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This is really a trivial question.

The 0-horn of a simplicial point $\Delta^0$ is not defined nor remarked in the books and papers I could find. So one expect if we could make some meaningful definition (like $0!=1$). The usual definition of $\Lambda^n_k$ for $n>1, 0\le k\le n$ simply does not make sense.

It is quite clear that the boundary $\partial\Delta^0$ should be $\emptyset$, the constant simplicial set with value $\emptyset$. However the 0-horn should be strict smaller than $\partial\Delta^0$, which is already the smallest simplicial set.

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    $\begingroup$ I think you answered your question in the last sentence: $\Lambda^0_0 \subseteq \partial \Delta^0 = \emptyset$. $\endgroup$
    – David Roberts ♦
    Commented Feb 7, 2014 at 0:34
  • $\begingroup$ @DavidRoberts I am not satisfied with the only possibility $Λ^0_0=∂Δ^0$. I think there should be better solution (like enlarge the category of simplicial sets). $\endgroup$
    – Ma Ming
    Commented Feb 7, 2014 at 0:36
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    $\begingroup$ If you work with presheaves over the augmented simplex category, then it's natural to define $\partial\Delta^0=\Delta^{-1}$ and $\Lambda^0_0$ to actually be the empty presheaf. I don't know if this is useful. $\endgroup$
    – Eric Wofsey
    Commented Feb 7, 2014 at 0:47
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    $\begingroup$ I think the 0-horn is best left undefined or non-existent (even in the augmented simplex category). After all, one of the most important properties of a horn is that it is a weak equivalence. That cannot be arranged for in the case of a 0-horn. $\endgroup$
    – Ricardo Andrade
    Commented Feb 7, 2014 at 0:51
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    $\begingroup$ The join is really an operation on augmented simplicial sets so the comment by Eric is of importance. $\endgroup$
    – Tim Porter
    Commented Feb 7, 2014 at 8:56

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(As already mentioned, we have to specify whether we're talking about simplicial sets or about augmented simplicial sets; I will be talking about simplicial sets.)

I agree with Ricardo Andrade: the simplex $\Delta^0$ has no horns.

The horns of a simplex $\Delta^n$ correspond to the (maximal) proper faces of $\Delta^n$, and those can already be defined within the simplex category $\Delta$. As there is no proper subobject of $\Delta^0$ in the category $\Delta$, the simplex $\Delta^0$ has no horns.

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  • $\begingroup$ then why we do not define it to be $\emptyset$? $\endgroup$
    – Ma Ming
    Commented Jun 25, 2014 at 23:24
  • $\begingroup$ Of course you may define anything you want. But if you want the horns of a simplex $\Delta^n$ to bijectively correspond to the maximal proper subobjects of $\Delta^n$ in the simplex category $\Delta$, then the simplex $\Delta^0$ has no horns. $\endgroup$
    – Daniel Gerigk
    Commented Jun 26, 2014 at 17:32
  • $\begingroup$ (Remember that $\emptyset$ is not contained in the simplex category $\Delta$.) $\endgroup$
    – Daniel Gerigk
    Commented Jun 26, 2014 at 17:35

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