It is possible to give a single-sorted (arrows-only) definition of a 1-category. For instance, see this nlab page. The basic idea is to identify objects with their identity morphisms.
Is it possible to give a single-sorted definition of an infinity-category? (The above nlab page seems to think it is, but does not elaborate much.)
The difficulty now is that there is no highest level of morphism. One idea I had was to remove the following conditions from the single-sorted definition of a 1-category:
$$s(s(x))=s(x)=t(s(x)) \\ t(t(x))=t(x)=s(t(x))$$
(where $s$ and $t$ denote source and target operations) for these seem to characterise 1-morphisms -- they do not even hold for identity 2-morphisms.
If it makes things easier, I am mostly interested in the case where the only n-morphisms for $n\geq2$ are identities. That, is I want to think about a 1-category as an infinity-category, but it is important that these higher identities are present, which is why using the definition of a single-sorted 1-category won't do.