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1. • CommentRowNumber1.
   • CommentAuthorMirco Richter
   • CommentTimeMay 14th 2012
   • (edited May 14th 2012)
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   Author: Mirco Richter
   Format: MarkdownItexIn their paper on the join of augmented simplicial sets Tim and P. J. Ehlers stated $\Delta[p] \odot \Delta[q] \simeq \Delta[p+q+1]$ where $\odot$ is the join of augmented simplicial sets. They said that the proof is easy, but before spending my time on finding a proof by myself, maybe Tim (or someone else) can give me the basic steps. I guess the starting point is the monoidal structure on $\Delta_a$, but...

   In their paper on the join of augmented simplicial sets Tim and P. J. Ehlers stated

   $\Delta[p] \odot \Delta[q] \simeq \Delta[p+q+1]$

   where $\odot$ is the join of augmented simplicial sets. They said that the proof is easy, but before spending my time on finding a proof by myself, maybe Tim (or someone else) can give me the basic steps. I guess the starting point is the monoidal structure on $\Delta_a$, but…

2. • CommentRowNumber2.
   • CommentAuthorTim_Porter
   • CommentTimeMay 15th 2012
   • (edited May 15th 2012)
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   Author: Tim_Porter
   Format: MarkdownItexThe proof uses that join with ? is left adjoint to an internal hom(?,-). That is not the `usual' hom, but uses the Dec. Note that this all is happening in ASS i.e. augmented simplicial sets. (There is a copy of Phil Ehlers PhD thesis on the nLab at the page with his name, [[Phil Ehlers]]. see page 53, but you need to check up on conventions used.)

   The proof uses that join with ? is left adjoint to an internal hom(?,-). That is not the ‘usual’ hom, but uses the Dec. Note that this all is happening in ASS i.e. augmented simplicial sets. (There is a copy of Phil Ehlers PhD thesis on the nLab at the page with his name, Phil Ehlers. see page 53, but you need to check up on conventions used.)

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 15th 2012
   • (edited May 15th 2012)
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   Author: Urs
   Format: MarkdownItexIt also follows immediately with the [formula](http://ncatlab.org/nlab/show/join%20of%20simplicial%20sets#DayConvolution) by Day convolution and the "[[co-Yoneda lemma]]".

   It also follows immediately with the formula by Day convolution and the “co-Yoneda lemma”.

4. • CommentRowNumber4.
   • CommentAuthorTim_Porter
   • CommentTimeMay 15th 2012
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   Author: Tim_Porter
   Format: MarkdownItex@Urs Do you know of other 'shapes' for n-cells for which a similar result holds (of course an even simpler one is cubes)?

   @Urs Do you know of other ’shapes’ for n-cells for which a similar result holds (of course an even simpler one is cubes)?

5. • CommentRowNumber5.
   • CommentAuthorMirco Richter
   • CommentTimeMay 15th 2012
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   Author: Mirco Richter
   Format: MarkdownItexI'm wondering that beside the join and the 'walking (co)monoid' in the bar construction, it seems that there is no use for the monoidal structure of $\Delta_a$.

   I’m wondering that beside the join and the ’walking (co)monoid’ in the bar construction, it seems that there is no use for the monoidal structure of $\Delta_a$.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeMay 15th 2012
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   Author: Urs
   Format: MarkdownItex> beside the join and the 'walking (co)monoid' in the bar construction, it seems that there is no use Right, what did the Romans ever do for us?! ;-) Seriously, the join _is_ that monoidal structure, Yoneda extended to all simplicial sets.

   beside the join and the ’walking (co)monoid’ in the bar construction, it seems that there is no use

   Right, what did the Romans ever do for us?! ;-)

   Seriously, the join is that monoidal structure, Yoneda extended to all simplicial sets.

7. • CommentRowNumber7.
   • CommentAuthorTim_Porter
   • CommentTimeMay 15th 2012
   • (edited May 15th 2012)
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   Author: Tim_Porter
   Format: MarkdownItexNot at all, the ordinal subdivision is very useful in homotopy coherence (see the paper by Phil and me on subdivision or that by Cordier and me on homotopy coherent category theory.) This could be linked to the bar construction of course but that is like saying that addition in the natural numbers is just used for algebra and arithmetic .... (I dislike the 'walking comonoid' terminology as it does not 'speak to me' about what is going on... I also am a fairly keen walker! :-)).

   Not at all, the ordinal subdivision is very useful in homotopy coherence (see the paper by Phil and me on subdivision or that by Cordier and me on homotopy coherent category theory.) This could be linked to the bar construction of course but that is like saying that addition in the natural numbers is just used for algebra and arithmetic …. (I dislike the ’walking comonoid’ terminology as it does not ’speak to me’ about what is going on… I also am a fairly keen walker! :-)).

8. • CommentRowNumber8.
   • CommentAuthorMirco Richter
   • CommentTimeMay 15th 2012
   • (edited May 15th 2012)
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   Author: Mirco Richter
   Format: MarkdownItex@Tim: Can you give a link to those papers? @Urs: The join is the reflection of that monoidal structure in the presheaf category (here of course simpl. sets). Or is it common to call that 'the same' ?

   @Tim: Can you give a link to those papers?

   @Urs: The join is the reflection of that monoidal structure in the presheaf category (here of course simpl. sets). Or is it common to call that ’the same’ ?

9. • CommentRowNumber9.
   • CommentAuthorTim_Porter
   • CommentTimeMay 15th 2012
   • (edited May 15th 2012)
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   Author: Tim_Porter
   Format: MarkdownItex@Mirco The material is in the Phd thesis.(link at [[Phil Ehlers]]) You know the first paper the second was published in Adv. in Math. but there is not an arxiv version as that paper had a tormented history!

   @Mirco The material is in the Phd thesis.(link at Phil Ehlers) You know the first paper the second was published in Adv. in Math. but there is not an arxiv version as that paper had a tormented history!