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I further expanded Moore complex (more propositions on normalization and on homotopy groups). Also tried to polish it a bit more.
I have tried to polish the section on the hypercrossed complex structure at Moore complex.
Tim, please have a look.
I think that there is a conflict of terminology. In the section on hypercrossed complexes, the entry says that the Moore complex is not a chain complex of groups although further up the page it is said to be one. Anyhow chain complex in its entry requires the ambient category to be preadditive. so the Moore complex, in that light, would not be a chain complex. This is a pity!!!! I think the better terminology is to say a chain complex of groups is a graded group with a differential. Then the Moore complex is a chain complex of groups. which need not be abelian.
From the point of view of the Moore complex, it would probably be better to redistribute this section largely to the entry on hypercrossed complexes, as there it can be developed more fully. Here in Moore complex, the mention could be a bit more limited. It is hidden at the bottom and is very unlike the abelian case so perhaps should be restructured. The entry on hypercrossed complexes does need a lot of work!
I may come back to this later this afternoon, but must see to something else for a while.
Anyhow chain complex in its entry requires the ambient category to be preadditive. so the Moore complex, in that light, would not be a chain complex. This is a pity!!!!
I have changed at chain complex the condition to “has a zero morphism”. At zero morphism there is a discussion that $(Set_*, smash)$-enrichment is sufficient.
I also removed at Moore complex the statement that it is not a chain complex in the non-abelian case.
From the point of view of the Moore complex, it would probably be better to redistribute this section largely to the entry on hypercrossed complexes,
I don’t think there is much to be re-distributed, but if you find time to add to hypercrossed complex, that would be appreciated.
Great.
I will try to do that, then we can see if we want to thin that discussion at Moore complex a bit.
There is a related entry at normal complex of groups. I have added a line to this effect.
Actually crossed complex looks to need some TLC. I think some material is (essentially) duplicated there, and the structure is strange.
I have tidied up Moore complex a bit more. I still feel that the non-abelian case may be better off being separated. Incidently several of the abelian results are special cases of their non-abelian counterparts.
Thanks.
I am hesitant to separate the entry into two, because the term “Moore complex” is used by many authors specifically for the abelian case (Goerss-Jardine, notably). We’d be in danger of getting in a messy redirect situation if we split the entry.
But once we have more results on the nonabelian case listed, we could restructure the entry, with one big section on the abelian, and one big section on the nonabelian case.
Good point about the redirects.
I have done a bit of work on crossed complex, but not yet to resolve the structural problem. (Examples in definitions, examples not in the section labelled examples etc. Don’t worry I will have a look at it and try to make some headway. (I should also do some work at 2-crossed complexes and related entries.
I have worked a bit on streamlining the sections
for readability and pedagogical purposes. For instance I have split the lemma that the differentials are all well defined into two (one for the alternating chain complex, one for its reduction modulo degeneracies), and I have given the notation for the subgroup generated from the degenerace cells its own numbered definition. Finally I made explicit as a corollary that, by 2-out-of-3, the projection of the alternating face chain complex onto the quotient by the degenerates is a quasi-isomorphism.
Finally I have started an Examples section and added a discussion of the chains on the 1-simplex.
Have also made the statement about the degenerate cells splitting off as a direct summand more explicit in a corollary.
added pointer to:
also spelled out (here) the example of the induced dg-algebra structure on $N_\bullet \, \mathbb{Z}[W \mathbb{Z}_2]$
(which could be simplified further if there is a good answer to this question)
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