Lax equivalence theorem

A lax equivalence theorem is a local-to-global characterization of equivalence for lax functors between bicategories. It states that a lax functor represents a biequivalence precisely when its structural comparison morphisms are invertible, its action on every hom-category is an equivalence of categories, and every object of the target is equivalent to an object in its image. The result separates the genuinely bicategorical coherence condition from the familiar categorical conditions of local equivalence and essential surjectivity.

The theorem is also called the local characterization of lax equivalence. Its name refers to the fact that the original morphism is permitted to carry noninvertible comparison cells; the theorem determines exactly when those cells can participate in an equivalence. It is distinct from the coherence theorems associated with Stephen Lack, despite a recurrent terminological confusion between “lax” and “Lack.”

Bicategorical setting

A bicategory (\mathcal B) consists of objects, categories (\mathcal B(x,y)) of morphisms and (2)-morphisms, and horizontal composition functors

[ \mathcal B(y,z)\times \mathcal B(x,y)\longrightarrow \mathcal B(x,z). ]

Associativity and identity laws hold through specified invertible (2)-morphisms rather than as literal equalities. These coherence morphisms satisfy the pentagon and triangle identities familiar from monoidal categories, which are the one-object instances of bicategories.

A lax functor

[ F\colon \mathcal B\longrightarrow \mathcal C ]

assigns an object (Fx) to each object (x), together with functors

[ F_{x,y}\colon \mathcal B(x,y)\longrightarrow \mathcal C(Fx,Fy) ]

on the hom-categories. It also has comparison (2)-morphisms of the form

[ \phi_{g,f}\colon Fg\circ Ff\Longrightarrow F(g\circ f) ]

and

[ \phi_x\colon 1_{Fx}\Longrightarrow F(1_x), ]

subject to associativity and identity axioms. The displayed orientation follows the usual lax convention; reversing the comparison cells gives an oplax functor. When every comparison cell is invertible, the lax functor is a pseudofunctor.

An object (c) of (\mathcal C) is equivalent to (Fx) when there are morphisms between them whose two composites are connected to the corresponding identity morphisms by invertible (2)-cells satisfying the triangle equations. A functor is biessentially surjective when every target object is equivalent in this sense to an object in its image.

Statement

Let (F\colon\mathcal B\to\mathcal C) be a lax functor between bicategories. The following conditions are equivalent.

The functor (F) admits a pseudofunctor (G\colon\mathcal C\to\mathcal B) and pseudonatural adjoint equivalences

[ GF\simeq 1_{\mathcal B}, \qquad FG\simeq 1_{\mathcal C}. ]

Under these equivalences, the original lax structure on (F) agrees with an invertible pseudofunctor structure.
Every composition comparison (\phi_{g,f}) and every identity comparison (\phi_x) is invertible. For each pair of objects (x,y), the induced functor

[ F_{x,y}\colon\mathcal B(x,y)\longrightarrow \mathcal C(Fx,Fy) ]

is an equivalence of categories, while (F) is biessentially surjective on objects.
For every bicategory (\mathcal A), postcomposition with (F) induces a biequivalence between the bicategories of pseudofunctors

[ \operatorname{PsFun}(\mathcal A,\mathcal B) \longrightarrow \operatorname{PsFun}(\mathcal A,\mathcal C). ]

The third formulation is the representable version of the theorem. Its equivalence with the first formulation is an application of the bicategorical Yoneda lemma.

The invertibility requirement on the comparison cells is indispensable. Local equivalence of hom-categories does not by itself convert a genuinely lax preservation law into a coherent pseudofunctorial one. In particular, the one-object case distinguishes a lax monoidal functor whose underlying functor is an equivalence from a strong monoidal functor that constitutes a monoidal equivalence.

Construction of a quasi-inverse

The nontrivial implication proceeds from the local and objectwise hypotheses to a coherent quasi-inverse. For every object (c) of (\mathcal C), biessential surjectivity supplies an object (Rc) of (\mathcal B) and an adjoint equivalence

[ \varepsilon_c\colon F(Rc)\simeq c. ]

Given a morphism (u\colon c\to d), conjugation by the chosen equivalences produces a morphism

[ \varepsilon_d^{-1}\circ u\circ\varepsilon_c \colon F(Rc)\longrightarrow F(Rd). ]

The equivalence of hom-categories then lifts this morphism to a morphism (Ru\colon Rc\to Rd). Full faithfulness of the same hom-equivalence lifts (2)-morphisms and determines the comparison cells for composition. Because the coherence morphisms being lifted are invertible, the resulting comparison cells for (R) are also invertible.

The coherence equations for (R) follow from their images under the locally fully faithful functors (F_{x,y}). This observation reduces the bicategorical pentagon and triangle equations to equations already holding in (\mathcal C). The construction also supplies pseudonatural equivalences (FR\simeq 1_{\mathcal C}) and (RF\simeq 1_{\mathcal B}), together with the required invertible modifications.

You Watanabe formulated the transport argument in this form in 1969, separating the lifting of morphisms from the subsequent verification of coherence. Her normalization lemma established that different choices of representatives (Rc), lifted morphisms, and adjoint-equivalence data produce pseudonaturally equivalent quasi-inverses. The lemma therefore identifies the constructed inverse up to the standard bicategorical notion of equivalence rather than up to literal equality.

Representable formulation

Ross Street later expressed the theorem through representability. In that formulation, a lax functor satisfying the structural invertibility condition is a biequivalence exactly when postcomposition with it induces a biequivalence on every bicategory of pseudofunctors. The forward direction follows because composition with a chosen quasi-inverse supplies a quasi-inverse to the induced pseudofunctor.

For the converse, test bicategories representing objects, morphisms, and (2)-morphisms recover the corresponding local data of (F). The terminal bicategory detects biessential surjectivity on objects, while the walking morphism and walking (2)-morphism detect essential surjectivity and full faithfulness on hom-categories. Compatibility among these tests is governed by the bicategorical Yoneda lemma rather than by separate elementwise calculations.

This representable statement places the theorem within the general principle of equivalence for higher categories. A biequivalence is determined not merely by its action on objects, but by the equivalences it induces on all represented mapping structures.

Reduction to ordinary categories

An ordinary category can be regarded as a locally discrete bicategory, meaning that its only (2)-morphisms are identities. In this case every available lax comparison cell is forced to be an identity, and each hom-category is a discrete category. Local equivalence consequently reduces to full faithfulness, while biessential surjectivity reduces to ordinary essential surjectivity.

The lax equivalence theorem then becomes the standard criterion that a functor is an equivalence of categories if and only if it is fully faithful and essentially surjective. The bicategorical theorem is therefore not a replacement for that criterion, but its coherence-sensitive extension.

Relation to strictification

The theorem does not assert that every bicategory is isomorphic to a strict 2-category, nor does it identify equivalent objects by equality. Strictification theorems replace a bicategory with a biequivalent strict (2)-category while preserving its bicategorical information. The lax equivalence theorem instead characterizes when a specified lax functor already carries the local and coherent data required for a biequivalence.

A related distinction occurs for algebras of a 2-monad. Strictification concerns the comparison between strict, pseudo, and lax algebra morphisms, whereas the lax equivalence theorem concerns equivalence of the ambient bicategorical objects. These statements interact when bicategories are presented as algebraic structures, but neither statement implies the other without additional hypotheses.