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.



How to define $\Lambda^0_0$, 0-horn of a simplicial point
(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.


