# Strictification of $\mathcal{V}$-pseudofunctors

Let $$\mathcal{B}$$ be a bicategory. Section 4.10 of Gordon, Power and Street's paper "Coherence for Tricategories" states that there is a bicategory $$\textbf{st}\mathcal{B}$$ and a biequivalence $$\eta_{\mathcal{B}}: \mathcal{B} \rightarrow \textbf{st}\mathcal{B}$$ such that, for any $$2$$-category $$\mathcal{C}$$, precomposition with $$\eta_{\mathcal{B}}$$ induces a bijection between $$2$$-functors from $$\textbf{st}\mathcal{B}$$ to $$\mathcal{C}$$ and pseudofunctors from $$\mathcal{B}$$ to $$\mathcal{C}$$.

Further, Corollary 3.6 in Alexander Campbell's article "How strict is strictification?" states that this bijection extends to an isomorphism $$\textbf{Gray}(\textbf{st}\mathcal{B},\mathcal{C}) \simeq \textbf{Bicat}(\mathcal{B},\mathcal{C})$$ of $$2$$-categories. The left-hand side is the $$\operatorname{Hom}$$-$$2$$-category in the $$\textbf{Gray}$$-category $$\textbf{Gray}$$, and thus consists of $$2$$-functors, pseudonatural transformations and modifications; the right-hand side is the $$\operatorname{Hom}$$-$$2$$-category in the tricategory of bicategories, pseudofunctors, pseudonatural transformations and modifications.

I am interested in the generalization of the above $$2$$-isomorphism to the setting of $$\mathcal{V}$$-bicategorical structures, where $$\mathcal{V}$$ is a closed, symmetric, (co)complete monoidal category.

More precisely, let $$\mathcal{V}$$-$$\textbf{Cat}$$ be the closed monoidal $$2$$-category of (small) $$\mathcal{V}$$-categories, $$\mathcal{V}$$-functors and $$\mathcal{V}$$-transformations. Following the definitions given by Garner and Shulman in "Enriched categories as a free cocompletion", we obtain the notions of bicategories, pseudofunctors, pseudonatural transformations and modifications enriched in $$\mathcal{V}$$-$$\textbf{Cat}$$ (as well as their strict variants), assembling into a tricategory $$\mathcal{V}$$-$$\mathbf{Bicat}$$.

We may now ask whether the above coherence results for bicategories also hold for $$\mathcal{V}$$-bicategories: given a $$\mathcal{V}$$-bicategory $$\mathcal{B}$$, is there a $$\mathcal{V}$$-$$2$$-category $$\textbf{st}\mathcal{B}$$ together with a $$\mathcal{V}$$-biequivalence $$\eta_{\mathcal{B}}: \mathcal{B} \rightarrow \textbf{st}\mathcal{B}$$ giving rise to $$\mathcal{V}$$-$$2$$-isomorphisms between $$\mathcal{V}$$-$$\textbf{Gray}(\textbf{st}\mathcal{B},\mathcal{C})$$ and $$\mathcal{V}$$-$$\textbf{Bicat}(\mathcal{B,C})$$, for any $$\mathcal{V}$$-$$2$$-category $$\mathcal{C}$$? If this is the case, is there a reference for the statement (preferably with some sketch of proof)? Or is it perhaps generally accepted as folklore?

I would also be interested in less general statements. For instance, if I understand correctly, in the bicategorical setting this statement is somewhat simpler if we assume $$\mathcal{C} = \textbf{Cat}$$. Is the above statement true in the case $$\mathcal{C} = \mathcal{V}$$-$$\textbf{Cat}$$? (The $$\mathcal{V}$$-pseudofunctors we consider then become a special case of modules studied in "Enriched categories as a free cocompletion") Restricting further, is either of the statements (general $$\mathcal{C}$$ or $$\mathcal{C} = \mathcal{V}$$-$$\textbf{Cat}$$) true if $$\mathcal{B}$$ is a $$\mathcal{V}$$-monoidal category, perhaps by some general results on pseudomonoids in monoidal $$2$$-categories? Or even further, is any of the above statements true in the case when $$\mathcal{V}$$ is locally presentable?

In section 4 of my paper Not every pseudoalgebra is equivalent to a strict one, I sketched a proof that for any monoidal 2-category $$\mathcal{W}$$ with small sums preserved on both sides by its tensor product (which includes $$\mathcal{V}\text{-Cat}$$ for nice enough $$\mathcal{V}$$), there is a 2-monad on the 2-category of $$\mathcal{W}$$-graphs whose strict algebras and pseudo algebras are, respectively, strict $$\mathcal{W}$$-categories and (unbiased) $$\mathcal{W}$$-bicategories.
Moreover, under appropriate hypotheses on $$\mathcal{W}$$ (one of which, the existence of an appropriate factorization system, holds for $$\mathcal{V}\text{-Cat}$$), the general 2-monadic coherence theorem holds for this 2-monad. This means that the inclusion of the 2-category of pseudoalgebras into that of strict algebras has a strict left 2-adjoint, and the components of the adjunction unit are equivalences (thus every $$\mathcal{W}$$-bicategory is equivalent to its strictification). This universal property lies in between the two mentioned in your first two paragraphs: instead of an isomorphism of 2-categories of strict/pseudo functors, pseudonatural transformations, and modifications, we have an isomorphism of 1-categories of strict/pseudo functors and icons, which in particular entails a bijection between sets of strict/pseudo functors.
• Thank you! Is $\mathcal{V}$ closed and cocomplete sufficient for $\mathcal{V}$-$\textbf{Cat}$ to satisfy the condition about small sums? In case $\mathcal{V}=\textbf{Set}$, pseudofunctors from a $\mathcal{V}$-$2$-category to $\mathcal{V}$-$\textbf{Cat}$ can be strictified to equivalent $\mathcal{V}$-$2$-functors, by a result of Power. Does this hold for general (nice) $\mathcal{V}$? So that we obtain a $\mathcal{V}$-biequivalence between $\mathcal{V}$-$\textbf{Gray}(\textbf{st}\mathcal{B}, \mathcal{V}\textbf{-Cat})$ and $\mathcal{V}\textbf{-Bicat}(\mathcal{B},\mathcal{V}\textbf{-Cat})$? Jun 5 at 14:01
• Yes, $\mathcal{V}\text{-Cat}$ has small sums preserved by its tensor product as soon as $\mathcal{V}$ has an initial object preserved by its tensor product, and the latter follows from closedness. Jun 6 at 3:28
• Is your second question not just a specialization of the original question to $\mathcal{C} = \mathcal{V}\text{-Cat}$? In which case, I don't have an answer other than that of my final paragraph. Jun 6 at 3:29