An ordinary groupoid $\mathcal{G}$ gives an $\infty$-groupoid $N(\mathcal{G})$ via the nerve. This is a Kan complex with unique horn fillers (it is $1$-truncated: $\pi_n = 0$ for $n \geq 2$).

The fundamental groupoid $\Pi_1(X)$ of a topological space $X$ is an ordinary groupoid whose nerve is the $1$-truncation of $\operatorname{Sing}(X)$.

For a group $G$, the nerve $N(BG)$ is a Kan complex with $\pi_0 = *$, $\pi_1 = G$, and $\pi_n = 0$ for $n \geq 2$. This is the Eilenberg--MacLane space $K(G, 1)$.

More generally, for a topological group $G$, the classifying space $BG$ is an $\infty$-groupoid with $\pi_1 = \pi_0(G)$, $\pi_2 = \pi_1(G)$, etc.

For an abelian group $A$ and $n \geq 1$, the Eilenberg--MacLane space $K(A, n)$ is an $\infty$-groupoid with $\pi_k = 0$ for $k \neq n$ and $\pi_n = A$. It represents cohomology:

$[X, K(A, n)] = H^n(X; A)$

These can be constructed as simplicial abelian groups via the Dold--Kan correspondence applied to the chain complex $A[-n]$ concentrated in degree $n$.

The homotopy hypothesis holds in the following precise senses:

1. Quillen equivalence: $\mathbf{sSet}_{\mathrm{KQ}} \simeq_Q \mathbf{Top}$ via $|\cdot| \dashv \operatorname{Sing}$.

2. Homotopy groups agree: $\pi_n(X) \cong \pi_n(|X|)$ for Kan complexes $X$.

3. Mapping spaces agree: $\operatorname{Map}_{\mathrm{Kan}}(X, Y) \simeq \operatorname{Map}_{\mathrm{Top}}(|X|, |Y|)$.

4. Classification: Homotopy types are classified by homotopy groups and Postnikov invariants, which are the same data as $\infty$-groupoids with their higher structure.

The $\infty$-category $\mathcal{S}$ of spaces is the central object of higher category theory. It can be constructed as:

1. The coherent nerve $N_\Delta(\operatorname{Kan})$ of the simplicial category of Kan complexes.
2. The $\infty$-categorical localization $N(\mathbf{sSet})[W^{-1}]$.
3. The quasi-category of objects in the Kan--Quillen model structure.

$\mathcal{S}$ plays the role in $\infty$-category theory that $\mathbf{Set}$ plays in ordinary category theory. It is the "base" for enrichment, and functors $\mathcal{C} \to \mathcal{S}$ are the $\infty$-categorical analogues of presheaves.

The $\infty$-category $\mathcal{S}$ is the free colimit-completion of the terminal $\infty$-category $\{*\}$:

$\mathcal{S} \simeq \operatorname{Ind}(\{*\}) \simeq \operatorname{Fun}(\{*\}^{\mathrm{op}}, \mathcal{S})$

This says that $\mathcal{S}$ is the "free presentable $\infty$-category on one generator." The generator is the point $* \in \mathcal{S}$, and every space is a colimit of copies of $*$.

A presheaf on a quasi-category $\mathcal{C}$ valued in $\mathcal{S}$ is a functor $F: \mathcal{C}^{\mathrm{op}} \to \mathcal{S}$. The $\infty$-category of presheaves is $\operatorname{Fun}(\mathcal{C}^{\mathrm{op}}, \mathcal{S})$, which is always a presentable $\infty$-category.

For $\mathcal{C} = N(\mathcal{C}_0)$ an ordinary category, presheaves valued in $\mathcal{S}$ generalize classical presheaves (valued in $\mathbf{Set}$): instead of sets of sections, we have spaces of sections.

• $S^0 = \{0, 1\}$ is $0$-truncated (discrete, $\pi_k = 0$ for $k \geq 1$).
• $K(\mathbb{Z}, 1) = S^1$ is $1$-truncated ($\pi_1 = \mathbb{Z}$, $\pi_k = 0$ for $k \geq 2$).
• $S^2$ is NOT finitely truncated: $\pi_3(S^2) = \mathbb{Z}$ (Hopf), etc.
• $K(\mathbb{Z}, 2) = \mathbb{C}P^\infty$ is $2$-truncated.

The $n$-truncation functor $\tau_{\leq n}: \mathcal{S} \to \mathcal{S}_{\leq n}$ is left adjoint to the inclusion $\mathcal{S}_{\leq n} \hookrightarrow \mathcal{S}$.

Every space $X$ has a Postnikov tower:

$\cdots \to \tau_{\leq 2} X \to \tau_{\leq 1} X \to \tau_{\leq 0} X$

with $X \simeq \lim_n \tau_{\leq n} X$. Each stage $\tau_{\leq n} X \to \tau_{\leq n-1} X$ is a principal $K(\pi_n(X), n)$-fibration, classified by the $k$-invariant $k_n \in H^{n+1}(\tau_{\leq n-1} X; \pi_n(X))$.

The Postnikov tower completely determines the homotopy type of $X$ (given the $k$-invariants). This is the $\infty$-groupoid analogue of describing a group by generators and relations.

For a pointed $\infty$-groupoid $(X, x)$, the loop space $\Omega_x X = \operatorname{Map}_X(x, x)$ is an $\infty$-groupoid with $\pi_n(\Omega X) = \pi_{n+1}(X)$.

Conversely, for an $\infty$-group (a group object in $\mathcal{S}$, i.e., a connected Kan complex $G$ with group-like structure), the delooping $BG$ satisfies $\Omega BG \simeq G$. This gives a bijection between $\infty$-groups and pointed connected spaces.

For $n$-fold loop spaces: $\Omega^n(K(A, n)) \simeq A$ (discrete), reflecting the identification $K(A, n) = B^n A$.