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1. • CommentRowNumber1.
   • CommentAuthorEmily Riehl
   • CommentTimeMay 19th 2015
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   Author: Emily Riehl
   Format: MarkdownItexAt the nLab entry on [exact squares](http://ncatlab.org/nlab/show/exact+square), under the section on "generalizations", there is a comment that I take to mean that the internal notion of pointwise Kan extension definable in any 2-category with commas (stable under pasting with comma squares) is not the same as the enriched notion of pointwise Kan extension (defined in Kelly's book) in the case of the 2-category of $V$-categories. Where can I learn more about this? What is the difference?

   At the nLab entry on exact squares, under the section on “generalizations”, there is a comment that I take to mean that the internal notion of pointwise Kan extension definable in any 2-category with commas (stable under pasting with comma squares) is not the same as the enriched notion of pointwise Kan extension (defined in Kelly’s book) in the case of the 2-category of $V$-categories. Where can I learn more about this? What is the difference?

2. • CommentRowNumber2.
   • CommentAuthorThomas Holder
   • CommentTimeMay 19th 2015
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   Author: Thomas Holder
   Format: MarkdownItexSteve Lack's [2-companion](http://arxiv.org/abs/math.CT/0702535) has in sec. 2.2 (p.116 of the book version) a short remark on pointwise left Kan extensions where he says concerning the difference: _the problem is that for $\mathcal{V}$-categories A and B, the ($\mathcal{V}$)-functor category $[A,B]$ should really be regarded as a $\mathcal{V}$-category, but the 2-category $\mathcal{V}-\mathbf{Cat}$ can't see this extra structure. There are ways around this if B is sufficiently complete or cocomplete._ So this is probably already discussed in one of the classic texts on abstract extensions like Kelly-Street, Street-Walters or Street. There is also a [tac paper](http://www.tac.mta.ca/tac/volumes/29/27/29-27abs.html) by Koudenberg that might be helpful as the abstract promises a generalization relative to which the enriched and the 2-categorical concept coincide.

   Steve Lack’s 2-companion has in sec. 2.2 (p.116 of the book version) a short remark on pointwise left Kan extensions where he says concerning the difference:

   the problem is that for $\mathcal{V}$-categories A and B, the ($\mathcal{V}$)-functor category $[A,B]$ should really be regarded as a $\mathcal{V}$-category, but the 2-category $\mathcal{V}-\mathbf{Cat}$ can’t see this extra structure. There are ways around this if B is sufficiently complete or cocomplete.

   So this is probably already discussed in one of the classic texts on abstract extensions like Kelly-Street, Street-Walters or Street. There is also a tac paper by Koudenberg that might be helpful as the abstract promises a generalization relative to which the enriched and the 2-categorical concept coincide.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeMay 21st 2015
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   Author: Mike Shulman
   Format: MarkdownItexIn the remarks after Theorem 4.43 in Kelly's book we find: > Street in [71] has given a definition of "pointwise Kan extension" in any 2-category with finite limits; it agrees with our notion when $\mathbf{V} = Set$, but is strictly stronger for a general $\mathbf{V}$, and is hence not suited to our context. [71] is "Fibrations and Yoneda’s lemma in a 2-category". However, at the moment I don't remember seeing a justification of this statement (e.g. a particular $\mathbf{V}$ and a particular situation where the two definitions differ) anywhere.

   In the remarks after Theorem 4.43 in Kelly’s book we find:

   Street in [71] has given a definition of “pointwise Kan extension” in any 2-category with finite limits; it agrees with our notion when $\mathbf{V} = Set$, but is strictly stronger for a general $\mathbf{V}$, and is hence not suited to our context.

   [71] is “Fibrations and Yoneda’s lemma in a 2-category”. However, at the moment I don’t remember seeing a justification of this statement (e.g. a particular $\mathbf{V}$ and a particular situation where the two definitions differ) anywhere.

4. • CommentRowNumber4.
   • CommentAuthorRoald
   • CommentTimeMay 23rd 2015
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   Author: Roald
   Format: MarkdownItexIn the case of enriching over $\mathbf{Cat}$, I would say that the problem is that the comma object of a pair of 2-functors into a 2-category $C$ doesn't "see" the cells of $C$. For an example in which this gives trouble take $B$ to be the "walking cell", i.e. the 2-category generated by a single cell, and $M$ the "walking pair of parallel arrows", i.e. $B$ with its cell removed. If I checked things correctly then the 2-functor $l \colon B \to M$ given by collapsing onto the source-object of $M$ is the enriched left Kan extension of the 2-functors that map the terminal 2-category onto the source-objects of $B$ and $M$ respectively. This Kan extension is however not pointwise in the target-object of $B$ in the sense of Street. Some details are in Example 2.24 of my [thesis](http://arxiv.org/abs/1304.4079). If you want to make formal this "doesn't see the cells", then the language of double categories is useful. In the double category of 2-categories, the comma object of 2-functors $f$ and $g$ into $C$ coincides with the tabulation (= the double limit of a horizontal morphism) of the representable 2-profunctor $C(f, g)$. We could say that the comma object "captures enough of $C$" whenever the defining cell of this tabulation is opcartesian. Finally, you can rewrite Dubuc/Kelly's notion of enriched Kan extension in terms of "biclosed equipments", like is done on the nLab [here](http://ncatlab.org/nlab/show/2-category+equipped+with+proarrows#limits). By "unfolding" the resulting notion it can be considered as a notion of pointwise Kan extension in any double category $\mathcal{K}$. The main result of my [paper](http://www.tac.mta.ca/tac/volumes/29/27/29-27abs.html) implies that, if $\mathcal{K}$ is an equipment that has opcartesian tabulations, then pointwise left Kan extensions along companions in $\mathcal{K}$ coincide with pointwise left Kan extensions in the vertical 2-category contained in $\mathcal{K}$; the latter in Street's sense. Hope this helps!

   In the case of enriching over $\mathbf{Cat}$, I would say that the problem is that the comma object of a pair of 2-functors into a 2-category $C$ doesn’t “see” the cells of $C$.

   For an example in which this gives trouble take $B$ to be the “walking cell”, i.e. the 2-category generated by a single cell, and $M$ the “walking pair of parallel arrows”, i.e. $B$ with its cell removed.

   If I checked things correctly then the 2-functor $l \colon B \to M$ given by collapsing onto the source-object of $M$ is the enriched left Kan extension of the 2-functors that map the terminal 2-category onto the source-objects of $B$ and $M$ respectively. This Kan extension is however not pointwise in the target-object of $B$ in the sense of Street. Some details are in Example 2.24 of my thesis.

   If you want to make formal this “doesn’t see the cells”, then the language of double categories is useful. In the double category of 2-categories, the comma object of 2-functors $f$ and $g$ into $C$ coincides with the tabulation (= the double limit of a horizontal morphism) of the representable 2-profunctor $C(f, g)$. We could say that the comma object “captures enough of $C$” whenever the defining cell of this tabulation is opcartesian.

   Finally, you can rewrite Dubuc/Kelly’s notion of enriched Kan extension in terms of “biclosed equipments”, like is done on the nLab here. By “unfolding” the resulting notion it can be considered as a notion of pointwise Kan extension in any double category $\mathcal{K}$. The main result of my paper implies that, if $\mathcal{K}$ is an equipment that has opcartesian tabulations, then pointwise left Kan extensions along companions in $\mathcal{K}$ coincide with pointwise left Kan extensions in the vertical 2-category contained in $\mathcal{K}$; the latter in Street’s sense.

   Hope this helps!

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeJun 5th 2015
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   Author: Mike Shulman
   Format: MarkdownItexNice example Roald, thanks! I added it to [Kan extension](http://ncatlab.org/nlab/show/Kan+extension#In2cat).

   Nice example Roald, thanks! I added it to Kan extension.