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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeAug 31st 2012
• (edited Oct 22nd 2012)
• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeOct 22nd 2012

Now there is a little bit more of content at abelianization.

• CommentRowNumber3.
• CommentAuthorzskoda
• CommentTimeOct 22nd 2012

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.

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeOct 22nd 2012

I added a few words of explanation to abelianization.

• CommentRowNumber5.
• CommentAuthorTobyBartels
• CommentTimeOct 23rd 2012

I added some remarks than one can abelianise more than groups. Some questions that I don't have time to figure out right now:

• What conditions must hold in a monoidal category (or multicategory) to form abelianisations of its monoid objects?
• What relation holds between the Lie algebra of the abelianisation of the Lie group $G$ and the abelianisation of the Lie algebra of $G$?

Also I added the example of the first homology group.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeOct 24th 2012

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.

• CommentRowNumber7.
• CommentAuthorTim_Porter
• CommentTimeOct 24th 2012

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.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeOct 24th 2012

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.

• CommentRowNumber9.
• CommentAuthorTobyBartels
• CommentTimeOct 24th 2012

I think you forgot to add the clause that $X$ be connected.

Yep! Or take the connected component containing the point.

• CommentRowNumber10.
• CommentAuthorTim_Porter
• CommentTimeOct 24th 2012
• (edited Oct 24th 2012)

@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.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeOct 24th 2012

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.

• CommentRowNumber12.
• CommentAuthorTim_Porter
• CommentTimeOct 24th 2012

Yes, possibly tangent category is slightly better. There is already an entry on Fox derivatives so I should try to bridge the two topics.