4
$\begingroup$

Consider the 2-category $[S, H] $ of 2-functors $S\to H $ (in which, obviously, $S$ and $H$ are 2-categories). And consider a (possibly fully faithful) functor $T: S\to Z $ For simplicity, let's assume that every 2-functor $S\to H $ admits a pointwise (right) Kan extension along $ T $ (for instance, we may assume that $H$ is complete). So we get a 2-functor

$Ran _T : [S, H]\to [Z,H] .$

Consider, now, the 2-category $[S, H]_{PS} $ of 2-functors and pseudonatural transformations... My questions is whether there is an extension of the 2-functor $ Ran _ T $ to a 2-functor $ RAN_ T : [S, H]_{PS}\to [Z,H] _ {PS} $ such that $RAN_T$ restricted to $ [S,H] $ is isomorphic to $Ran _ T $.

I'm almost sure that it is possible to do so using the universal property of the Kan extension and the results given in S Lack's Codescent objects and coherence.

But I would like to know if there is a natural way of constructing such a 2-functor. And, furthermore, it would help me a lot some indications of literature about it.

Thank you in advance!

$\endgroup$
2
$\begingroup$

After thinking (and discussing with professor Steve Lack), I am sure that this extension of the Kan extension does not exist. Such an extension would give a 2-functor $ RAN_T : [S,H]_{PS}\to [Z,H]_{PS} $. And, of course, this 2-functor would take pseudonatural isomorphic diagrams to pseudonatural isomorphic diagrams. But there are pseudonatural isomorphic diagrams whose Kan extensions aren't pseudonatural isomorphic.

Therefore there is no such (strict) extension (as I originally asked for in this question).

The following example is an adaptation of an example given by professor Steve Lack.

Let $\nabla 2 $ be the localization of the category $ 2 $ (w.r.t. all morphisms). And let $ \ast $ be the category with only one object and only one morphism (the identity). If S is the 2-category consisting of a parallel pair of arrows ($\alpha : a\to b $, $ \beta : a\to b $) and Z is the category consisting of a parallel pair of arrows with an equalizer. $S$ and $Z$ have no nontrivial 2-cells, we can consider the following diagrams

$X: S\to Cat $ and $Y: S\to Cat $

such that $ X(a)= Y(a) = \ast $, $ X(b) = Y(b) = \nabla 2 $, $X(\alpha ) = X (\beta ) $ and $ Y(\alpha ) \not = Y( \beta ) $.

$X$ is pseudonaturally isomorphic to $Y$. However the Kan extensions of $X$ and $Y$ are not pseudonaturally isomorphic (because $[Z, Cat] _{PS} (Ran_T X, Ran_T Y ) = \emptyset$ ).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.