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I was wondering what a good source for the properties (or even the existence) of the abelianisation of a (2-) groupoid would be? A naive construction would certainly be to abelianise the automorphisms (this is also mentioned here) but maybe that isn't the best construction?

As for basic properties I'm curious about: is this an exact (or rather $2$-exact) functor and adjoint to the inclusion-functor of groupoids into abelian groupoids (by which I should probably mean abelian objects in the category of categories?)

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    $\begingroup$ Depending on exactly how abelian you want the result to be, you can take the suspension spectrum, which more generally is the free “abelian group” on an $\infty$-groupoid in a suitable sense. $\endgroup$
    – Qiaochu Yuan
    Commented Sep 1, 2022 at 22:29

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