In small dimensions:

• $\delta^0, \delta^1: [0] \to [1]$ are the two inclusions: $\delta^0(0) = 1$ and $\delta^1(0) = 0$.
• $\sigma^0: [1] \to [0]$ is the unique map.
• $\delta^0, \delta^1, \delta^2: [1] \to [2]$: for instance, $\delta^1$ maps $0 \mapsto 0, 1 \mapsto 2$, skipping $1$.
• There are $\binom{m+n}{n}$ morphisms $[m] \to [n]$ in total: these correspond to choosing which elements of $[n]$ lie in the image (with multiplicities).

The number of morphisms $[m] \to [n]$ equals $\binom{m+n}{n}$ because each order-preserving map $[m] \to [n]$ is determined by a weakly increasing sequence of $m+1$ elements from $[n]$.

The standard $n$-simplex is the representable presheaf

$\Delta[n] = \operatorname{Hom}_\Delta(-, [n]): \Delta^{\mathrm{op}} \to \mathbf{Set}.$

Its $m$-simplices are $\Delta[n]_m = \operatorname{Hom}_\Delta([m], [n])$, the set of order-preserving maps $[m] \to [n]$. By Yoneda, morphisms $\Delta[n] \to X$ in $\mathbf{sSet}$ correspond bijectively to $n$-simplices of $X$.

For $n = 0$: $\Delta[0]$ is a single point. For $n = 1$: $\Delta[1]$ has two $0$-simplices (vertices), one non-degenerate $1$-simplex (the edge), and all higher simplices are degenerate. For $n = 2$: $\Delta[2]$ is a "filled triangle" with three vertices, three edges, and one non-degenerate $2$-simplex.

The boundary $\partial\Delta[n]$ is the simplicial subset of $\Delta[n]$ consisting of all non-surjective maps $[m] \to [n]$. In other words, it contains all simplices of $\Delta[n]$ except the unique non-degenerate $n$-simplex $\mathrm{id}: [n] \to [n]$ and its degeneracies that require this top simplex.

For $n = 1$: $\partial\Delta[1]$ consists of just the two vertices, with no edge connecting them. For $n = 2$: $\partial\Delta[2]$ is the boundary of a triangle (three edges and three vertices, but no interior face).

The inclusion $\partial\Delta[n] \hookrightarrow \Delta[n]$ is a fundamental cofibration in the theory of simplicial sets.

The simplicial circle $S^1$ can be modeled as the quotient $\Delta[1]/\partial\Delta[1]$, identifying the two vertices of the interval. Explicitly, $S^1$ has one $0$-simplex $v$, one non-degenerate $1$-simplex $\sigma$ with $d_0(\sigma) = d_1(\sigma) = v$, and all higher simplices are degenerate.

The geometric realization $|S^1|$ is homeomorphic to the topological circle.

Consider $\Delta[2]$. Its non-degenerate simplices are:

• Three $0$-simplices: $e_0, e_1, e_2$ (the constant maps to $0, 1, 2$)
• Three $1$-simplices: $d^{01}, d^{02}, d^{12}$ (the three edge inclusions)
• One $2$-simplex: $\mathrm{id}_{[2]}$

All other simplices are degenerate. For instance, $s_0(e_0) \in \Delta[2]_1$ is the degenerate edge at vertex $0$.

The total number of $m$-simplices in $\Delta[n]$ is $\binom{m+n}{n}$, but only $\binom{n+1}{m+1}$ of these are non-degenerate (for $m \leq n$).

For any set $S$, the discrete (or constant) simplicial set has $X_n = S$ for all $n$, with all face and degeneracy maps being the identity. This defines a fully faithful functor $\mathbf{Set} \hookrightarrow \mathbf{sSet}$.

Geometrically, a discrete simplicial set is a collection of points with no higher-dimensional simplices.

The product $X \times Y$ of simplicial sets is defined levelwise:

$(X \times Y)_n = X_n \times Y_n$

with face and degeneracy maps applied componentwise: $d_i(x, y) = (d_i x, d_i y)$, $s_j(x, y) = (s_j x, s_j y)$.

The product $\Delta[1] \times \Delta[1]$ models a square. It has four $0$-simplices, and its geometric realization is a square (divided into triangles).

The coproduct (disjoint union) is also levelwise: $(X \sqcup Y)_n = X_n \sqcup Y_n$.

Thus $\Delta[0] \sqcup \Delta[0]$ is a simplicial set with two $0$-simplices and no non-degenerate higher simplices: geometrically, two disjoint points.

The simplicial $n$-sphere can be defined as $S^n = \Delta[n]/\partial\Delta[n]$ (the quotient collapsing the boundary to a point).

For $n = 0$: $S^0 = \Delta[0] \sqcup \Delta[0]$ (two points). For $n = 1$: $S^1$ has one vertex and one non-degenerate $1$-simplex (a loop). For $n = 2$: $S^2$ has one vertex, no non-degenerate edges, and one non-degenerate $2$-simplex.

The geometric realization $|S^n|$ is homeomorphic to the topological $n$-sphere.

Every simplicial set $X$ is canonically the colimit of its simplices:

$X \cong \operatorname{colim}_{(\Delta[n] \to X) \in \Delta/X} \Delta[n]$

where $\Delta/X$ is the category of simplices of $X$ (the comma category). This is a formal consequence of being a presheaf: every presheaf is a colimit of representables.

This decomposition is the simplicial analogue of the CW decomposition of a space into cells.

Let $X = \Delta[1]$ and $Y = \Delta[1]$. The identity map $\mathrm{id}: \Delta[1] \to \Delta[1]$ and the constant map at vertex $0$ are related by the simplicial homotopy $H: \Delta[1] \times \Delta[1] \to \Delta[1]$ defined by $(s, t) \mapsto \min(s, t)$ (in the order-preserving sense). This witnesses that $\Delta[1]$ is contractible.

More generally, $\Delta[n]$ is contractible for all $n$: there is a simplicial homotopy from $\mathrm{id}_{\Delta[n]}$ to the constant map at vertex $0$, given by $(x, t) \mapsto \min(x, t \cdot n)$ (appropriately formalized via the cone construction).

The nerve of a category (discussed in the next section) is a $2$-coskeletal simplicial set. This means that a simplicial set is the nerve of a category if and only if its simplices of dimension $\leq 2$ satisfy the appropriate compatibility conditions and all higher simplices are uniquely determined.

Concretely, an $n$-simplex of a nerve $N(\mathcal{C})$ for $n \geq 2$ is a composable chain of $n$ morphisms, which is completely determined by its faces (the sub-chains of length $n-1$).