Simplicial sets, injectivity

Question

Let $X\colon \Delta^{op}\to \mathrm{Set}$ be a simplicial object with no nondegenerate cells except in degree $0$. Why the (composition of) degeneracy map $t_n\colon X_0\to X_n$ is injective? Is this $t_n$ unique? Why?

Answer 1

For any simplicial set, because of the simplicial identity $d_j s_j = 1$, each degeneracy map $s_j$ is injective, and therefore any composite of degeneracy maps is injective.

For any 0-simplex $v$, you can use the simplicial identity $s_i s_{j} = s_{j+1} s_i$ for $i<j+1$ to show that $t_n v = s_{n-1} \circ \dots \circ s_1 \circ s_0 v$, if $t_n$ is as in your question.

Edit: using the simplicial identity, one can show that any composite of degeneracies can be written as a composite $s_{i_1} s_{i_2} ...$ where the subscripts strictly decrease: $i_j > i_{j+1}$ for each $j$. Then use the fact that on $X_k$, the possible degeneracies are $s_0$, ..., $s_k$, to deduce that if you start with $X_0$, there is only one possible composite of degeneracy maps.