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Now there is a little bit more of content at abelianization.
Interesting, I did not know that the abelianization is a left adjoint!
Abelianization is defined for a number of algebraic structures and for number of categories with structure. For rings/algebras it is related to the Feynman-Maslov calculus and Kapranov’s noncommutative geometry, as it is emphasised in his original paper listed there. Among kind of categorifications there is the abelianization functor from triangulated categories to abelian categories, for which some time ago I created a stub Verdier’s abelianization functor, which also pops out when using the search button for “abelianization”. I have for now put the link at abelianization.
I added a few words of explanation to abelianization.
I added some remarks than one can abelianise more than groups. Some questions that I don't have time to figure out right now:
Also I added the example of the first homology group.
I added some remarks
Thanks.
Also I added the example of the first homology group.
I think you forgot to add the clause that $X$ be connected. I have added it and also pointed to singular homology - relation to homotopy groups for further details.
There is a neat example of what I call relative abelianisation. In the category of groups/G, the abilan group objects are the $G$-modules, and the abelianisation is the derived module construction given by Dick Crowell. I will try to put something on this but have to finish a letter of two first.
There is a neat example of what I call relative abelianisation.
BTW, we discuss this at module in As stabilized overcategories.
In the category of groups/G, the abilan group objects are the G-modules,
I have added that to this entry at Modules - In terms of stabilized overcategories - Modules over a group.
I think you forgot to add the clause that $X$ be connected.
Yep! Or take the connected component containing the point.
@Urs I had forgotten that was there. The interesting extra fact is to give the left adjoint. This leads to the Fox derivatives.
BTW I seem to remember that Quillen attributes the idea to Jon Beck and I checked in his thesis and it (and a lot of other stuff) is there. But I don’t know that he was the first to notice it. Possibly Mike Barr had an influence there. I changed mention of the origins to allow for this.
The interesting extra fact is to give the left adjoint. This leads to the Fox derivatives.
By the way, we have a bit of discussion of the left adjoint and its relation to derivations at tangent category, tangent (infinity,1)-category and Kähler differentials.
When or if you add stuff on Fox derivatives, I’d suggest to do it in a subsection of either tangent category or Kähler differentials.
Yes, possibly tangent category is slightly better. There is already an entry on Fox derivatives so I should try to bridge the two topics.
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