# Is a pseudonatural transformation of strict 2-functors to Cat isomorphic to a 2-natural transformation?

Let $\mathcal{C}$ be a strict 2-category. A corollary of the bicategorical Yoneda lemma says that any pseudofunctor $\mathcal{C} \to \operatorname{Cat}$ is pseudonaturally equivalent to a strict 2-functor. I would like to know if the "next level" of strictification is true; namely, is it true that any pseudonatural transformation of strict 2-functors $\mathcal{C} \to \operatorname{Cat}$ is isomorphic (via an invertible modification) to a 2-natural transformation?

The specific case of this that I am interested is when $\mathcal{C}$ is the delooping of a monoidal category: the first statement says that any left $\mathcal{C}$-module can be made into a "strict" left $\mathcal{C}$-module, while the second would imply that any morphism of left $\mathcal{C}$-modules can be strictified.

No, it is not true. For a counterexample, let C be the delooping of the group Z/2 (regarded as a discrete monoidal category). A strict C-module is then a category equipped with an involution, a strict C-module morphism is a functor preserving the involution strictly, and a pseudo C-module morphism preserves the involution up to coherent isomorphism. In particular, a strict C-module morphism must map an object fixed by the involution to another fixed object, but a pseudo C-module morphism can map a fixed object to one which is only fixed up to isomorphism. Thus, if A is a C-module with fixed objects and B is a C-module with no fixed objects, there can be pseudo C-module morphisms from A to B, but there cannot be any strict C-module morphisms from A to B.

It is, however, true that any strict (or even pseudo) 2-functor $F:C\to Cat$ is pseudonaturally equivalent to a strict 2-functor $F':C\to Cat$ which has the property that any pseudonatural transformation $F'\to G$ is isomorphic to a strict 2-natural transformation. An $F'$ with this property is called flexible (if you're a 2-category theorist) or cofibrant (if you're a homotopy theorist). The possibility of flexible replacement follows from generalities about 2-monads: there is a strict 2-monad T on the 2-category $Cat^{ob(C)}$ for which strict T-algebras are strict 2-functors $C\to Cat$, strict T-morphisms are strict 2-natural transformations, and pseudo T-morphisms are pseudonatural transformations. The general coherence theorems of Power and Lack (see the papers "A general coherence result" and "Codescent objects and coherence") apply to this 2-monad and specialize to the statement I quoted above. The homotopy theorist can instead construct a model structure on the 2-category of strict 2-functors and strict 2-natural transformations in which the cofibrant objects are cofibrant/flexible; see Lack's paper "Homotopy-theoretic aspects of 2-monads."

• Thanks, this is a terrific answer! I was not aware of these "general coherence theorems" until now, but they sound potentially relevant to some related stuff I'm trying to figure out as well. – Evan Jenkins Aug 6 '10 at 5:59