Simplicial sets of dimension $\leq 0$ admit a simple classification:
Proposition 1.1.5.1. The evaluation functor restricts to an equivalence of categories
\[ \operatorname{ev}_0: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set}\quad \quad X_{\bullet } \mapsto X_0 \]
\[ \{ \textnormal{Simplicial sets of dimension $\leq 0$} \} \simeq \operatorname{Set}. \]
We will give a proof of Proposition 1.1.5.1 at the end of this section. First, we make some general remarks which apply to simplicial objects of any category $\operatorname{\mathcal{C}}$.
Construction 1.1.5.2. Let $\operatorname{\mathcal{C}}$ be a category. For each object $C \in \operatorname{\mathcal{C}}$, we let $\underline{C}_{}$ denote the constant functor $\operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \{ C \} \hookrightarrow \operatorname{\mathcal{C}}$ taking the value $C$. We regard $\underline{C}_{}$ as a simplicial object of $\operatorname{\mathcal{C}}$, which we will refer to as the constant simplicial object with value $C$.
Remark 1.1.5.3. Let $C$ be an object of the category $\operatorname{\mathcal{C}}$. The constant simplicial object $\underline{C}_{}$ can be described concretely as follows:
For each $n \geq 0$, we have $\underline{C}_{n} = C$.
The face and degeneracy operators
are the identity maps from $C$ to itself.
\[ d^{n}_{i}: \underline{C}_{n} \rightarrow \underline{C}_{n-1} \quad \quad s^{n}_{i}: \underline{C}_{n} \rightarrow \underline{C}_{n+1} \]
Example 1.1.5.4. Let $S = \{ s \} $ be a set containing a single element. Then $\underline{S}$ 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 $\underline{C}_{}$ of Construction 1.1.5.2 can be characterized by a universal mapping property:
Proposition 1.1.5.5. Let $\operatorname{\mathcal{C}}$ be a category and let $C$ be an object of $\operatorname{\mathcal{C}}$. For any simplicial object $X_{\bullet }$ of $\operatorname{\mathcal{C}}$, evaluation at the object $[0] \in \operatorname{{\bf \Delta }}^{\operatorname{op}}$ induces a bijection
\[ \operatorname{Hom}_{\operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{\mathcal{C}})}( \underline{C}_{}, X_{\bullet } ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, X_0 ). \]
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 $f_{\bullet }: \underline{C}_{} \rightarrow X_{\bullet }$. 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
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
where the commutativity of the square on the right follows from the observation that $\alpha (m)$ is equal to the composition $[m] \xrightarrow {\beta } [n] \xrightarrow { \alpha (n) } [0]$. $\square$
Example 1.1.5.6. Let $X_{\bullet }$ be a simplicial set and let $S = X_0$ be the set of vertices of $X_{\bullet }$. It follows from Proposition 1.1.5.5 that there is a unique morphism of simplicial sets $f: \underline{S} \rightarrow X_{\bullet }$ which is the identity map on $0$-simplices. Using Proposition 1.1.4.12, we see that this map is an isomorphism from $\underline{S}$ to the $0$-skeleton $\operatorname{sk}_0(X_{\bullet })$. In particular, $f$ is a monomorphism, which is an isomorphism if and only if $X_{\bullet }$ has dimension $\leq 0$.
Remark 1.1.5.7. Let $\operatorname{\mathcal{C}}$ be a category. Proposition 1.1.5.5 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 a final object of the category $\operatorname{{\bf \Delta }}$ (and therefore an initial object of the category $\operatorname{{\bf \Delta }}^{\operatorname{op}}$).
Corollary 1.1.5.8. Let $\operatorname{\mathcal{C}}$ be a category. Then the evaluation functor admits a left adjoint, given on objects by the formation of constant simplicial objects $C \mapsto \underline{C}_{}$ described in Construction 1.1.5.2.
\[ \operatorname{ev}_{0}: \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}\quad \quad X_{\bullet } \mapsto X_0 \]
Corollary 1.1.5.9. Let $\operatorname{\mathcal{C}}$ be a category. Then the construction $C \mapsto \underline{C}_{}$ 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
is a bijection. This is clear, since $\theta $ is right inverse to the evaluation map
which is bijective by virtue of Proposition 1.1.5.5. $\square$
We now specialize to the case where $\operatorname{\mathcal{C}}= \operatorname{Set}$ is the category of sets.
Definition 1.1.5.10. 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 $X_{\bullet } \simeq \underline{S}$; here $\underline{S}$ denotes the constant simplicial set of Construction 1.1.5.2.
Specializing Corollary 1.1.5.9 to the case $\operatorname{\mathcal{C}}= \operatorname{Set}$, we obtain the following:
Corollary 1.1.5.11. The construction $S \mapsto \underline{S}$ 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 1.1.5.12. Let $S$ be a set. We will often abuse notation by identifying $S$ with the constant simplicial set $\underline{S}$ of Construction 1.1.5.2. (by virtue of Corollary 1.1.5.11, this is mostly harmless).
Remark 1.1.5.13. The fully faithful embedding preserves (small) limits and colimits (since limits and colimits of simplicial sets are computed levelwise; see Remark 1.1.0.8). It follows that the collection of discrete simplicial sets is closed under the formation of (small) limits and colimits in $\operatorname{Set_{\Delta }}$.
\[ \operatorname{Set}\hookrightarrow \operatorname{Set_{\Delta }}\quad \quad S \mapsto \underline{S} \]
Proposition 1.1.5.14. Let $X_{\bullet }$ be a simplicial set. The following conditions are equivalent:
The simplicial set $X_{\bullet }$ is discrete (Definition 1.1.5.10). That is, $X_{\bullet }$ is isomorphic to a constant simplicial set $\underline{S}$.
For every morphism $\alpha : [m] \rightarrow [n]$ in the category $\operatorname{{\bf \Delta }}$, the induced map $X_{n} \rightarrow X_{m}$ is a bijection.
For every positive integer $n$, the $0$th face operator $d^{n}_0: X_{n} \rightarrow X_{n-1}$ is a bijection.
The simplicial set $X_{\bullet }$ has dimension $\leq 0$, in the sense of Definition 1.1.3.1. That is, $X_{\bullet }$ does not contain any nondegenerate $n$-simplices for $n > 0$.
Proof. The implication $(1) \Rightarrow (2)$ follows from Remark 1.1.5.3, and the implication $(2) \Rightarrow (3)$ is immediate. To prove that $(3) \Rightarrow (4)$, we observe that if the face operator $d^{n}_0: X_{n} \rightarrow X_{n-1}$ is bijective, then the degeneracy operator $s^{n-1}_0: X_{n-1} \rightarrow X_{n}$ is also bijective (since it is a right inverse of $d^{n}_0$). In particular, $s^{n-1}_0$ is surjective, so every $n$-simplex of $X_{\bullet }$ is degenerate. The implication $(4) \Rightarrow (1)$ follows from Example 1.1.5.6. $\square$
Proof of Proposition 1.1.5.1. By virtue of Proposition 1.1.5.14, it will suffice to show that the construction $X_{\bullet } \mapsto X_0$ induces an equivalence of categories
This follows immediately from Corollary 1.1.5.11. $\square$