Is there a sort of lax Yoneda Lemma for 2-categories? Here is what I seem to have proven (although I have not checked all the details):

If $\mathcal{C}$ is a (weak) 2-category, $A$ is an object of $\mathcal{C}$, and $F : \mathcal{C} \to \mathsf{Cat}$ is a lax functor, then the category of lax transformations $\mathrm{Hom}(A,-) \to \mathcal{C}$ together with modifications is isomorphic to the category $F(A)$. (Not just equivalent, right?) If $F$ is strong, then every lax transformation $\mathrm{Hom}(A,-) \to \mathcal{C}$ is already strong.

Is this correct?

  • 4
    $\begingroup$ ncatlab.org/nlab/show/… has a weaker result, which suggests Lax(Hom(A, -), F) ≅ FA might not be the case in general $\endgroup$
    – user54748
    May 22 '15 at 14:05

Let $\mathcal{C} $ be a $2$-category and $A$ be an object of $\mathcal{C} $. In most of the cases, there are lax natural transformations $\mathcal{C}(A,-)\to\mathcal{C}(A,-) $ which are not $2$-natural or even pseudonatural. For instance, take the $2$-category generated by two parallel arrows $f,g: A\to B $ with a $2$-cell $f\to g$. We have that $\mathcal{C}(A,A) = \ast $ and $\mathcal{C}(A,B) = 2 $. Observe that, by the Enriched Yoneda Lemma, the category $[\mathcal{C}(A, -), \mathcal{C}(A, -)] _ {Strc} $ is isomorphic to $\mathcal{C} (A, A) $ and, therefore, terminal (only one object and the identity). However, $[\mathcal{C}(A, -), \mathcal{C}(A, -)] _ {Lax} $ has another object (besides the identity). This proves that $\mathcal{C} (A, A) $ is not isomorphic to $[\mathcal{C}(A, -), \mathcal{C}(A, -)] _ {Lax} $. And this nontrivial lax natural transformation is not isomorphic to the identity - therefore $[\mathcal{C}(A, -), \mathcal{C}(A, -)] _ {Lax} $ is not equivalent to $[\mathcal{C}(A, -), \mathcal{C}(A, -)] _ {Strict} $. And, by the bicategorical Yoneda Lemma, we conclude that $[\mathcal{C}(A, -), \mathcal{C}(A, -)] _ {Lax} $ is not equivalent to $[\mathcal{C}(A, -), \mathcal{C}(A, -)] _ {Psd}\simeq [\mathcal{C}(A, -), \mathcal{C}(A, -)] _ {Strict} $. This fact contradicts your conjecture. The Lax version of the Yoneda Lemma should be weaker.


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