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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 Δ k[n]\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 Δ k[n]\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

    [Δ[n],X]dK n:={(x 0,...,x k1,x k+1,...,x n|d ix j=d j1x i,i¬=k,j¬=k,i<j}[\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 ( [Λ k[n],X][\Lambda^k[n],X] is then in bijection with K nK_n where d id_i denotes the i-th face map and d:=d 0×...×d k 1×d k+1×...×d nd:=d_0\times...\times d_{k_1}\times d_{k+1}\times...\times d_n). ”

    So this condition is just a combinatoric description of [Λ n k,X][\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 × s,t 0in[Δ[n],X]\times^{0\le i\le n}_{s,t}[\Delta[n],X] where s:=d nd n1d n4...s:=d_n\star d_{n-1}\star d_{n-4}\star... and t:=d n1 n3d n5t:=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!

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