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1.2.1 Construction of the Nerve

We begin with a few definitions.

Construction 1.2.1.1. For every integer $n \geq 0$, let us view the linearly ordered set $[n] = \{ 0 < 1 < \cdots < n-1 < n \} $ as a category (where there is a unique morphism from $i$ to $j$ when $i \leq j$). For any category $\operatorname{\mathcal{C}}$, we let $\operatorname{N}_{n}( \operatorname{\mathcal{C}})$ denote the set of all functors from $[n]$ to $\operatorname{\mathcal{C}}$. Note that for any nondecreasing map $\alpha : [m] \rightarrow [n]$, precomposition with $\alpha $ determines a map of sets $\operatorname{N}_{n}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{m}(\operatorname{\mathcal{C}})$. We can therefore view the construction $[n] \mapsto \operatorname{N}_{n}(\operatorname{\mathcal{C}})$ as a simplicial set. We will denote this simplicial set by $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ and refer to it as the nerve of $\operatorname{\mathcal{C}}$.

Remark 1.2.1.2 (The Classifying Space of a Category). Let $\operatorname{\mathcal{C}}$ be a category. Then the topological space $| \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) |$ is called the classifying space of the category $\operatorname{\mathcal{C}}$.

Remark 1.2.1.3. Let $\operatorname{\mathcal{C}}$ be a category and let $n \geq 0$. Elements of $\operatorname{N}_{n}(\operatorname{\mathcal{C}})$ are given by

\[ C_0 \xrightarrow {f_1} C_1 \xrightarrow {f_2} C_2 \rightarrow \cdots \xrightarrow {f_ n} C_ n \]

in the category $\operatorname{\mathcal{C}}$. In other words, we can identify elements of $\operatorname{N}_{n}(\operatorname{\mathcal{C}})$ with $n$-tuples $(f_1, \ldots , f_ n)$ of morphisms of $\operatorname{\mathcal{C}}$ having the property that, for $0 < i < n$, the domain of $f_{i+1}$ coincides with the codomain of $f_{i}$.

Example 1.2.1.4. Let $\operatorname{\mathcal{C}}$ be a category. Then:

  • Vertices of the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ can be identified with objects of the category $\operatorname{\mathcal{C}}$.

  • Edges of the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ can be identified with morphisms in the category $\operatorname{\mathcal{C}}$.

  • Let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$, regarded as an edge of the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Then the faces of $f$ are given by the codomain $d_0 f = Y$ and the domain $d_1 f = X$, respectively.

  • Let $X$ be an object of $\operatorname{\mathcal{C}}$, which we regard as a vertex of the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Then the degenerate edge $s_0(X)$ is the identity morphism $\operatorname{id}_{X}: X \rightarrow X$.

Remark 1.2.1.5 (Face Operators on $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$). Let $\operatorname{\mathcal{C}}$ be a category and suppose we are given an $n$-simplex $\sigma $ of the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ for some $n > 0$, which we identify with a diagram

\[ C_0 \xrightarrow {f_1} C_1 \xrightarrow {f_2} C_2 \rightarrow \cdots \xrightarrow {f_ n} C_ n. \]

Then:

  • The $0$th face $d_0 \sigma \in \operatorname{N}_{n-1}(\operatorname{\mathcal{C}})$ can be identified with the diagram

    \[ C_1 \xrightarrow {f_2} C_2 \xrightarrow {f_3} C_3 \rightarrow \cdots \xrightarrow {f_ n} C_ n \]

    obtained from $\sigma $ by “deleting” the object $C_0$ (and the morphism $f_1$ with domain $C_0$).

  • The $n$th face $d_ n \sigma \in \operatorname{N}_{n-1}(\operatorname{\mathcal{C}})$ can be identified with the diagram

    \[ C_0 \xrightarrow {f_1} C_1\rightarrow \cdots \rightarrow C_{n-2} \xrightarrow { f_{n-1} } C_{n-1} \]

    obtained from $\sigma $ by “deleting” the object $C_ n$ (and the morphism $f_ n$ with codomain $f_ n$).

  • For $0 < i < n$, the $i$th face $d_ i \sigma \in \operatorname{N}_{n-1}(\operatorname{\mathcal{C}})$ can be identified with the diagram

    \[ C_0 \xrightarrow {f_1} C_1 \rightarrow \cdots \rightarrow C_{i-1} \xrightarrow { f_{i+1} \circ f_ i} C_{i+1} \rightarrow \cdots \xrightarrow {f_ n} C_ n \]

    obtained by “deleting” the object $C_{i}$ (and composing the morphisms $f_{i}$ and $f_{i+1}$).

Remark 1.2.1.6 (Degeneracy Operators on $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$). Let $\operatorname{\mathcal{C}}$ be a category and suppose we are given an $n$-simplex $\sigma $ of the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ which we identify with a diagram

\[ C_0 \xrightarrow {f_1} C_1 \xrightarrow {f_2} C_2 \rightarrow \cdots \xrightarrow {f_ n} C_ n. \]

Then, for $0 \leq i \leq n$, we can identify $s_{i} \sigma \in \operatorname{N}_{n+1}(\operatorname{\mathcal{C}})$ with the diagram

\[ C_0 \xrightarrow {f_1} \cdots \xrightarrow {f_{i-1}} C_{i-1} \xrightarrow {f_ i} C_{i} \xrightarrow { \operatorname{id}_{C_ i} } C_ i \xrightarrow { f_{i+1} } C_{i+1} \rightarrow \cdots \xrightarrow {f_ n} C_ n \]

obtained from $\sigma $ by “inserting” the identity morphism $\operatorname{id}_{ C_{i} }$.

Remark 1.2.1.7. Let $\operatorname{\mathcal{C}}$ be a category and let $\sigma $ be an $n$-simplex of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, corresponding to a diagram

\[ C_0 \xrightarrow {f_1} C_1 \xrightarrow {f_2} C_2 \rightarrow \cdots \xrightarrow {f_ n} C_ n. \]

Then $\sigma $ is degenerate (Definition 1.1.3.2) if and only if some $f_ i$ is an identity morphism of $\operatorname{\mathcal{C}}$ (in which case we must have $C_{i-1} = C_{i}$).

Remark 1.2.1.8. Let $I$ be a set equipped with a partial ordering $\leq _ I$. Then we can regard $I$ as a category whose objects are the elements of $I$, with morphisms given by

\[ \operatorname{Hom}_{I}(i,j) = \begin{cases} \ast & \text{ if } i \leq _ I j \\ \emptyset & \text{otherwise.} \end{cases} \]

We will denote the nerve of this category by $\operatorname{N}_{\bullet }(I)$, and refer to it as the nerve of the partially ordered set $I$. For each $n \geq 0$, we can identify $n$-simplices of $\operatorname{N}_{\bullet }(I)$ with monotone functions $[n] \rightarrow I$: that is, with nondecreasing sequences $(i_0 \leq _ I i_1 \leq _ I \cdots \leq _ I i_ n)$ of elements of $I$.

Example 1.2.1.9. For each $n \geq 0$, the nerve $\operatorname{N}_{\bullet }( [n] )$ can be identified with the standard $n$-simplex $\Delta ^{n}$ of Construction 1.1.2.1.

Remark 1.2.1.10. The construction $\operatorname{\mathcal{C}}\mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ determines a functor $\operatorname{N}_{\bullet }: \operatorname{Cat}\rightarrow \operatorname{Set_{\Delta }}$ from the category $\operatorname{Cat}$ of (small) categories to the category $\operatorname{Set_{\Delta }}$ of simplicial sets. This is a special case of the construction described in Variant 1.1.5.3. More precisely, we can identify $\operatorname{N}_{\bullet }$ with the functor $\operatorname{Sing}_{\bullet }^{Q}$, where $Q: \operatorname{{\bf \Delta }}\rightarrow \operatorname{Cat}$ is the functor which carries each object $[n] \in \operatorname{{\bf \Delta }}$ to itself, regarded as a category. It follows from Proposition 1.1.6.18 that this functor admits a left adjoint, which we will study in §1.3.6.