Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
I have added to Postnikov tower paragraphs on the relative version, (definition and construction in simplicial sets).
I also added the remark that the relative Postnikov tower is the tower given by the (n-connected, n-truncated) factorization system as varies, hence is the tower of n-images of a map in . And linked back from these entries.
I have added that
In Goerss-Jardine there is a nice model for the relative Postnikov sections of a fibration of simplicial sets, which is reproduced in the entry.
One may also ask the question:
Given a chain map between chain complexes, what is its factorization such that under Dold-Kan this models a given relative Postnikov stage of the corresponding simplicial map of Kan complexes?
I suppose the following works: Given a chain map
then its -image factorization
is modeled by
where
and
with the maps to and from it the obvious ones.
This is elementary and straightforward checking, unless I am making a simple mistake.
What’s a citable reference for this?
Looks plausible, but I don’t recall having seen this written out.
Okay, thanks.
That abelian group
of course has a slicker expression:
but I found the coproduct expression more useful for checking that the maps all work out (hopefully).
There was a mistake left in #4. The following should work:
Let be a chain map between chain complexes and let .
Then the following diagram of abelian groups commutes:
Moreover, the middle vertical sequence is a chain complex , and hence the diagram gives a factorization of into two chain maps
This is a model for the (n+1)-image factorization of in that on homology groups the following holds:
are isomorphisms;
is the image factorization of ;
are isomorphisms.
1 to 7 of 7