Composition in Quasi-categories

Definition

In an $(\infty ,1)$-category modeled as a quasi-category, composition is defined by the extension property of inner simplicial horns.

Inner Horns

Let ${\Delta }^n$ be the standard $n$-simplex. For $0<k<n$, the inner horn ${\Lambda }_k^n$ is the union of all faces of ${\Delta }^n$ except the $k$-th face. For defining the composition of two morphisms, the relevant horn is ${\Lambda }_1^2\subset {\Delta }^2$, which consists of two edges meeting at a vertex.

Composition Law

A simplicial set $S$ is a quasi-category if every map $f_0:{\Lambda }_k^n\to S$ for $0<k<n$ extends to a map $f:{\Delta }^n\to S$.

Given morphisms $u:x\to y$ and $v:y\to z$ in $S$, they define a map ${\Lambda }_1^2\to S$. The existence of a filler ${\Delta }^2\to S$ provides:

1. An edge $w:x\to z$ corresponding to the face $d_1$.
2. A 2-simplex $\sigma$ such that $d_2\sigma =u$, $d_0\sigma =v$, and $d_1\sigma =w$. The edge $w$ is a composite of $u$ and $v$.

Non-uniqueness and Contractibility

Composition is not a functional operation but a property. While a composite $w$ is not unique, the space of composites—defined as the fiber of the restriction map $Map({\Delta }^2,S)\to Map({\Lambda }_1^2,S)$—is a contractible Kan complex. This ensures that any two composites are equivalent in a higher-categorical sense.

Higher Categories

The definition generalizes to higher dimensions. A filler for ${\Lambda }_k^n$ in $S$ for $n>2$ ensures that composition is associative up to higher-dimensional coherences. In a Kan complex, which models an infinity-groupoid, all horns (including outer horns ${\Lambda }_0^n$ and ${\Lambda }_n^n$) have fillers, implying all morphisms are invertible.

Related Concepts

• Simplicial Set: The underlying combinatorial structure.
• Kan Complex: A simplicial set where all horns have fillers.
• Higher Category Theory: The study of structures where composition is defined up to higher cells.

References

lurie2009-higher-topos-theory - J. Lurie, “Higher Topos Theory,” Annals of Mathematics Studies, vol. 170, Princeton University Press, 2009. joyal2008-notes-quasi-categories - A. Joyal, “Notes on Quasi-categories,” 2008. groth2010-short-course-infinity-categories - M. Groth, “A Short Course on $\infty$-categories,” 2010.