nLab Segal type

 Idea

Segal types are the (infinity,1)-categorical version of precategories in simplicial type theory

Definition

In simplicial type theory, a Segal type is a type $A$ such that given elements $x:A$, $y:A$, and $z:A$ and morphisms $f:\mathrm{hom}_A(x, y)$ and $g:\mathrm{hom}_A(x, y)$, the type

$$\sum_{h:\mathrm{hom}_A(x, z)} \left\langle \Delta^2 \to A \vert_{[x, y, z, f, g, h]}^{\partial \Delta^2} \right\rangle$$

is a contractible type, where $\Delta^2$ is the $2$-simplex probe shape and $\partial \Delta^2$ is its boundary.

Related concepts

• hom type
• discrete Segal type
• Rezk type
• isomorphism in a Segal type
• Segal space
• op modality

References

• Emily Riehl, Michael Shulman, A type theory for synthetic $\infty$-categories $[$arXiv:1705.07442$]$