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Since objects in categories are only well-identified up to the equivalences between them, one way to ask what you ask is, I suppose, whether there is a process that takes an $n$-category and somehow universally for some $k \leq n$ turns its $r \leq k$-morphisms into equivalences.
I am not sure if I have seen this discussed much, but I suppose one may certainly consider it. (One example that vaguely comes to mind is maybe Kan’s Ex functor. )
If you look at just groupoids, then in spaces (via the homotopy hypothesis) this is the Whitehead tower.
If you focus on one object at a time, then this is looping.
There is an interesting class of simplicial groups that I have wondered about the higher categorical interpretations of their structure. and which may show another aspect of this. The idea is that they are general up to level n and then from that point on each box has a unique thin filler. (This in this context means in the subgroup generated by degenerate elements.) In the case of n= 1 (or 2 depending on how you count!) these simplicial groups have crossed complexes as their Moore complexes. These structures are nested (of course) and the inclusions have left adjoints. Of course, this is not quite what Skave is asking about as these are all ‘groupoids’ but I wonder what the n-categorical analogue might look like. Perhaps the Whitehead tower is related to this, but I am not at all sure. These homotopy types would have trivial Whitehead products from some point on, but possibly a lot of other information could be gleaned from them. The left adjoint would therefore give a universal way of killing the Whitehead products at a given level. (Does that make sense?)
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