(Infinity,1)-Category

Let $\mathcal{C}$ be an $\infty$-category.

Definition. An $(\infty,1)$-category is an $\infty$-category $\mathcal{C}$ in which every $k$-morphism for $k \geq 2$ is invertible (up to $(k+1)$-morphisms, and so on coherently). That is, the only non-invertible cells are the 1-morphisms; all higher cells serve as homotopical coherence data.

In the quasicategory model (the standard model due to Joyal and Lurie), an $(\infty,1)$-category is a simplicial set $X$ satisfying the inner horn-filling condition: for every $n \geq 2$ and $0 < k < n$, every map $\Lambda^n_k \to X$ extends to a map $\Delta^n \to X$. Objects are $0$-simplices, morphisms are $1$-simplices, and $n$-simplices for $n \geq 2$ witness composition and higher coherences. Composition is determined up to a contractible space of choices.

Proposition. The nerve $N(\mathcal{C})$ of an ordinary category $\mathcal{C}$ is a quasicategory in which all inner horns have unique fillers. Conversely, a quasicategory $X$ is isomorphic to the nerve of an ordinary category if and only if all inner horn fillers are unique.

Proof sketch. The nerve $N(\mathcal{C})_n = \mathrm{Fun}([n], \mathcal{C})$. Inner horn filling corresponds to composition, and uniqueness of fillers corresponds to strict associativity and unitality.

Proposition. For any two objects $x, y$ in a quasicategory $X$, the mapping space $\mathrm{Map}_X(x, y)$ is a Kan complex. An $(\infty,1)$-category in which all mapping spaces are contractible or empty is equivalent to a poset. An $(\infty,1)$-category in which all mapping spaces are discrete is equivalent to an ordinary category.

Proposition. An $(\infty,1)$-category in which all $1$-morphisms are invertible is an $\infty$-groupoid, i.e., a Kan complex. Under the homotopy hypothesis, the homotopy theory of $\infty$-groupoids is equivalent to the homotopy theory of topological spaces.

Proposition. The standard categorical notions — adjunctions, limits, colimits, Kan extensions, and natural transformations — all admit $(\infty,1)$-categorical analogues, characterized by the same universal properties but with diagrams commuting up to coherent homotopy.

Examples.

• Spaces: The $(\infty,1)$-category $\mathcal{S}$ of spaces (Kan complexes), with mapping spaces given by $\mathrm{Map}_{\mathcal{S}}(X,Y) = Y^X$. This is the terminal presentable $(\infty,1)$-category under colimit-preserving functors.
• Stable categories: The derived $(\infty,1)$-category $\mathcal{D}(R)$ of a ring $R$, obtained as the $\infty$-categorical localization of the category of chain complexes of $R$-modules at the quasi-isomorphisms.
• $(\infty,1)$-topoi: The $(\infty,1)$-category of sheaves of spaces on a site is the central example in higher topos theory.