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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeApr 15th 2021
• (edited Apr 15th 2021)

Added a remark amplifying that the 0-simplex really has no horn, and that one must not think it could be defined to be the empty set (saw long and unresolved MO discussion of this point…)

• CommentRowNumber2.
• CommentAuthorJonasFrey
• CommentTimeApr 15th 2021
• (edited Apr 15th 2021)

I agree, but interestingly, Lurie explicitly comments on the 0-0-horn in Kerodon, and clarifies that it’s empty. See 1.1.2.13 in https://kerodon.net/tag/000K.

… but then he excludes 0 when stating the Kan condition. See 1.1.9.1 here : https://kerodon.net/tag/002G

I don’t really see the point of introducing the 0-0-horn just to exclude it whenever horns are used, but maybe he has his reasons.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeApr 16th 2021

Thanks for the remark, I hadn’t seen that.

Looking at 1.1.9.1

(incidentally: make links here like this: [1.1.2.13](https://kerodon.net/tag/000K))

… he might just want to stick to that general formula to define horns.

But I am going to part company on this point with species of South American rock cavies, related to capybaras and guinea pigs, and insist that there is no horn in a 0-simplex – not to mess with the long and widely established and fully reasonable definition that a Kan complex has fillers for all horns.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeApr 16th 2021
• (edited Apr 16th 2021)

Made more explicit that excluding horns of the 0-simplex is mandatory if the usual definition of Kan complex is not to be broken.

Then, while I was at it, I restructured some of the entry slightly and then considerably expanded its Idea-section, mentioning also the relation to cylinder boundary inclusions of topological spaces

• CommentRowNumber5.
• CommentAuthorDmitri Pavlov
• CommentTimeApr 16th 2021
• (edited Apr 16th 2021)

The boundary ∂Δ^n is defined for all n≥0. In particular, ∂Δ^0=∅.

The horn Λ^n_k is obtained by removing the kth face (0≤k≤n) from the boundary ∂Δ^n.

In particular, Λ^n_k has one less nondegenerate simplex than ∂Δ^n.

But for n=0 there are no (-1)-dimensional faces to remove, and in any case ∂Δ^0 has no simplices at all.

Example 1.1.2.13 in Kerodon is just so weird and makes no sense from any practical or theoretical viewpoint.

Goerss and Jardine require that n≥1 for the horn Λ^n_k (the last paragraph of Section I.1).

I am really curious what induced Jacob to deviate from Goerss and Jardine.

Curiously, the most literal reading of Kan’s original 1956 paper on Kan complexes Abstract homotopy. III would seem to indicate that Kan does allow horns to be empty and his definition of a Kan complex seemingly requires Kan complexes to be nonempty!

• CommentRowNumber6.
• CommentAuthorHurkyl
• CommentTimeNov 5th 2021