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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorHurkyl
• CommentTimeJun 19th 2022
• (edited Jun 19th 2022)

Should this topic be renamed to something like “$E_\infty$ group” or some similar thing? I haven’t seen “abelian” used elsewhere to describe this notion.

IMO that choice of name is potentially misleading. For example, it could also refer to a model of the usual finite product theory of abelian groups: i.e. an object of the $\infty$-category of by connective chain complexes of abelian groups modulo quasi-isomorphism. This is actually specifically what I would have expected from the term.

This example is, in some sense, also “more commutative” than being a grouplike $E_\infty$ monoid, which makes the description of being “maximally abelian” misleading as well.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJun 19th 2022

Yes, the entry should clarify this. I have expanded out as follows:

An ordinary group is either an abelian group or not. For an ∞-group there is an infinite tower of notions ranging from completely general non-abelian ∞-groups, over braided $\infty$-groups, sylleptic $\infty$-groups …, to ever more abelian groups By an abelian ∞-group (not an established term) one may want to mean an ∞-group which is maximally abelian, in this sense.

Technically, the level of abelianness may be encoded (see at May recognition theorem) by the $E_n$-operads as $n$ ranges from 1 to $\infty$: On the non-abelian end, a general ∞-group is equivalently a groupal algebra over $E_1$, also known as the associative operad, hence is a groupal A-∞ algebra; while at the abelian end a groupal $E_\infty$-space is a connective spectrum.

Notice that referring to connective spectra as “abelian $\infty$-groups” (which is not standard) matches the established terminology for non-abelian cohomology (which is standard): The qualifier “non-abelian” in non-abelian cohomology is in contrast to Whitehead-generalized cohomology theories which are represented by spectra.

In a more restrictive sense one may say that plain abelian cohomology is just ordinary cohomology theory, subsuming only those Whitehead-generalized cohomology theories which are represented specifically by Eilenberg-MacLane spectra. Under the Dold-Kan correspondence these are equivalently chain complexes of abelian groups. One may think of these as being yet more commutative than general spectra and might want to reserve the term “abelian $\infty$-group” for them.