotopy class $[\operatorname{id}_ X] \in \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,X)$ is a two-sided identity with respect to the composition law of Construction 1.4.5.1. That is, for every morphism $f: W \rightarrow X$ in $\operatorname{\mathcal{C}}$ and every morphism $g: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$, we have $[\operatorname{id}_ X] \circ [f]= [f]$ and $[g] \circ [ \operatorname{id}_ X]= [g]$.
Proof.
We first prove $(1)$. Let $u: W \rightarrow Y$ be a composition of $f$ and $g$, let $v: X \rightarrow Z$ be a composition of $g$ and $h$, and let $w: W \rightarrow Z$ be a composition of $f$ and $v$. Then $( [h] \circ [g] ) \circ [f] = [w]$ and $[h] \circ ([g] \circ [f]) = [h] \circ [u]$. It will therefore suffice to show that $w$ is a composition of $u$ and $h$. Choose a $2$-simplex $\sigma _0$ of $\operatorname{\mathcal{C}}$ which witnesses $v$ as a composition of $g$ and $h$, a $2$-simplex $\sigma _2$ of $\operatorname{\mathcal{C}}$ which witnesses $w$ as a composition of $f$ and $v$, and a $2$-simplex $\sigma _3$ of $\operatorname{\mathcal{C}}$ which witnesses $u$ as a composition of $f$ and $g$. Then the sequence $( \sigma _0, \bullet , \sigma _2, \sigma _3)$ determines a map of simplicial sets $\tau _0: \Lambda ^3_1 \rightarrow \operatorname{\mathcal{C}}$ (Proposition 1.2.4.7), which we depict informally as a diagram
\[ \xymatrix@C =70pt@R=70pt{ & X \ar [r]^-{g} \ar [drr]^{v} & Y \ar@ {-->}[dr]^{h} & \\ W \ar [ur]^{f} \ar@ {-->}[urr]^{u} \ar@ {-->}[rrr]^{w} & & & Z. } \]
Using our assumption that $\operatorname{\mathcal{C}}$ is an $\infty $-category, we can extend $\tau _0$ to a $3$-simplex $\tau $ of $\operatorname{\mathcal{C}}$. Then the $2$-simplex $d^{3}_1(\tau )$ witnesses $w$ as a composition of $u$ and $h$.
We now prove $(2)$. Fix an object $X \in \operatorname{\mathcal{C}}$ and a morphism $g: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$; we will show that $[g] \circ [\operatorname{id}_ X] = [g]$ (the analogous identity $[\operatorname{id}_ X] \circ [f] = [f]$ follows by a similar argument). For this, it suffices to observe that the degenerate $2$-simplex $s^{1}_0(g)$ witnesses $g$ as a composition of $\operatorname{id}_ X$ and $g$.
$\square$
Definition 1.4.5.3 (The Homotopy Category). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We define a category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as follows:
The objects of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ are the objects of $\operatorname{\mathcal{C}}$.
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, we let $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y)$ denote the collection of homotopy classes of morphisms from $X$ to $Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$ (as in Construction 1.4.5.1).
For every object $X \in \operatorname{\mathcal{C}}$, the identity morphism from $X$ to itself in $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is given by the homotopy class $[\operatorname{id}_ X]$.
Composition of morphisms is defined as in Construction 1.4.5.1.
We will refer to $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as the homotopy category of the $\infty $-category $\operatorname{\mathcal{C}}$.
Example 1.4.5.4. Let $\operatorname{\mathcal{C}}$ be an ordinary category. Then the homotopy category of the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ can be identified with $\operatorname{\mathcal{C}}$. In particular, for each $n \geq 0$, the homotopy category $\mathrm{h} \mathit{\Delta ^ n}$ can be identified with $[n] = \{ 0 < 1 < \cdots < n \} $.
Example 1.4.5.5. Let $X$ be a topological space, and regard the singular simplicial set $\operatorname{Sing}_{\bullet }(X)$ as an $\infty $-category. Then the homotopy category $\mathrm{h} \mathit{\operatorname{Sing}}_{\bullet }(X)$ can be identified with the fundamental groupoid $\pi _{\leq 1}(X)$. More precisely, we can regard the contents of §1.4, when specialized to $\infty $-categories of the form $\operatorname{Sing}_{\bullet }(X)$, as providing a construction of the fundamental groupoid of $X$. By virtue of Exercise 1.4.3.4 and Example 1.4.4.5, the resulting category $\mathrm{h} \mathit{\operatorname{Sing}}_{\bullet }(X)$ matches the informal description of $\pi _{\leq 1}(X)$ given in the introduction to §1.4.5.
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Beware that we have now introduced two different definitions of the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$:
The homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ of Definition 1.4.5.3, defined by an explicit construction using the assumption that $\operatorname{\mathcal{C}}$ is an $\infty $-category.
The homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ of Notation 1.3.6.3, defined for any simplicial set $\operatorname{\mathcal{C}}$ by a universal mapping property.
We conclude this section by showing that these definitions are equivalent (Proposition 1.4.5.7).
Construction 1.4.5.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\sigma : \Delta ^{n} \rightarrow \operatorname{\mathcal{C}}$ be an $n$-simplex of $\operatorname{\mathcal{C}}$. For $0 \leq i \leq n$, let $C_{i}$ denote the object of $\operatorname{\mathcal{C}}$ given by the image of the $i$th vertex of $\Delta ^ n$. For $0 \leq i \leq j \leq n$, let $f_{ij}: C_ i \rightarrow C_ j$ denote the image under $\sigma $ of the edge of $\Delta ^ n$ joining the $i$th vertex to the $j$th vertex, and let $[f_{ij}] \in \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }( C_ i, C_ j )$ denote the homotopy class of $f_{ij}$. Then we can regard $( \{ C_ i \} _{0 \leq i \leq n}, \{ [f_{ij} ] \} _{0 \leq i \leq j \leq n} )$ as a functor from the linearly ordered set $[n]$ to the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. Let $u(\sigma )$ denote the corresponding $n$-simplex of $\operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$. Then the construction $\sigma \mapsto u(\sigma )$ determines a map of simplicial sets
\[ u: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ). \]
The comparison map of Construction 1.4.5.6 has the following universal property:
Proposition 1.4.5.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $u: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ be as in Construction 1.4.5.6. Then $u$ exhibits $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as a homotopy category of the simplicial set $\operatorname{\mathcal{C}}$, in the sense of Definition 1.3.6.1. In other words, for every category $\operatorname{\mathcal{D}}$, the composite map
\[ \operatorname{Hom}_{ \operatorname{Cat}}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ), \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) ) \xrightarrow { \circ u} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\mathcal{C}}, \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) ) \]
is a bijection.
Proof.
Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ be a morphism of simplicial sets. Then $F$ induces a functor of homotopy categories $G: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{ \operatorname{N}}_{\bullet }(\operatorname{\mathcal{D}}) \simeq \operatorname{\mathcal{D}}$ (where the second identification comes from Example 1.4.5.4). By construction, the morphism of simplicial sets
\[ \operatorname{\mathcal{C}}\xrightarrow {u} \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ) \xrightarrow { \operatorname{N}_{\bullet }(G) } \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) \]
coincides with $F$ on the vertices and edges of $\operatorname{\mathcal{C}}$, and therefore coincides with $F$ (since a simplex of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is determined by its $1$-dimensional facets; see Remark 1.3.1.3). We leave it to the reader to verify that $G$ is the unique functor with this property.
$\square$