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 type theory

There are multiple different formalisms of simplicial homotopy type theory; two of them are given in Gratzer, Weinberger, & Buchholtz 2024 and in Riehl & Shulman 2017, and in each formalism there is a different way to define the hom type.

Directed interval via axioms

In simplicial homotopy type theory where the directed interval primitive $\mathbb{I}$ is defined via axioms as a bounded total order, the hom type is defined as the dependent sum type

$$\mathrm{hom}_A(x, y) \coloneqq \sum_{f:\mathbb{I} \to A} (f(0) =_A x) \times (f(1) =_A y)$$

Type theory with shapes formalism

In simplicial homotopy type theory in the type theory with shapes formalism, 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 $\Delta^1$.

In dependent type theory

More generally, hom types can be defined for any type $A$ and any bounded poset $I$ with a bottom element $0:I$ and a top element $1:I$, as the type

$$\mathrm{hom}_{A, I}(x, y) \coloneqq \sum_{f:I \to A} (f(0) =_A x) \times (f(1) =_A y)$$

 Related concepts

• simplicial homotopy type theory
• directed homotopy type theory
• identity type
• function type
• directed univalence
• composite of morphisms
• pair of composable morphisms
• Segal type, Rezk type
• covariant type family
• heterogeneous hom type

References

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

• Daniel Gratzer, Jonathan Weinberger, Ulrik Buchholtz, Directed univalence in simplicial homotopy type theory (arXiv:2407.09146)