nLab locally univalent bicategory

Definition

In an intensional type theory such as homotopy type theory, a bicategory is locally univalent if every hom-category is a univalent category.

More specifically, it is a bicategory $C$ for which the canonical functions

$$idtoiso^{2,1}(f, g):(f = g) \to inv2cell(f, g)$$

is an equivalence of types for all objects $a:Ob(C)$, $b:Ob(C)$, and morphisms $f:Mor_C(a,b)$ and $g:Mor_C(a,b)$.

In set-level foundations

A locally univalent bicategory in set-level foundations is the same thing as a locally gaunt bicategory whose type of objects is a set.

See also

• bicategory

• univalent bicategory

References

• Benedikt Ahrens, Dan Frumin, Marco Maggesi, Niels van der Weide, Bicategories in Univalent Foundations (arXiv:1903.01152)