Exercise 3.2.4.8. Let $X$ be a simplicial set. Show that the following conditions are equivalent:
- $(1)$
The simplicial set $X$ is contractible: that is, the projection map $X \rightarrow \Delta ^0$ is a homotopy equivalence (Definition 3.2.4.1).
- $(2)$
The identity morphism $\operatorname{id}_{X}: X \rightarrow X$ is nullhomotopic.
- $(3)$
Every morphism of simplicial sets $f: X \rightarrow Y$ is nullhomotopic.
- $(4)$
Every morphism of simplicial sets $g: Z \rightarrow X$ is nullhomotopic.
In particular, these conditions are satisfied in the special case where $X = \Delta ^{n}$ is a standard simplex.