For any category $\mathcal{C}$, the nerve $N(\mathcal{C})$ is a quasi-category. The objects are the objects of $\mathcal{C}$, the morphisms are the morphisms of $\mathcal{C}$, and composition is strict (unique $2$-simplex fillers).

Every ordinary category embeds into the world of $\infty$-categories via the nerve.

Every Kan complex $X$ is a quasi-category. In this case, all morphisms are invertible up to homotopy: for any $f: x \to y$, there exists $g: y \to x$ with $g \circ f \simeq \mathrm{id}_x$ and $f \circ g \simeq \mathrm{id}_y$.

Kan complexes = $\infty$-groupoids = spaces. The homotopy hypothesis (Grothendieck) asserts this identification.

For any topological space $Y$, $\operatorname{Sing}(Y)$ is a Kan complex, hence a quasi-category. The objects are points of $Y$, morphisms are paths, $2$-simplices are homotopies between paths, and so on. This is the prototypical $\infty$-groupoid.

In a quasi-category $\mathcal{C}$, given morphisms $f: x \to y$ and $g: y \to z$, the inner horn $\Lambda^2_1 \to \mathcal{C}$ (with edges $f$ and $g$) has a filler -- a $2$-simplex $\sigma$ with $d_2(\sigma) = f$, $d_0(\sigma) = g$, and $d_1(\sigma) = h$ for some $h: x \to z$. This $h$ is "a" composite of $g$ and $f$.

The composite is not unique, but the space of all composites is contractible: any two composites are connected by a homotopy (a $3$-simplex), any two homotopies by a higher homotopy ($4$-simplex), and so on. This is the hallmark of $\infty$-categorical structure.

Given $f: w \to x$, $g: x \to y$, $h: y \to z$, we can form $h \circ (g \circ f)$ and $(h \circ g) \circ f$. These are not equal, but they are connected by a $3$-simplex (the filler of an inner $3$-horn $\Lambda^3_1$ or $\Lambda^3_2$). This $3$-simplex is an associator: a homotopy witnessing associativity.

Higher coherences (pentagon, etc.) are automatically encoded by fillers of higher-dimensional inner horns. This is the key advantage of quasi-categories: all coherence data is built into the definition.

A DG category $\mathcal{A}$ (a category enriched in chain complexes) gives rise to a quasi-category $N_{\mathrm{dg}}(\mathcal{A})$ via the DG nerve construction. Objects are the same, and the mapping spaces are the Dold--Kan images of the mapping chain complexes.

This provides a bridge between algebraic (DG) and homotopical (quasi-categorical) approaches to derived categories.

Every model category $\mathcal{M}$ has an underlying quasi-category $N(\mathcal{M})[W^{-1}]$, obtained by inverting the weak equivalences in the $\infty$-categorical sense (Dwyer--Kan localization). This $\infty$-category contains strictly more information than $\operatorname{Ho}(\mathcal{M})$: it remembers the full mapping spaces, not just their $\pi_0$.

The quasi-category $\mathcal{S}$ of spaces (also denoted $\operatorname{Kan}$ or $\infty\text{-}\operatorname{Gpd}$) has Kan complexes as objects and mapping spaces $\operatorname{Map}(X, Y)$ between them. This is the $\infty$-categorical analogue of $\mathbf{Set}$: it plays the role of the "universe" of $\infty$-groupoids.

$\mathcal{S}$ can be constructed as the coherent nerve of the simplicial category of Kan complexes.

There is a quasi-category $\operatorname{Cat}_\infty$ whose objects are (small) quasi-categories and whose morphisms are functors. This is the $\infty$-categorical analogue of $\mathbf{Cat}$.

$\operatorname{Cat}_\infty$ is itself a large quasi-category. It has a rich internal structure: mapping spaces $\operatorname{Fun}(\mathcal{C}, \mathcal{D})$ are themselves quasi-categories.

A functor $F: N(\mathcal{C}) \to N(\mathcal{D})$ between nerves of ordinary categories is exactly a functor $\mathcal{C} \to \mathcal{D}$ in the classical sense (since the nerve is fully faithful). A natural transformation $\alpha: F \Rightarrow G$ corresponds to a $1$-simplex in $\operatorname{Fun}(N(\mathcal{C}), N(\mathcal{D}))$.

In the functor quasi-category $\operatorname{Fun}(\mathcal{C}, \mathcal{D})$, a $1$-simplex from $F$ to $G$ is a map $\mathcal{C} \times \Delta[1] \to \mathcal{D}$ restricting to $F$ and $G$ at the endpoints. This is the $\infty$-categorical analogue of a natural transformation.

A natural equivalence is an invertible $1$-simplex in $\operatorname{Fun}(\mathcal{C}, \mathcal{D})$: a natural transformation $\alpha: F \Rightarrow G$ such that $\alpha_c: F(c) \to G(c)$ is an equivalence for every object $c$.