1.3 From Categories to Simplicial Sets
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 1.2.2.2), and simplicial sets of the form $\operatorname{Sing}_{\bullet }(X)$ have a special property: they are Kan complexes (Proposition 1.2.5.8). In this section, we will study a different class of simplicial sets, which arise instead from the theory of categories. In §1.3.1, 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 §1.3.3 that the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is fully faithful (Proposition 1.3.3.1). In §1.3.4, we show that a simplicial set $S$ 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 1.3.4.1). This lifting condition is similar to the Kan extension condition (Definition 1.2.5.1), but not identical to it: in §1.3.5, 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 1.3.5.2).
In §1.3.6, 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 $S$ a category $\mathrm{h} \mathit{S}$ which we call the homotopy category of $S$ (Definition 1.3.6.1). This category admits a particularly simple description in the case where the simplicial set $S$ has dimension $\leq 1$: in §1.3.7, we show that it can be identified with the path category of the directed graph $G$ corresponding to $S$ (under the equivalence of Proposition 1.1.6.9).
Structure
- Subsection 1.3.1: The Nerve of a Category
- Subsection 1.3.2: Example: Monoids as Simplicial Sets
- Subsection 1.3.3: Recovering a Category from its Nerve
- Subsection 1.3.4: Characterization of Nerves
- Subsection 1.3.5: The Nerve of a Groupoid
- Subsection 1.3.6: The Homotopy Category of a Simplicial Set
- Subsection 1.3.7: Example: The Path Category of a Directed Graph