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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 7th 2012
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   Author: Urs
   Format: MarkdownItexSomebody asks by email: > double categories are cartesian closed and it seems pretty reasonable that n-fold categories are cartesian closed - would you know of a reference for this? It would be good if there was some result that said categories internal to a suitable category E were cartesian closed. Does anyone easily have a pointer?

   Somebody asks by email:

   double categories are cartesian closed and it seems pretty reasonable that n-fold categories are cartesian closed - would you know of a reference for this? It would be good if there was some result that said categories internal to a suitable category E were cartesian closed.

   Does anyone easily have a pointer?

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeNov 7th 2012
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   Author: Todd_Trimble
   Format: MarkdownItex> Does anyone easily have a pointer? I don't, but it doesn't seem unreasonable to write out a proof for this type of thing. Without having written anything down, I'd guess that if $E$ is finitely complete and cartesian closed, then $Cat(E)$ is also finitely complete and cartesian closed. Then $n$-fold categories would be cartesian closed by induction.

   Does anyone easily have a pointer?

   I don’t, but it doesn’t seem unreasonable to write out a proof for this type of thing. Without having written anything down, I’d guess that if is finitely complete and cartesian closed, then is also finitely complete and cartesian closed. Then -fold categories would be cartesian closed by induction.

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeNov 7th 2012
   • (edited Nov 7th 2012)
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   Author: Todd_Trimble
   Format: MarkdownItexWell, I may as well have a go at writing down a sketch of a proof of the assertion from my previous comment, that if $E$ is finitely complete and cartesian closed, then $Cat(E)$ is also finitely complete and cartesian closed. First, let $E$ be finitely complete. Then the category of directed graphs $E^{\bullet \stackrel{\to}{\to} \bullet}$ is also finitely complete, and since $Cat(E)$ is monadic over $E^{\bullet \stackrel{\to}{\to} \bullet}$, it follows that $Cat(E)$ is also finitely complete. Now let $E$ be finitely complete and cartesian closed. Then $E^C$ is cartesian closed for any finite category $C$ (by adapting [Mike's proof](http://ncatlab.org/nlab/show/cartesian+closed+category#exponentials_of_cartesian_closed_categories_30), using only finite ends). This applies in particular to the case where $C$ is a suitable truncation of the simplex category, say where $C$ is opposite to the category of nonempty ordinals up to cardinality 3. Now $Cat(E)$ is a full subcategory of $E^C$, and it should be simple to see directly that it is an exponential ideal of $E^C$; in particular, it's cartesian closed. (When I say "easy to see directly", I mean that we don't need to consider colimits in $Cat(E)$ or its being a reflective subcategory of $E^C$ -- just use the formula for exponentials in $E^C$ to suggest the correct construction of exponentials in $Cat(E)$.) Does this seem reasonable?

   Well, I may as well have a go at writing down a sketch of a proof of the assertion from my previous comment, that if is finitely complete and cartesian closed, then is also finitely complete and cartesian closed.

   First, let be finitely complete. Then the category of directed graphs is also finitely complete, and since is monadic over , it follows that is also finitely complete.

   Now let be finitely complete and cartesian closed. Then is cartesian closed for any finite category (by adapting Mike’s proof, using only finite ends). This applies in particular to the case where is a suitable truncation of the simplex category, say where is opposite to the category of nonempty ordinals up to cardinality 3. Now is a full subcategory of , and it should be simple to see directly that it is an exponential ideal of ; in particular, it’s cartesian closed. (When I say “easy to see directly”, I mean that we don’t need to consider colimits in or its being a reflective subcategory of – just use the formula for exponentials in to suggest the correct construction of exponentials in .)

   Does this seem reasonable?

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeNov 7th 2012
   • (edited Nov 7th 2012)
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   Author: Urs
   Format: MarkdownItexThanks, Todd! I have alerted my correspondent of your messages, thanks. Maybe some of this could be copied to the entry _[n-fold category](http://ncatlab.org/nlab/show/n-fold+category#CartesianClosure)_.

   Thanks, Todd! I have alerted my correspondent of your messages, thanks.

   Maybe some of this could be copied to the entry n-fold category.

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeNov 7th 2012
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   Author: Mike Shulman
   Format: MarkdownItexThe closest thing I can think of to a reference for this is the remark following B2.3.15 in the Elephant. But surely someone, somewhere, must have written it down before...

   The closest thing I can think of to a reference for this is the remark following B2.3.15 in the Elephant. But surely someone, somewhere, must have written it down before…

6. • CommentRowNumber6.
   • CommentAuthorDavidRoberts
   • CommentTimeNov 8th 2012
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   Author: DavidRoberts
   Format: MarkdownItexEhresmann's original proof, which I alluded to on the page [[internal category]], is really quite awful. It's working with generalised sketches, and for some reason either this precludes using the simple machinery from Todd's #3, or Ehresmann just wasn't in that frame of mind.

   Ehresmann’s original proof, which I alluded to on the page internal category, is really quite awful. It’s working with generalised sketches, and for some reason either this precludes using the simple machinery from Todd’s #3, or Ehresmann just wasn’t in that frame of mind.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeNov 8th 2012
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   Author: Urs
   Format: MarkdownItexCan you add a pointer, or a page number, or something, to identify the alluded-to proof in Ehresmann's work`?

   Can you add a pointer, or a page number, or something, to identify the alluded-to proof in Ehresmann’s work‘?

8. • CommentRowNumber8.
   • CommentAuthorDavidRoberts
   • CommentTimeNov 8th 2012
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   Author: DavidRoberts
   Format: MarkdownItexCan do.

   Can do.

9. • CommentRowNumber9.
   • CommentAuthorTodd_Trimble
   • CommentTimeNov 12th 2012
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   Author: Todd_Trimble
   Format: MarkdownItexI got around to writing out a proof at [[internal category]]. Please feel free to check for accuracy. I see that someone already had written there > If the ambient category $\mathcal{C}$ is a [[cartesian closed category]], then the category $Cat(\mathcal{C})$ of categories internal to $\mathcal{C}$ is also cartesian closed. I have added in a finite completeness assumption (which is a bit of a no-brainer; at a minimum, existence of pullbacks should be assumed).

   I got around to writing out a proof at internal category. Please feel free to check for accuracy.

   I see that someone already had written there

   If the ambient category is a cartesian closed category, then the category of categories internal to is also cartesian closed.

   I have added in a finite completeness assumption (which is a bit of a no-brainer; at a minimum, existence of pullbacks should be assumed).

10. • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 12th 2012
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    Author: DavidRoberts
    Format: MarkdownItexOk, edited to include the references alluded to in #6.

    Ok, edited to include the references alluded to in #6.