Kerodon

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

1.2 The Nerve of a Category

In §1.1, we introduced the theory of simplicial sets and discussed its relationship to the theory of topological spaces. Every topological space $X$ determines a simplicial set $\operatorname{Sing}_{\bullet }(X)$ (Construction 1.1.7.1), and simplicial sets of the form $\operatorname{Sing}_{\bullet }(X)$ have a special property: they are Kan complexes (Proposition 1.1.9.8). In this section, we will study a different class of simplicial sets, which arise instead from the theory of categories. In §1.2.1, we associate to every category $\operatorname{\mathcal{C}}$ a simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, called the nerve of $\operatorname{\mathcal{C}}$. We show in §1.2.2 that the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is fully faithful (Proposition 1.2.2.1). In §1.2.3, we show that a simplicial set $S_{\bullet }$ belongs to the essential image of the functor $\operatorname{\mathcal{C}}\mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ if and only if it satisfies a certain lifting condition (Proposition 1.2.3.1). This lifting condition is similar to the Kan extension condition (Definition 1.1.9.1), but not identical to it: in §1.2.4, we show that a simplicial set of the form $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a Kan complex if and only if every morphism in $\operatorname{\mathcal{C}}$ is invertible (Proposition 1.2.4.2).

In §1.2.5, we show that the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ has a left adjoint, which associates to each simplicial set $S_{\bullet }$ a category $\mathrm{h} \mathit{S}_{\bullet }$ which we call the homotopy category of $S_{\bullet }$ (Definition 1.3.6.3). This category admits a particularly simple description in the case where the simplicial set $S_{\bullet }$ has dimension $\leq 1$: in §1.2.6, we show that it can be identified with the path category of the directed graph $G$ corresponding to $S_{\bullet }$ (under the equivalence of Proposition 1.1.5.9).

Structure

  • Subsection 1.2.1: Construction of the Nerve
  • Subsection 1.2.2: Recovering a Category from its Nerve
  • Subsection 1.2.3: Characterization of Nerves
  • Subsection 1.2.4: The Nerve of a Groupoid
  • Subsection 1.2.5: The Homotopy Category of a Simplicial Set
  • Subsection 1.2.6: Example: The Path Category of a Directed Graph