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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 21st 2012
• (edited Nov 21st 2012)

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 $n$ varies, hence is the tower of n-images of a map in $\infty Grpd$. And linked back from these entries.

• CommentRowNumber2.
• CommentAuthorjim_stasheff
• CommentTimeNov 22nd 2012
There shuld also be reference to the old name Moore-Postnikov systems etc.

J. C. MOORE, Semisimplicial complexes and Postnikov systems, Symposium Internacional
de Topologia Algebraica, Mexico City, 1958, pp. 232-247.
• CommentRowNumber3.
• CommentAuthorTim_Porter
• CommentTimeNov 22nd 2012

I have added that

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeMar 15th 2017
• (edited Mar 16th 2017)

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

$V_\bullet \overset{f_\bullet}{\longrightarrow} W_\bullet$

then its $n+1$-image factorization

$V_\bullet \longrightarrow im_{n+1}(f_\bullet) \longrightarrow W_\bullet$

is modeled by

$\array{ \vdots && \vdots && \vdots \\ \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\mathrlap{\partial_{W}}} && \downarrow^{\mathrlap{\partial_{W}}} \\ V_{n+2} &\overset{f_{n+2}}{\longrightarrow}& W_{n+2} &\overset{=}{\longrightarrow}& W_{n+2} \\ \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\mathrlap{\partial_{W}}} && \downarrow^{\mathrlap{\partial_{W}}} \\ V_{n+1} &\overset{f_{n+1}}{\longrightarrow}& Y &\overset{}{\longrightarrow}& W_{n+1} \\ \downarrow^{\mathrlap{\partial_{V}}} && \downarrow &(pb)& \downarrow^{\mathrlap{\partial_{W}}} \\ V_n &\longrightarrow& X &\longrightarrow& W_n \\ \downarrow^{\mathrlap{\partial_V}} && \downarrow && \downarrow^{\mathrlap{\partial_W}} \\ V_{n-1} &\overset{=}{\longrightarrow}& V_{n-1} &\overset{f_{n-1}}{\longrightarrow}& W_{n-1} \\ \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\mathrlap{\partial_W}} \\ V_{n-2} &\overset{=}{\longrightarrow}& V_{n-2} &\overset{f_{n-2}}{\longrightarrow}& W_{n-2} \\ \\ \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\mathrlap{\partial_W}} \\ \vdots && \vdots && \vdots }$

where

$X \coloneqq \underset{v_{n-1} \in V_{n-1}}{\sqcup} \left\{ f_n(v_n) \vert \partial v_n = v_{n-1} \right\}$

and

$Y \coloneqq \{w_{n+1} \vert \partial_W w_{n+1} = f(a), \partial_V a = 0 \}$

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?

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeMar 16th 2017

Looks plausible, but I don’t recall having seen this written out.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeMar 16th 2017
• (edited Mar 16th 2017)

Okay, thanks.

That abelian group

$X \coloneqq \underset{v_{n-1} \in V_{n-1}}{\sqcup} \left\{ f_n(v_n) \vert \partial v_n = v_{n-1} \right\}$

of course has a slicker expression:

$X \simeq coker\left( ker(\partial_V) \cap ker(f_n) \to V_n \right)$

but I found the coproduct expression more useful for checking that the maps all work out (hopefully).

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeMar 16th 2017
• (edited Mar 16th 2017)

There was a mistake left in #4. The following should work:

Let $f_\bullet \colon V_\bullet \longrightarrow W_\bullet$ be a chain map between chain complexes and let $n \in \mathbb{N}$.

Then the following diagram of abelian groups commutes:

$\array{ \vdots && \vdots && \vdots \\ \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\mathrlap{\partial_{W}}} && \downarrow^{\mathrlap{\partial_{W}}} \\ V_{n+2} &\overset{f_{n+2}}{\longrightarrow}& W_{n+2} &\overset{=}{\longrightarrow}& W_{n+2} \\ \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\mathrlap{ \partial_W } } && \downarrow^{\mathrlap{\partial_{W}}} \\ V_{n+1} &\overset{f_{n+1}}{\longrightarrow}& \left\{ w_{n+1} | \exists v_n \colon \partial w_{n+1} = f_n(v_n), \partial_V v_n = 0, \right\} &{\longrightarrow}& W_{n+1} \\ \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\partial_W} && \downarrow^{\mathrlap{\partial_{W}}} \\ V_n &\overset{ (f_n, \partial_V) }{\longrightarrow}& \underset{v_{n-1}}{\sqcup} \left\{ f_n(v_n) \vert \partial_V v_n = v_{n-1} \right\} &\overset{ }{\longrightarrow}& W_n \\ \downarrow^{\mathrlap{\partial_V}} && \downarrow^{\mathrlap{(f_n(v_n),\partial_V v_n) \mapsto \partial_V v_n}} && \downarrow^{\mathrlap{\partial_W}} \\ V_{n-1} &\overset{=}{\longrightarrow}& V_{n-1} &\overset{f_{n-1}}{\longrightarrow}& W_{n-1} \\ \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\mathrlap{\partial_W}} \\ V_{n-2} &\overset{=}{\longrightarrow}& V_{n-2} &\overset{f_{n-2}}{\longrightarrow}& W_{n-2} \\ \\ \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\mathrlap{\partial_W}} \\ \vdots && \vdots && \vdots }$

Moreover, the middle vertical sequence is a chain complex $im_{n+1}(f)_\bullet$, and hence the diagram gives a factorization of $f_\bullet$ into two chain maps

$f_\bullet \;\colon\; V_\bullet \longrightarrow im_{n+1}(f)_\bullet \longrightarrow W_\bullet \,.$

This is a model for the (n+1)-image factorization of $f$ in that on homology groups the following holds:

1. $H_{\bullet \lt n}(V) \overset{\simeq}{\to} H_{\bullet \lt n}(im_{n+1}(f))$ are isomorphisms;

2. $H_n(V) \to H_n(im_{n+1}(f)) \hookrightarrow H_n(W)$ is the image factorization of $H_n(f)$;

3. $H_{\bullet \gt n}(im_{n+1}(f)) \overset{\simeq}{\to} H_{\bullet \gt n}(W)$ are isomorphisms.