Biequivalence

Biequivalence is a notion of equivalence between bicategories that generalizes equivalence of ordinary categories to the setting of 2‑categories. A pseudo‑functor \(F\colon \mathcal{B}\to\mathcal{C}\) is a biequivalence if it satisfies two main conditions. First, for every pair of objects \(X,Y\) in \(\mathcal{B}\), the induced functor on hom‑categories \(\mathcal{B}(X,Y)\to\mathcal{C}(FX,FY)\) is an equivalence of ordinary categories. This ensures that the 1‑morphisms and 2‑morphisms are preserved up to equivalence. Second, the functor \(F\) is essentially surjective on objects: for every object \(Z\) in \(\mathcal{C}\) there exists an object \(X\) in \(\mathcal{B}\) together with an equivalence \(Z\cong FX\) in \(\mathcal{C}\). Together these requirements mean that \(F\) is locally an equivalence and essentially surjective on objects.

Biequivalences compose and have quasi‑inverse functors in the bicategorical sense; two bicategories are biequivalent precisely when they become isomorphic in the homotopy category of bicategories. A biequivalence preserves many structural properties such as being monoidal, cartesian, or closed, provided the functor respects the additional structure. Moreover, biequivalent bicategories have equivalent homotopy 2‑types: their classifying spaces are homotopy equivalent.

Typical examples include the diagram of bicategories of categories, functors, and natural transformations, or the bicategory of monoidal categories, lax monoidal functors, and monoidal natural transformations. In higher category theory, biequivalence generalizes to \(\infty\)-equivalence of \(\infty\)-bicategories, and in homotopy theory it plays a role in comparing different models of 2‑groupoids or 2‑types. By providing a flexible notion of equivalence that respects both 1‑morphisms and 2‑morphisms, biequivalence is a central tool for working with higher categorical structures.

References:

- Bénabou, J., *Introduction to Bicategories*, Hermann, 1967.

- Lack, S., “A 2‑category of 2‑categories”, J. Homotopy Relat. Struct., 2004.

- Riehl, E., “2–Category Theory”, Cambridge Studies in Advanced Mathematics, 2016.