nLab composer

Idea

In a 1-category, compositions are by definition unique. However, in a general simplicial set the notion of composition of 1-simplices, such as required for the simplicial set to be a quasi-category, is more subtle. The notion of composers [Land 2021, Def. 1.2.1] reflects some of this.

Definition

A composer $C$ is a simplicial set with the right lifting property against all spine inclusions $I^n \to \Delta^n$.

In particular since $I^2=\Lambda^2 _1$ is the (2,1)-horn, and since a pair of composable 1-simplices $f,g$ determines an image of $I^2$, this means that given a composer there is a notion of composition $g \circ f$ as the restriction of the extension of the spine $\sigma|_{\{0,2\}}$.

Examples

Example

Since spine inclusions are anodyne it follows that every quasi-category is a composer. This is one way to make precise how there is a notion of composition on (higher) morphisms in a quasi-category.

References

• Markus Land, Def. 1.2.1 in: Introduction to Infinity-Categories, Birkhäuser (2021) [doi:10.1007/978-3-030-61524-6]