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1. I created a seperate Kan object which was desribed in internal infinity-groupoid before. The combinatoric part in the motivation is not needed, yet.

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeJan 10th 2012

What does “where sSet is regarded as a Set-enriched category” mean? It hardly seems necessary…

Also, I don’t understand the third equivalent condition for X to be Kan. What is $\Delta^k[n]$? Can you spell it out more? I didn’t know there was any way to define a Kan complex in terms of some map being a bijection.

• CommentRowNumber3.
• CommentAuthorStephan A Spahn
• CommentTimeJan 10th 2012
• (edited Jan 10th 2012)

What does “where sSet is regarded as a Set-enriched category” mean? It hardly seems necessary…

Yes, this is not necessary. I wrote it this way to motivate the generalization.

Also, I don’t understand the third equivalent condition for X to be Kan. What is $\Delta^k[n]$? Can you spell it out more? I didn’t know there was any way to define a Kan complex in terms of some map being a bijection.

I’ m sorry, there I did not type what I meant: the correct condition would be

$[\Delta[n],X]\xrightarrow{d} K_n:=\{(x_0,...,x_{k-1},x_{k+1},...,x_n|d_i x_j =d_{j-1} x_i, i\neg = k, j\neg =k, i\lt j\}$ is an epimorphism ( $[\Lambda^k[n],X]$ is then in bijection with $K_n$ where $d_i$ denotes the i-th face map and $d:=d_0\times...\times d_{k_1}\times d_{k+1}\times...\times d_n$). ”

So this condition is just a combinatoric description of $[\Lambda^k_n,X]$. The intention behind this is to think of the horn as a collection of n consecutive (n-1)-simplices which the Kan condition then closes by adding a further (n-1)-cell.

I deleted this condition since it is not used in the generalization.

Maybe one could add some hints to conditions making a Kan objecct an internal ∞-groupoid. The nlab is down at the moment, so I can’t look up what there already stands in regard to this. Maybe one can use some description of objects of composable n-cells -someting like $\times^{0\le i\le n}_{s,t}[\Delta[n],X]$ where $s:=d_n\star d_{n-1}\star d_{n-4}\star...$ and $t:=d_{n-1}\star_{n-3}\star d_{n-5}$ are the pasting of degeneracies- too.

(Obviously something is wrong with the formatting)

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeJan 10th 2012

Okay, I see; thanks!