We need to develop this language in order to define and work with $\infty$-categories.
Simplicial sets#
Definition 1. The category $\Delta$ has objects $[n] = {0, \dotsc, n}$ for all nonnegative integers $n$, and has morphisms $$\operatorname{Hom} _ \Delta([m], [n]) = { \text{non-decreasing maps } [m] \to [n] }.$$
Note that every linearly ordered set $P$ can be viewed as a category by letting $$\operatorname{Hom}(x, y) = \begin{cases} \ast & x \le y \br \emptyset & \text{otherwise}. \end{cases}$$
Definition 2. A simplicial set is a functor $X_\bullet \colon \Delta^\mathrm{op} \to \mathsf{Sets}$. We write $X_n = X_\bullet([n])$, and the category of all simplicial sets is denoted by $\mathsf{Set}_\Delta$.
Note that $\mathsf{Set}_\Delta$ has all limits and colimits. We can also define simplicial objects in a general category $\mathcal{C}$ as functors $\Delta^\mathrm{op} \to \mathcal{C}$.
A simplicial set comes with certain maps. There are these maps $$ d^i \colon [n] \to [n+1]; \quad d^i(j) = \begin{cases} j & j \lt i \br j+1 & j \le e \end{cases} $$ and also $$ s^i \colon [n+1] \to [n]; \quad s^i(j) = \begin{cases} j & j \le i \br j-1 & j \gt i \end{cases}. $$ These induce maps $$ d_i \colon X_n \to X_ {n-1}; \quad s_i \colon X_n \to X_ {n+1}. $$
The data of $X_ \bullet$ can be alternatively described as the collection of $X_n$ with these maps $d_i, s_j$ satisfying the identities
- $d_i \circ d_j = d_ {j-1} \circ d_i$ for $i \gt j$,
- $s_i \circ s_j = s_j \circ s_i$,
- $d_j \circ s_j = \ldots$.
Examples of simplicial sets#
Here are some examples of simplicial sets.
Example 3. There is this simplicial set $\Delta^n$ called the standard simplex. The easiest way of defining it is by defining it as $$[m] \mapsto \operatorname{Hom}_ \Delta([m], [n]).$$ By formal nonsense, we can see that $$\operatorname{Hom}_ {\mathsf{Set}_ \Delta}(\Delta^n, X_ \bullet) = X_n.$$
Example 4. For every category $\mathcal{C}$, you can define a simplicial set $N(\mathcal{C})_ \bullet$ by $$ N(\mathcal{C})_ n = \operatorname{Hom}_ \mathsf{Cat}([n], \mathcal{C}) $$ by regarding the linearly ordered set $[n]$ as a category.
Example 5. For $0 \le i \le n$, we can define the horn $\Lambda_i^n$ as $$\Lambda_i^n([m]) = { \alpha \colon [m] \to [n] : [n] \subsetneq \operatorname{im}(\alpha) \cup {i} }. $$ The picture is that we remove the interior of $\Delta^n$ and also the face opposite the vertex $i$.
Example 6. We can also define the boundary $\partial \Delta^n$ as $$\partial \Delta^n([m]) = { \alpha \colon [m] \to [n] \text{ not surjective} }.$$
Example 7. For each topological space $X$, we can define its singular simplicial set $\operatorname{Sing}(X) _ \bullet$ as $$\operatorname{Sing}(X) _n = \operatorname{Hom} _ \mathsf{Top}(\lvert \Delta^n \rvert, X)$$ where $\lvert \Delta^n \rvert$ is the topological simplex $$\lvert \Delta^n \rvert = { (t_0, \dotsc, t_n) \in \mathbb{R}^n : {\textstyle\sum} t_i = 1, t_i \ge 0 }.$$ The face and degeneracy maps are induced from the face and degeneracy maps on the topological simplex.
This topological space $\lvert \Delta^n \rvert$ is called that because of the following construction.
Proposition 8. There is an left adjoint $$ \lvert - \rvert \colon \mathsf{Set} _ \Delta \to \mathsf{Top}$$ to the singular simplex functor $\operatorname{Sing} _ \bullet \colon \mathsf{Top} \to \mathsf{Set} _ \Delta$.
Proof.
This follows from abstract nonsense. We can define $$\lvert X \rvert = \varinjlim_ {\Delta^n \to X} \lvert \Delta^n \rvert.$$
We can also construct this as a CW complex by gluing all $n$-simplices at one step and doing this for all $n$.
The mapping simplicial set#
All small limits and colimits exist in $\mathsf{Set} _ \Delta$, and they can be computed levelwise. More concretely, for a functor $F \colon \mathcal{C} \to \mathsf{Set} _ \Delta$, its colimit is $$ \left( \varinjlim _ {C \in \mathcal{C}} F _ \bullet(C) \right) _ n = \varinjlim _ {C \in \mathcal{C}} F_n(C) $$ and its limit satisfies a similar formula.
Given two simplicial sets $X_ \bullet$ and $Y_ \bullet$, you can also form the mapping simplicial set. We can define $$\operatorname{Fun}(X_ \bullet, Y_ \bullet) _n = \operatorname{Hom} _ {\mathsf{Set} _ \Delta}(\Delta^n \times X, Y).$$
Theorem 9 (exponential law). For simplicial sets $X_ \bullet, Y_ \bullet, Z_ \bullet$, there is a natural isomorphism $$\operatorname{Hom} _ {\mathsf{Set} _ \Delta}(X_ \bullet, \operatorname{Fun}(Y_ \bullet, Z_ \bullet)) \cong \operatorname{Hom} _ {\mathsf{Set} _ \Delta}(X_ \bullet \times Y_ \bullet, Z_ \bullet).$$
Proof.
We note that every $X_ \bullet$ can be written as a colimit of $\Delta^n$ along all maps $\Delta^n \to X_ \bullet$. So we reduce to the case when $X = \Delta^n$,
Using this, we can construct the evaluation map $$ \mathrm{ev} \colon \operatorname{Fun}(S_ \bullet, T_ \bullet) \times S_ \bullet \to T_ \bullet $$ using this exponential adjunction. conversely, we can use this evaluation map to describe the exponential map by looking at $$\begin{align*} \operatorname{Hom} _ {\mathsf{Set} _ \Delta}(X_ \bullet, \operatorname{Fun}(Y_ \bullet, Z_ \bullet)) &\to \operatorname{Hom} _ {\mathsf{Set} _ \Delta}(X_ \bullet \times Y_ \bullet, \operatorname{Fun}(Y_ \bullet, Z_ \bullet) \times Y_ \bullet) \br &\xrightarrow{\mathrm{ev}} \operatorname{Hom} _ {\mathsf{Set} _ \Delta}(X_ \bullet \times Y_ \bullet, Z_ \bullet). \end{align*}$$
Kan complexes#
We want to specify a class of simplicial sets that behaves well so that we can do homotopy theory.
Definition 10. A simplicial set $X_ \bullet \in \mathsf{Set} _ \Delta$ is a Kan complex when it satisfies the horn-lifting condition: for every $0 \le i \le n$ and every $\Lambda_i^n \to X_ \bullet$, there is an extension to $\Delta^n \to X_ \bullet$.
Here is a hard theorem of Joyal: in the definition, we can relax $0 \le i \le n$ to $0 \le i \lt n$.
Example 11. The standard complex $\Delta^n$ is not a Kan complex. This is because we can map $\Lambda_0^2 \to \Delta^n$ sending $0$ to $0$, $1$ to $1$, and $2$ to $0$. Then we can’t extend it because there is no edge from $1$ to $0$.
Example 12. For every topological space $X$, the singular simplicial set $\operatorname{Sing}(X) _ \bullet$ is a Kan complex. This is because there is a retract $\lvert \Delta^n \rvert \to \lvert \Lambda_i^n \rvert$.
The adjunction gives a map $$ \lvert \operatorname{Sing}(X) _ \bullet \rvert \to X. $$ This is always a weak homotopy equivalence.
Theorem 13. If $X_ \bullet, Y_ \bullet \in \mathsf{Set} _ \Delta$ are simplicial sets, and $Y_ \bullet$ is a Kan complex, then $\operatorname{Fun}(X_ \bullet, Y_ \bullet)$ is a Kan complex as well.
Proof.
The idea is that if we have lifting for these horn inclusions, then we can use it to prove lifting for inclusions $\Lambda_i^n \times X \hookrightarrow \Delta^n \times X$. Using inductive arguments, you can reduce it to proving lifting for inclusions $\Lambda_i^n \times \Delta^m \cup \Delta^n \times \partial \Delta_m \hookrightarrow \Delta^n \times \Delta^m$, and this is combinatorics.
There is a nice homotopy theory of Kan complexes.
Theorem 14. Let $X_ \bullet, Y_ \bullet$ be simplicial sets, and let $f, g \colon X_ \bullet \to Y_ \bullet$ be two maps. A homotopy between $f$ and $g$ is a map $X_ \bullet \times \Delta^1 \to Y_ \bullet$ that restricts to $f$ at ${0} \subset \Delta^1$ and $g$ at ${1} \subset \Delta^2$.
Definition 15. For $X_ \bullet$ a simplicial set, we define $\pi_0(X)$ to be the set of vertices $X_0$ quotiented by the relation generated by edges connecting two vertices.
Definition 16. We say that $f, g \colon X_ \bullet \to Y_ \bullet$ are homotopic if they are joined by a sequence of homotopies, or equivalently, are in the same connected component $\pi_0 \operatorname{Fun}(X_ \bullet, Y_ \bullet)$.
Proposition 17. If $X_ \bullet$ and $Y_ \bullet$ are Kan complexes, then the maps $f, g$ are homotopic if and only if there is a homotopy from $f$ to $g$.
Proof.
It suffices to show that having a homotopy is an equivalence relation. Recall that $Z_ \bullet = \operatorname{Fun}(X_ \bullet, Y_ \bullet)$ is a Kan complex. Reflexivity comes from just using $\Delta^1 \to \Delta^0$. For symmetry, we use $\Lambda_0^2 \to Z$ sending $0 \mapsto f$, $1 \mapsto g$, $2 \mapsto f$, and then lifting gives a edge from $g$ to $f$. For transitivity, we can use $\Lambda_1^2 \to Z$.
Definition 18. The category $h\mathsf{Kan}$ is the $1$-category with objects Kan complexes and morphisms being $\pi_0 \operatorname{Fun}(X_ \bullet, Y_ \bullet)$.
This is just the category of spaces up to homotopy.
There is this construction that takes in a simplicial set and turns it to a category. The idea is to take the colimit of the linearly ordered category over all simplices: $$ h(X_ \bullet) = \varinjlim_ {\Delta^n \to X_ \bullet} [n]. $$ The colimit is taken in the $1$-category of $1$-categories, so we get more morphisms than just $X_1$.
Proposition 19. If $X_ \bullet$ is Kan, then the associated category $h(X_ \bullet)$ is a groupoid, i.e., all morphisms are isomorphisms. The set of objects is just $X_0$, and the maps from $x$ to $y$ are maps $\Delta^1 \to X$ starting at $x$ and ending at $y$, up to some equivalence relation.