Thanks for the examples! It seems like standard Borel spaces can be much more general than what I thought.

Interesting. I was hoping for a more topological condition though (something that doesn't mention the Borel sets).

What is $b_{n+k,n+l}$? Also it seems to me that if $A$ isn't symmetric then B won't be neither since the top left $n \times n$ block of $B$ is exactly $A$.

By $(S_\mu V)^*$ do you just mean the dual vector space to $S_\mu V$? If so why is that not a satisfying definition, do you expect a simpler description of this vector space?

Why is that particularly problematic? For example, the tensor product of nontrivial rings can sometimes be trivial, but that doesn't mean it's not useful to define and study the tensor product in full generality.

@HJRW No, graph objects in the category of groups would have a group structure on their vertex set and edge set, that's not what graph of groups are. We need groupoids since they can be interpreted (up to equivalence) as sets with a group attached to each element.

I've never heard of homotopy colimits; that seems interesting. Could you elaborate on how the fundamental groupoid of a graph of groups can be interpreted as a homotopy colimit?

One way to think about graphs of groups is as quotients of graphs where you "remember" the stabilizers of the vertices and edges in case the action is not free, similar to how orbifolds are locally quotients of manifolds by finite groups.