nLab covariant type family

1. Definition

In simplicial homotopy type theory, given a type $A$ and a type family

$$x:A \vdash B(x) \; \mathrm{type}$$

let us define the type family

$$x:A, y:A, f:\mathrm{hom}_A(x, y), u:B(x), v:B(y) \vdash \mathrm{hom}_{B(f)}(u, v) \; \mathrm{type}$$

as

$$\mathrm{hom}_{B(f)}(u, v) \coloneqq \left\langle\prod_{t:\mathbb{2}} C(f(t)) \vert_{[u,v]}^{\partial \Delta^1} \right\rangle$$

where $\mathbb{2}$ is the directed interval primitive, $\Delta^1$ is the $1$-simplex probe shape, and $\partial \Delta^1$ is its boundary.

Then a covariant type family over $A$ is a type family

$$x:A \vdash B(x) \; \mathrm{type}$$

such that given elements $x:A$, $y:A$, $f:\mathrm{hom}_A(x, y)$, and $u:B(x)$, the type

$$\sum_{v:B(y)} \mathrm{hom}_{B(f)}(u, v)$$

is a contractible type.

2. Properties

Theorem 2.1. Given a Segal type $A$, a type family

$$x:A \vdash B(x) \; \mathrm{type}$$

is a covariant type family if and only if for all elements $f:\mathrm{hom}_A(x, y)$, and $u:B(x)$, there is a unique lifting of $f$ that starts at $u$; or equivalently, that the type

$$\sum_{v:B(y)} \left\langle\prod_{t:\mathbb{2}} C(f(t)) \vert_{u}^{0} \right\rangle$$

is contractible.

3.  Related concepts

• hom type

• directed univalence

• contravariant type family

4. References

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