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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 21st 2012
   • (edited Nov 21st 2012)
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   Author: Urs
   Format: MarkdownItexI have added to _[[Postnikov tower]]_ paragraphs on the _relative_ version, (_[definition](http://ncatlab.org/nlab/show/Postnikov+system#ForSimplicialSetsRelativeVersion)_ and _[construction](http://ncatlab.org/nlab/show/Postnikov+system#RelativePostnikovTowerConstruction)_ 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.

   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.

2. • CommentRowNumber2.
   • CommentAuthorjim_stasheff
   • CommentTimeNov 22nd 2012
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   Author: jim_stasheff
   Format: TextThere 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.

   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.
3. • CommentRowNumber3.
   • CommentAuthorTim_Porter
   • CommentTimeNov 22nd 2012
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   Author: Tim_Porter
   Format: MarkdownItexI have added that

   I have added that

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMar 15th 2017
   • (edited Mar 16th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexIn 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?

   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?

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeMar 16th 2017
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   Author: Mike Shulman
   Format: MarkdownItexLooks plausible, but I don't recall having seen this written out.

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

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeMar 16th 2017
   • (edited Mar 16th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexOkay, 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).

   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).

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeMar 16th 2017
   • (edited Mar 16th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThere 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]] [[commuting diagram|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-image|(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]]; 1. $H_n(V) \to H_n(im_{n+1}(f)) \hookrightarrow H_n(W)$ is the [[image|image factorization]] of $H_n(f)$; 1. $H_{\bullet \gt n}(im_{n+1}(f)) \overset{\simeq}{\to} H_{\bullet \gt n}(W)$ are [[isomorphisms]].

   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.