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1. • CommentRowNumber1.
   • CommentAuthorbezem
   • CommentTimeFeb 16th 2015
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   Author: bezem
   Format: TextThe webside http://ncatlab.org/nlab/show/algebraic+Kan+complex defines: "An algebraic Kan complex is a Kan complex equipped with a choice of horn fillers for all horns. A morphism of algebraic Kan complexes is a morphism of the underlying Kan complexes that sends chosen fillers to chosen fillers. This defines the category AlgKan of algebraic Kan complexes." and refers to the paper by Thomas Nikolaus, Algebraic models for higher categories, arXiv/1003.1342. Thomas writes to me that AlgKan is not a Cartesian closed category. It seems that the requirement that morphisms send chosen fillers to chosen fillers is rather strong, and causes difficulties in defining a well-behaved exponent in AlgKan. Would there be a case for a notion that could be called "functional Kan complex", where one has filler functions, but the morphisms are just as in the case of simplicial sets? In particular, morphisms do not have to send chosen fillers to chosen fillers (they will send fillers to fillers anyway, by naturality, but not necessarily to the chosen ones). I am well aware that, *under AC and classical logic* this would be equivalent to the usual notion of Kan simplicial set. However, with "functional Kan" one would have the same benefits regarding AC as mentioned on http://ncatlab.org/nlab/show/algebraic+Kan+complex. Constructively, "functional Kan" is perhaps more natural and a little stronger than "Kan". In some cases it is interesting to work with weaker axioms (then the results have greater generality). For example, it can be hoped that the category of "functional Kan" simplicial sets can be proved to be Cartesian closed without using AC. (There is little hope to eliminate classical logic here.) Marc

   The webside http://ncatlab.org/nlab/show/algebraic+Kan+complex defines:
   
   "An algebraic Kan complex is a Kan complex equipped with a choice of horn fillers for all horns.
   
   A morphism of algebraic Kan complexes is a morphism of the underlying Kan complexes that sends chosen fillers to chosen fillers.
   
   This defines the category AlgKan of algebraic Kan complexes."
   
   and refers to the paper by Thomas Nikolaus, Algebraic models for higher categories, arXiv/1003.1342.
   
   Thomas writes to me that AlgKan is not a Cartesian closed category. It seems that the requirement that morphisms send chosen fillers to chosen fillers is rather strong, and causes difficulties in defining a well-behaved exponent in AlgKan. Would there be a case for a notion that could be called "functional Kan complex", where one has filler functions, but the morphisms are just as in the case of simplicial sets? In particular, morphisms do not have to send chosen fillers to chosen fillers (they will send fillers to fillers anyway, by naturality, but not necessarily to the chosen ones). I am well aware that, *under AC and classical logic* this would be equivalent to the usual notion of Kan simplicial set. However, with "functional Kan" one would have the same benefits regarding AC as mentioned on http://ncatlab.org/nlab/show/algebraic+Kan+complex. Constructively, "functional Kan" is perhaps more natural and a little stronger than "Kan". In some cases it is interesting to work with weaker axioms (then the results have greater generality). For example, it can be hoped that the category of "functional Kan" simplicial sets can be proved to be Cartesian closed without using AC. (There is little hope to eliminate classical logic here.)
   
   Marc
2. • CommentRowNumber2.
   • CommentAuthorRichard Williamson
   • CommentTimeFeb 17th 2015
   • (edited Feb 17th 2015)
   • PermaLink
   Author: Richard Williamson
   Format: MarkdownItexIn my opinion, the question of equipping a category of algebraic Kan complexes, or something similar, with a closed monoidal structure is a very important and interesting one. I don't know whether you have seen it, but I made a few remarks on this in a cubical setting [here](http://nforum.mathforge.org/discussion/3386/cransgray-tensor-product/?Focus=51644#Comment_51644). In particular, I _do_ think that the category of cubical algebraic Kan complexes should be able to be equipped with a closed monoidal structure. However, if one experiments a little, it is clear that it is a subtle and interesting matter, closely related to pinning down how we wish to think of algebraic Kan complexes as $\infty$-groupoids. This underlies the point that the category of algebraic Kan complexes, where morphisms preserve the chosen fillers, is very different from the ordinary category of cubical sets. The monoidal structure that I think should exist on cubical algebraic Kan complexes would not be cartesian, though, but I would not expect it to be: the 'correct' monoidal structure on higher categories seems to me to be Gray-like, not cartesian. It may be that the category of algebraic Kan complexes itself may not quite be able to be equipped with the closed monoidal structure that I have in mind: one might need to tweak it a little. But the morphisms will still preserve fillers. Regarding your suggestion for a category of 'functional Kan complexes', one could of course consider it, but I would strongly disagree that the objects of this category can be thought of as algebraic $\infty$-groupoids. It is precisely the fact that morphisms preserve the chosen fillers that is significant about the category of algebraic Kan complexes, and which differentiates its behaviour from the usual category of simplicial/cubical sets. This fact is I think not emphasised enough about algebraic Kan complexes: the fact that we choose fillers on the objects doesn't really change anything from a categorical point of view. Indeed, as of course you will know, the latter is really the _only_ way that one can make the definition of a Kan complex in a strong constructive foundations (it is the question of the meaning of the existential quantifier). I also think that one should be able to show closure of the non-cartesian monoidal structure on cubical 'functional Kan complexes' in a constructive way: at least, I think it possible that the most one will need is to be able to argue by cases on finite sets of natural numbers. Definitely the axiom of choice should not be necessary. I didn't quite follow your remark about fillers being sent to fillers by naturality: could you elaborate upon it?

   In my opinion, the question of equipping a category of algebraic Kan complexes, or something similar, with a closed monoidal structure is a very important and interesting one.

   I don’t know whether you have seen it, but I made a few remarks on this in a cubical setting here. In particular, I do think that the category of cubical algebraic Kan complexes should be able to be equipped with a closed monoidal structure. However, if one experiments a little, it is clear that it is a subtle and interesting matter, closely related to pinning down how we wish to think of algebraic Kan complexes as $\infty$-groupoids. This underlies the point that the category of algebraic Kan complexes, where morphisms preserve the chosen fillers, is very different from the ordinary category of cubical sets.

   The monoidal structure that I think should exist on cubical algebraic Kan complexes would not be cartesian, though, but I would not expect it to be: the ’correct’ monoidal structure on higher categories seems to me to be Gray-like, not cartesian.

   It may be that the category of algebraic Kan complexes itself may not quite be able to be equipped with the closed monoidal structure that I have in mind: one might need to tweak it a little. But the morphisms will still preserve fillers.

   Regarding your suggestion for a category of ’functional Kan complexes’, one could of course consider it, but I would strongly disagree that the objects of this category can be thought of as algebraic $\infty$-groupoids. It is precisely the fact that morphisms preserve the chosen fillers that is significant about the category of algebraic Kan complexes, and which differentiates its behaviour from the usual category of simplicial/cubical sets. This fact is I think not emphasised enough about algebraic Kan complexes: the fact that we choose fillers on the objects doesn’t really change anything from a categorical point of view. Indeed, as of course you will know, the latter is really the only way that one can make the definition of a Kan complex in a strong constructive foundations (it is the question of the meaning of the existential quantifier).

   I also think that one should be able to show closure of the non-cartesian monoidal structure on cubical ’functional Kan complexes’ in a constructive way: at least, I think it possible that the most one will need is to be able to argue by cases on finite sets of natural numbers. Definitely the axiom of choice should not be necessary.

   I didn’t quite follow your remark about fillers being sent to fillers by naturality: could you elaborate upon it?

3. • CommentRowNumber3.
   • CommentAuthorbezem
   • CommentTimeFeb 17th 2015
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   Author: bezem
   Format: TextSorry, that remark was wrong. F.e., if m : X ->Y is a morphism and the horn is Lambda^2_0, then m(fill(e1,e2)) fills m(e1),m(e2). What could be said is that fillings are sent to fillings, but not necessarily the chosen filling. One cannot say anything about the fillers in Y in general.

   Sorry, that remark was wrong. F.e., if m : X ->Y is a morphism and the horn is Lambda^2_0, then m(fill(e1,e2)) fills m(e1),m(e2). What could be said is that fillings are sent to fillings, but not necessarily the chosen filling. One cannot say anything about the fillers in Y in general.
4. • CommentRowNumber4.
   • CommentAuthorRichard Williamson
   • CommentTimeFeb 17th 2015
   • PermaLink
   Author: Richard Williamson
   Format: MarkdownItexThank you, I see what you mean now!

   Thank you, I see what you mean now!

5. • CommentRowNumber5.
   • CommentAuthorPeter Heinig
   • CommentTimeJun 5th 2017
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   Author: Peter Heinig
   Format: MarkdownItex(replying to this in order not to have to start a dicussion on account of such a small change) Edited both [[filler]] and [[Kan complex]]; links added in the latter, building on a contribution of Tim Porter in the former.

   (replying to this in order not to have to start a dicussion on account of such a small change)

   Edited both filler and Kan complex; links added in the latter, building on a contribution of Tim Porter in the former.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJun 6th 2017
   • (edited Jun 6th 2017)
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   Author: Urs
   Format: MarkdownItex[edit: After working through my backlog, I see now that others here have expressed the same as below before. Peter, please wait for a while with creating many more entries, as it's going a little against the grain of the way we usually proceed. ] Peter, we usually do not create a separate page for each synonym. It seems more efficient to have all the information in one place and instead have the synomyms be _redirects_ to that page. If in a given entry XYZ you add line of the form [[!redirects ABC]] then the URL https://ncatlab.org/nlab/show/ABC will take the user to the same place as https://ncatlab.org/nlab/show/XYZ I suggest _[[filler]]_ and _[[diagonal fill-in]]_ should all redirect to _[[lift]]_. At the very least "diagonal fill-in" should be but a redirect to "filler".

   [edit: After working through my backlog, I see now that others here have expressed the same as below before. Peter, please wait for a while with creating many more entries, as it’s going a little against the grain of the way we usually proceed. ]

   Peter,

   we usually do not create a separate page for each synonym. It seems more efficient to have all the information in one place and instead have the synomyms be redirects to that page.

   If in a given entry XYZ you add line of the form

     [[!redirects ABC]]
   

   then the URL

     https://ncatlab.org/nlab/show/ABC
   

   will take the user to the same place as

     https://ncatlab.org/nlab/show/XYZ
   

   I suggest filler and diagonal fill-in should all redirect to lift. At the very least “diagonal fill-in” should be but a redirect to “filler”.

7. • CommentRowNumber7.
   • CommentAuthorPeter Heinig
   • CommentTimeJun 18th 2017
   • (edited Jun 18th 2017)
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   Author: Peter Heinig
   Format: MarkdownItexMerged [[diagonal fill-in]] into [[filler]], according to the HowTo and the comments, which seems an improvement. Merging into [[lift]] would require more care. If it is strongly preferred to do so, too, will do this sometime in the future.

   Merged diagonal fill-in into filler, according to the HowTo and the comments, which seems an improvement. Merging into lift would require more care. If it is strongly preferred to do so, too, will do this sometime in the future.

8. • CommentRowNumber8.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 19th 2017
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   Author: Todd_Trimble
   Format: MarkdownItexI added some organizational structure to [[filler]], and a few more words and links.

   I added some organizational structure to filler, and a few more words and links.