Is there something similar you can say for strict 2-categories?
In a personal attempt to understand strict 2-categories, I ended up constructing a speculative conceptual tool (whose validity remains to be seen) that I call the boundary of a 2-morphism. I've written up some raw notes here:
The basic idea is that given morphisms $f,g:x\to y$ and a 2-morphism $\alpha:f\Rightarrow g$, we define its boundary as an endomorphism
$$\partial\alpha\circ f = g.$$
When the source of the 2-morphism is an identity morphism, then we have
$$\partial\alpha = t(\alpha),$$
which seems to relate things well to cross modules when all morphisms are invertible.
I'm curious if there is anything like a crossed module, but where we're not dealing with groups and morphisms are not invertible. What I'm trying to cook up seems like it might be related to such a thing if it exists.
Any thoughts and/or any comments on my notes would be greatly appreciated.
PS: Apologies in advance if my writing is not very clear. I'm not a mathematician, but am trying to teach myself some basic higher category theory.