It is a standard fact that two morphisms $f,g \in S_1$ (with the same vertexes) in an $\infty$-category $S$ are homotopic in the sense that
there is a $2$-simplex $\sigma \in S_2$ such that $d_0\sigma=id$, $d_1\sigma = g$, $d_2\sigma=f$
if and only if they are homotopic in the sense that
there is a map $\Delta^1 \times \Delta^1 \to S$ which is a homotopy in the classical sense.
I believe this ought be true also for the case $n >1,$ and I think the proof of the equivalence should not be much different, at least at a first thought. But although I have seen the above statement written in the literature, I have not found the statement for $n>1,$ so maybe I am not seeing some obvious thing that fails for higher morphisms?
Ultimately my questions are two:
- Is the above characterization of homotopies valid also in higher dimensions i.e. for every $n$?
- Is my thought that the proof is the same correct? Do you have a reference?
Note: For clarity:
what I mean by the statement for $n>1$ is the idea that morally they should be homotopic when they can be seen as faces of a common $n+1$-simplex
So maybe this would mean something along the lines of:
$f,g \in S_n$ are homotopic in the sense that
there is a sequence $f_0,f_1, \dots f_{n-1} \in S_n$ satisfying $f_0 = f$ and $f_{n-1}=g$ and with the property that there exists a $\sigma \in S_{n+1}$ such that $d_0\sigma = f_0, \dots, d_{n-1}\sigma=g $ and $d_n \sigma = id$
if and only if they are homotopic in the sense that
there is a map $\Delta^n \times \Delta^1 \to S$ which is a homotopy in the classical sense