Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

1.2.5 The Homotopy Category of a Simplicial Set

We now show that the functor $\operatorname{\mathcal{C}}\mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ of Construction 1.2.1.1 admits a left adjoint (Corollary 1.3.6.7).

Definition 1.2.5.1. Let $\operatorname{\mathcal{C}}$ be a category. We will say that a map of simplicial sets $u: S_{\bullet } \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ exhibits $\operatorname{\mathcal{C}}$ as the homotopy category of $S_{\bullet }$ if, for every category $\operatorname{\mathcal{D}}$, the composite map

\[ \operatorname{Hom}_{\operatorname{Cat}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) ) \xrightarrow {\circ u} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S_{\bullet }, \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) ) \]

is bijective (note that the map on the left is always bijective, by virtue of Proposition 1.2.2.1).

Exercise 1.2.5.2. Let $X$ be a topological space and let $\tau _{\leq 1}(X)$ denote its fundamental groupoid. Show that there is a unique map of simplicial sets $u: \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{N}_{\bullet }( \tau _{\leq 1}(X) )$ with the following properties:

  • On $0$-simplices, $u$ carries each point $x \in X$ (regarded as a vertex of $\operatorname{Sing}_{\bullet }(X)$) to it (regarded as an object of $\tau _{\leq 1}(X)$).

  • On $1$-simplices, $u$ carries each path $p: [0,1] \rightarrow X$ (regarded as an edge of $\operatorname{Sing}_{\bullet }(X)$) to its homotopy class $[p]$ (regarded as a morphism of the cateogry $\tau _{\leq 1}(X)$).

Moreover, show that $u$ exhibits the fundamental groupoid $\tau _{\leq 1}(X)$ as a homotopy category of the singular simplicial set $\operatorname{Sing}_{\bullet }(X)$. For a generalization, see Proposition 1.3.6.2.

Notation 1.2.5.3. Let $S_{\bullet }$ be a simplicial set. It follows immediately from the definition that if there exists a category $\operatorname{\mathcal{C}}$ and a map $u: S_{\bullet } \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ which exhibits $\operatorname{\mathcal{C}}$ as a homotopy category of $S_{\bullet }$, then the category $\operatorname{\mathcal{C}}$ is unique up to isomorphism and depends functorially on $S_{\bullet }$. To emphasize this dependence, we will refer to $\operatorname{\mathcal{C}}$ as the homotopy category of $S_{\bullet }$ and denote it by $\mathrm{h} \mathit{S}_{\bullet }$.

Proposition 1.2.5.4. Let $S_{\bullet }$ be a simplicial set. Then there exists a category $\operatorname{\mathcal{C}}$ and a map of simplicial sets $u: S_{\bullet } \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ which exhibits $\operatorname{\mathcal{C}}$ as a homotopy category of $S_{\bullet }$.

Proof. Let $Q^{\bullet }$ denote the cosimplicial object of $\operatorname{Cat}$ given by the inclusion $\operatorname{{\bf \Delta }}\hookrightarrow \operatorname{Cat}$. Unwinding the definitions, we see that a homotopy category of $S_{\bullet }$ can be identified with a realization $| S_{\bullet } |^{Q}$, whose existence is a special case of Proposition 1.1.8.20. Alternatively, we can give a direct construction of the homotopy category $\mathrm{h} \mathit{S}_{\bullet }$:

  • The objects of $\mathrm{h} \mathit{S}_{\bullet }$ are the vertices of $S_{\bullet }$.

  • Every edge $e$ of $S_{\bullet }$ determines a morphism $[e]$ in $\mathrm{h} \mathit{S}_{\bullet }$, whose source is the vertex $d_1(e)$ and whose target is the vertex $d_0(e)$.

  • The collection of morphisms in $\mathrm{h} \mathit{S}_{\bullet }$ is generated under composition by morphisms of the form $[e]$, subject only to the relations

    \[ [ s_0(x) ] = \operatorname{id}_ x \text{ for $x \in S_0$ } \quad \quad [ d_1(\sigma ) ] = [ d_0(\sigma ) ] \circ [ d_2(\sigma ) ] \text{ for $\sigma \in S_2$. } \]
$\square$

Corollary 1.2.5.5. The nerve functor $\operatorname{N}_{\bullet }: \operatorname{Cat}\rightarrow \operatorname{Set_{\Delta }}$ admits a left adjoint, given on objects by the construction $S_{\bullet } \mapsto \mathrm{h} \mathit{S}_{\bullet }$.

Remark 1.2.5.6. Let $\operatorname{\mathcal{C}}$ be a category. Then the counit of the adjunction described in Corollary 1.2.5.5 induces an isomorphism of categories $\mathrm{h} \mathit{\operatorname{N}}_{\bullet }(\operatorname{\mathcal{C}}) \xrightarrow {\sim } \operatorname{\mathcal{C}}$ (this is a restatement of Proposition 1.2.2.1). In other words, every category $\operatorname{\mathcal{C}}$ can be recovered as the homotopy category of its nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.

Warning 1.2.5.7. Let $S_{\bullet }$ be a simplicial set. Our proof of Proposition 1.2.5.4 gives a construction of the homotopy category $\mathrm{h} \mathit{S}_{\bullet }$ by generators and relations. The result of this construction is not easy to describe. If $x$ and $y$ are vertices of $S_{\bullet }$, then every morphism from $x$ to $y$ in $\mathrm{h} \mathit{S}_{\bullet }$ can be represented by a composition

\[ [e_ n] \circ [e_{n-1}] \circ \cdots \circ [e_1], \]

where $\{ e_ i \} _{0 \leq i \leq n}$ is a sequence of edges satisfying

\[ d_1(e_1) = x \quad \quad d_0(e_ i) = d_1(e_{i+1}) \quad \quad d_0( e_ n ) = y. \]

In general, it can be difficult to determine whether or not two such compositions represent the same morphism of $\mathrm{h} \mathit{S}_{\bullet }$ (even for finite simplicial sets, this question is algorthmically undecidable). However, there are two situations in which the homotopy category $\mathrm{h} \mathit{S}_{\bullet }$ admits a simpler description:

  • Let $S_{\bullet }$ be a simplicial set of dimension $\leq 1$, which we can identify with a directed graph $G$ (Proposition 1.1.5.9). In this case, the homotopy category $\mathrm{h} \mathit{S}_{\bullet }$ is generated freely by the vertices and edges of the graph $G$: that is, it can be identified with the path category of $G$ (Proposition 1.2.6.5) which we study in §1.2.6.

  • Let $S_{\bullet }$ be an $\infty $-category. In this case, every morphism in the homotopy category $\operatorname{\mathcal{C}}= \mathrm{h} \mathit{S}_{\bullet }$ can be represented by a single edge of $S_{\bullet }$, rather than a composition of edges (in other words, the canonical map $u: S_{\bullet } \rightarrow \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})$ is surjective on edges), and two edges of $S_{\bullet }$ represent the same morphism in $\mathrm{h} \mathit{S}_{\bullet }$ if and only if they are homotopic (Definition 1.3.3.1). This leads to a more explicit description of the homotopy category $\operatorname{\mathcal{C}}$ (generalizing Exercise 1.2.5.2) which we will discuss in §1.3.5 (see Proposition 1.3.6.2).