Kerodon

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1.2 The Nerve of a Category

In §1.1, we reviewed the theory of simplicial sets and its relationship to the theory of topological spaces. Every topological space $X$ determines a simplicial set $\operatorname{Sing}_{\bullet }(X)$ (Construction 1.1.5.1), and simplicial sets of the form $\operatorname{Sing}_{\bullet }(X)$ have a special property: they are Kan complexes (Proposition 1.1.7.3). 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 1.1.7.1), 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).

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