Infinity-Category

Let $X$ be a simplicial set, i.e., a functor $X: \Delta^{\mathrm{op}} \to \mathbf{Set}$ where $\Delta$ is the simplex category.

Definition. An $\infty$-category (in the sense of a weak Kan complex or quasicategory) is a simplicial set $X$ satisfying the inner horn-filling condition: for every $n \geq 2$ and every $0 < k < n$, every map $\Lambda^n_k \to X$ from the $k$-th inner horn extends to a map $\Delta^n \to X$.

Here the $0$-simplices of $X$ are the objects, the $1$-simplices are the morphisms, and the higher simplices encode composition and coherence data. An inner horn $\Lambda^2_1 \to X$ specifies a composable pair of morphisms; a filler $\Delta^2 \to X$ provides a composite together with a witness of the composition. The inner horn condition ensures that composites exist but composites are unique up to a contractible space of choices.

More generally, the term “$\infty$-category” refers to any structure with objects, morphisms, 2-morphisms, and so on to all levels, with composition at each level satisfying associativity and unitality up to coherent higher morphisms.

Proposition. The following models of $\infty$-categories are equivalent (in the sense of Quillen-equivalent model categories): quasicategories (Joyal, Lurie), complete Segal spaces (Rezk), Segal categories (Hirschowitz–Simpson), and simplicial categories (Bergner). In particular, any theorem proved in one model transfers to all others.

Proposition. An ordinary category $\mathcal{C}$ embeds into $\infty$-categories via its nerve $N(\mathcal{C})$. The nerve is a quasicategory in which every inner horn has a unique filler. A quasicategory is a Kan complex (i.e., satisfies all horn-filling conditions, including outer horns) if and only if every morphism is invertible, in which case it models an $\infty$-groupoid.

Examples.

• Kan complexes: A Kan complex is a quasicategory in which every morphism is invertible. These model homotopy types of spaces.
• Nerves: For any ordinary category $\mathcal{C}$, the nerve $N(\mathcal{C})$ is a quasicategory.
• $(\infty,1)$-categories: The most commonly used variant, equivalent to the quasicategory model. All $k$-morphisms for $k \geq 2$ are invertible; higher structure encodes homotopical coherence.
• $(\infty,n)$-categories: More generally, an $(\infty,n)$-category has non-invertible cells up to dimension $n$, with all cells above dimension $n$ invertible. The case $n=1$ gives the $(\infty,1)$-category.