nLab hom type

Idea

In dependent type theory, a hom type on a type $A$ is a dependent type $\mathrm{hom}_A(x, y)$ which models the hom spaces of $\infty$-categories.

Definition

In simplicial homotopy type theory

In simplicial homotopy type theory, the hom type is defined as the extension type

$$\mathrm{hom}_A(x, y) \coloneqq \langle \Delta^1 \to A \vert_{[x,y]}^{\partial \Delta^1} \rangle$$

where $\Delta^1$ is the directed interval primitive $\mathbb{2}$ and $\partial \Delta^1$ is the boundary of $\Delta^1$.

 Related concepts

• simplicial homotopy type theory
• directed homotopy type theory
• identity type
• function type
• directed univalence
• Segal type, Rezk type
• covariant type family
• dependent hom type?

References

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