How to show that $\Delta[n]$ isn't Kan fibrant...?

Question

This is the problem: I have to prove that $\Delta[n]$ isn't Kan fibrant for n >=2. Does anyone how idea how to do it?

Answer 1

There are many ways. One is to note that $\Delta^n$ is the nerve of the category $\{ 0 \to 1 \to \cdots \to n \}$, and that the nerve of a category is a Kan complex if and only if it is a groupoid. Explicitly, consider the horn in $\Delta^n$ generated by the edges $0 \to 1$ and $0 \to 0$. A filler for this horn would have an edge $1 \to 0$, but by construction of $\Delta^n$ there is no such edge, so $\Delta^n$ cannot be a Kan complex.