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1.1.4 Discrete Simplicial Sets

Simplicial sets of dimension $\leq 0$ admit a simple classification:

Proposition 1.1.4.1. The evaluation functor

\[ \operatorname{ev}_0: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set}\quad \quad X_{\bullet } \mapsto X_0 \]

restricts to an equivalence of categories

\[ \{ \textnormal{Simplicial sets of dimension $\leq 0$} \} \simeq \operatorname{Set}. \]

We will give a proof of Proposition 1.1.4.1 at the end of this section. First, we make some general remarks which apply to simplicial objects of any category $\operatorname{\mathcal{C}}$.

Construction 1.1.4.2. Let $\operatorname{\mathcal{C}}$ be a category. For each object $C \in \operatorname{\mathcal{C}}$, we let $\underline{C}_{\bullet }$ denote the constant functor $\operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \{ C \} \hookrightarrow \operatorname{\mathcal{C}}$ taking the value $C$. We regard $\underline{C}_{\bullet }$ as a simplicial object of $\operatorname{\mathcal{C}}$, which we will refer to as the constant simplicial object with value $C$.

Remark 1.1.4.3. Let $C$ be an object of the category $\operatorname{\mathcal{C}}$. The constant simplicial object $\underline{C}_{\bullet }$ can be described concretely as follows:

  • For each $n \geq 0$, we have $\underline{C}_{n} = C$.

  • The face and degeneracy operators

    \[ d_{i}: \underline{C}_{n} \rightarrow \underline{C}_{n-1} \quad \quad s_{i}: \underline{C}_{n} \rightarrow \underline{C}_{n+1} \]

    are the identity maps from $C$ to itself.

Example 1.1.4.4. Let $S = \{ s \} $ be a set containing a single object. Then $\underline{S}_{\bullet }$ is a final object of the category of simplicial sets: that is, it is isomorphic to the standard simplex $\Delta ^0$.

The constant simplicial object $\underline{C}_{\bullet }$ of Construction 1.1.4.2 can be characterized by a universal mapping property:

Proposition 1.1.4.5. Let $\operatorname{\mathcal{C}}$ be a category and let $C$ be an object of $\operatorname{\mathcal{C}}$. For any simplicial object $X_{\bullet }$ of $\operatorname{\mathcal{C}}$, the canonical map

\[ \operatorname{Hom}_{\operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{\mathcal{C}})}( \underline{C}_{\bullet }, X_{\bullet } ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \underline{C}_0, X_0) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, X_0 ) \]

is a bijection.

Proof. Let $f: C \rightarrow X_0$ be a morphism in $\operatorname{\mathcal{C}}$; we wish to show that $f$ can be promoted uniquely to a map of simplicial objects $f_{\bullet }: \underline{C}_{\bullet } \rightarrow X_{\bullet }$. The uniqueness of $f_{\bullet }$ is clear. For existence, we define $f_{\bullet }$ to be the natural transformation whose value on an object $[n] \in \operatorname{{\bf \Delta }}^{\operatorname{op}}$ is given by the composite map

\[ \underline{C}_{n} = C \xrightarrow {f} X_0 \xrightarrow { X_{\alpha (n)} } X_{n}, \]

where $\alpha (n)$ denotes the unique morphism in $\operatorname{{\bf \Delta }}$ from $[n]$ to $[0]$. To prove the naturality of $f_{\bullet }$, we observe that for any nondecreasing map $\beta : [m] \rightarrow [n]$ we have a commutative diagram

\[ \xymatrix { \underline{C}_ n \ar@ {=}[r] \ar [d]^{ \underline{C}_{\beta } } & C \ar@ {=}[d] \ar [r]^{f} & X_0 \ar [r]^-{ X_{\alpha (n)} } \ar@ {=}[d] & X_{n} \ar [d]^{ X_{\beta } } \\ \underline{C}_ m \ar@ {=}[r] & C \ar [r]^{f} & X_0 \ar [r]^-{ X_{\alpha (m)} } & X_ m, } \]

where the commutativity of the square on the right follows from the observation that $\alpha (m)$ is equal to the composition $[m] \xrightarrow {\beta } [n] \xrightarrow { \alpha (n) } [0]$. $\square$

Remark 1.1.4.6. Let $\operatorname{\mathcal{C}}$ be a category. Proposition 1.1.4.5 can be rephrased as follows:

  • For any simplicial object $X_{\bullet }$ of $\operatorname{\mathcal{C}}$, the limit $\varprojlim _{ [n] \in \operatorname{{\bf \Delta }}^{\operatorname{op}} } X_{n}$ exists in the category $\operatorname{\mathcal{C}}$.

  • The canonical map $\varprojlim _{ [n] \in \operatorname{{\bf \Delta }}^{\operatorname{op}} } X_{n} \rightarrow X_0$ is an isomorphism.

These assertions follow formally from the observation that $[0]$ is an final object of the category $\operatorname{{\bf \Delta }}$ (and therefore an initial object of the category $\operatorname{{\bf \Delta }}^{\operatorname{op}}$).

Corollary 1.1.4.7. Let $\operatorname{\mathcal{C}}$ be a category. Then the evaluation functor

\[ \operatorname{ev}_{0}: \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}\quad \quad X_{\bullet } \mapsto X_0 \]

admits a left adjoint, given on objects by the formation of constant simplicial objects $C \mapsto \underline{C}_{\bullet }$ described in Construction 1.1.4.2.

Corollary 1.1.4.8. Let $\operatorname{\mathcal{C}}$ be a category. Then the construction $C \mapsto \underline{C}_{\bullet }$ determines a fully faithful embedding from $\operatorname{\mathcal{C}}$ to the category $\operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{\mathcal{C}})$ of simplicial objects of $\operatorname{\mathcal{C}}$.

Proof. Let $C$ and $D$ be objects of $\operatorname{\mathcal{C}}$; we wish to show that the canonical map

\[ \theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{\mathcal{C}})}( \underline{C}_{\bullet }, \underline{\operatorname{\mathcal{D}}}_{\bullet } ) \]

is a bijection. This is clear, since $\theta $ is right inverse to the evaluation map

\[ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{\mathcal{C}})}( \underline{C}_{\bullet }, \underline{\operatorname{\mathcal{D}}}_{\bullet } ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D ) \]

which is bijective by virtue of Proposition 1.1.4.5. $\square$

We now specialize to the case where $\operatorname{\mathcal{C}}= \operatorname{Set}$ is the category of sets.

Definition 1.1.4.9. Let $X_{\bullet }$ be a simplicial set. We will say that $X_{\bullet }$ is discrete if there exists a set $S$ and an isomorphism of simplicial sets $X_{\bullet } \simeq \underline{S}_{\bullet }$; here $\underline{S}_{\bullet }$ denotes the constant simplicial set of Construction 1.1.4.2.

Specializing Corollary 1.1.4.8 to the case $\operatorname{\mathcal{C}}= \operatorname{Set}$, we obtain the following:

Corollary 1.1.4.10. The construction $S \mapsto \underline{S}_{\bullet }$ determines a fully faithful embedding $\operatorname{Set}\hookrightarrow \operatorname{Set_{\Delta }}$. The essential image of this embedding is the full subcategory of $\operatorname{Set_{\Delta }}$ spanned by the discrete simplicial sets.

Notation 1.1.4.11. Let $S$ be a set. We will often abuse notation by identifying $S$ with the constant simplicial set $\underline{S}_{\bullet }$ of Construction 1.1.4.2. (by virtue of Corollary 1.1.4.10, this is mostly harmless). This abuse will occur most frequently in the special case where $S = \{ v \} $ consists of a single vertex $v$ of some other simplicial set $X_{\bullet }$: in this case, we view $\{ v\} $ as a simplicial subset of $X_{\bullet }$ which is abstractly isomorphic to $\Delta ^0$ (see Example 1.1.2.5).

Remark 1.1.4.12. The fully faithful embedding

\[ \operatorname{Set}\hookrightarrow \operatorname{Set_{\Delta }}\quad \quad S \mapsto \underline{S}_{\bullet } \]

preserves (small) limits and colimits (since limits and colimits of simplicial sets are computed levelwise; see Remark 1.1.1.13). It follows that the collection of discrete simplicial sets is closed under the formation of (small) limits and colimits in $\operatorname{Set_{\Delta }}$.

Proposition 1.1.4.13. Let $X_{\bullet }$ be a simplicial set. The following conditions are equivalent:

$(1)$

The simplicial set $X_{\bullet }$ is discrete (Definition 1.1.4.9). That is, $X_{\bullet }$ is isomorphic to a constant simplicial set $\underline{S}_{\bullet }$.

$(2)$

For every morphism $\alpha : [m] \rightarrow [n]$ in the category $\operatorname{{\bf \Delta }}$, the induced map $X_{n} \rightarrow X_{m}$ is a bijection.

$(3)$

For every positive integer $n$, the $0$th face map $d_0: X_{n} \rightarrow X_{n-1}$ is a bijection.

$(4)$

The simplicial set $X_{\bullet }$ has dimension $\leq 0$, in the sense of Definition 1.1.3.9. That is, $X_{\bullet }$ does not contain any nondegenerate $n$-simplices for $n > 0$.

Proof. The implication $(1) \Rightarrow (2)$ follows from Remark 1.1.4.3, and the implication $(2) \Rightarrow (3)$ is immediate. To prove that $(3) \Rightarrow (4)$, we observe that if the face map $d_0: X_{n} \rightarrow X_{n-1}$ is bijective, then the degeneracy operator $s_0: X_{n-1} \rightarrow X_{n}$ is also bijective (since it is a right inverse of $d_0$). In particular, $s_0$ is surjective, so every $n$-simplex of $X_{\bullet }$ is degenerate.

We complete the proof by showing that $(4) \Rightarrow (1)$. If $X_{\bullet }$ is a simplicial set of dimension $\leq 0$ and $S = X_0$ is the set of vertices of $X_{\bullet }$, then Proposition 1.1.3.11 supplies an isomorphism of simplicial sets $\coprod _{v \in S} \Delta ^{0} \simeq X_{\bullet }$, whose domain can be identified with the constant simplicial set $\underline{S}_{\bullet }$ (by virtue of Remark 1.1.4.12 and Example 1.1.4.4). $\square$

Proof of Proposition 1.1.4.1. By virtue of Proposition 1.1.4.13, it will suffice to show that the construction $X_{\bullet } \mapsto X_0$ induces an equivalence of categories

\[ \{ \text{Discrete simplicial sets} \} \rightarrow \operatorname{Set}. \]

This follows immediately from Corollary 1.1.4.10. $\square$