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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 nn varies, hence is the tower of n-images of a map in Grpd\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 f W V_\bullet \overset{f_\bullet}{\longrightarrow} W_\bullet

    then its n+1n+1-image factorization

    V im n+1(f )W V_\bullet \longrightarrow im_{n+1}(f_\bullet) \longrightarrow W_\bullet

    is modeled by

    V W W V n+2 f n+2 W n+2 = W n+2 V W W V n+1 f n+1 Y W n+1 V (pb) W V n X W n V W V n1 = V n1 f n1 W n1 V V W V n2 = V n2 f n2 W n2 V V W \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 }


    Xv n1V n1{f n(v n)|v n=v n1} X \coloneqq \underset{v_{n-1} \in V_{n-1}}{\sqcup} \left\{ f_n(v_n) \vert \partial v_n = v_{n-1} \right\}


    Y{w n+1| Ww n+1=f(a), Va=0} 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

    Xv n1V n1{f n(v n)|v n=v n1} 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:

    Xcoker(ker( V)ker(f n)V n) 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 :V W f_\bullet \colon V_\bullet \longrightarrow W_\bullet be a chain map between chain complexes and let nn \in \mathbb{N}.

    Then the following diagram of abelian groups commutes:

    V W W V n+2 f n+2 W n+2 = W n+2 V W W V n+1 f n+1 {w n+1|v n:w n+1=f n(v n), Vv n=0,} W n+1 V W W V n (f n, V) v n1{f n(v n)| Vv n=v n1} W n V (f n(v n), Vv n) Vv n W V n1 = V n1 f n1 W n1 V V W V n2 = V n2 f n2 W n2 V V W \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) im_{n+1}(f)_\bullet, and hence the diagram gives a factorization of f f_\bullet into two chain maps

    f :V im n+1(f) W . 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 ff in that on homology groups the following holds:

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

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

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

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