Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJun 30th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexat [[simplicial group]] I added/expanded the section <a href="http://ncatlab.org/nlab/show/simplicial+group#DeloopingAndBundle">delooping and simplicial principal bundles</a> I discuss this in revisionistic terms meant to exhibit the simple general underlying structure, and then try to spell out how it corresponds to vaarious explicit constructions in the literature, trying to point out page and verse in May's "Simplicial objects in algebraic topology" and discuss how that yields what I am discussing. By the way, did anyone ever find the time to make a sanity check of my query-box claim at [[decalage]] that forming decalage in sSet is nothing but forming the standard based path space object?

   at simplicial group I added/expanded the section delooping and simplicial principal bundles

   I discuss this in revisionistic terms meant to exhibit the simple general underlying structure, and then try to spell out how it corresponds to vaarious explicit constructions in the literature, trying to point out page and verse in May’s “Simplicial objects in algebraic topology” and discuss how that yields what I am discussing.

   By the way, did anyone ever find the time to make a sanity check of my query-box claim at decalage that forming decalage in sSet is nothing but forming the standard based path space object?

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeJun 30th 2010
   • (edited Jun 30th 2010)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexDon't worry, Urs -- you're completely sane. At least in this instance ;-) I think maybe your argument can be simplified, but I don't have time to look into this now. Maybe later.

   Don’t worry, Urs – you’re completely sane. At least in this instance ;-)

   I think maybe your argument can be simplified, but I don’t have time to look into this now. Maybe later.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJun 30th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks for the peer review, Todd! This motivated me to make the statement a formal proposition with formal proof at [[decalage]]. Also expanded and edited the entry further, adding in particular an Idea-section that uses this statement to give a good idea of the nature of decalage. I know -- thanks to chat with Danny Stevenson -- precisely one point in the literature where the (evident) observation that for G a simplicial group the thing called $W G$ is nothing but decalage of the thing called $\overline{W} G$ is stated more or less explicitly. That's on page 85 of Duskin's "Simplicial methods". Does anyone know of any other reference that does? I would like to list them withj page and verse on this statement in the nLab etry.

   Thanks for the peer review, Todd!

   This motivated me to make the statement a formal proposition with formal proof at decalage. Also expanded and edited the entry further, adding in particular an Idea-section that uses this statement to give a good idea of the nature of decalage.

   I know – thanks to chat with Danny Stevenson – precisely one point in the literature where the (evident) observation that for G a simplicial group the thing called $W G$ is nothing but decalage of the thing called $\overline{W} G$ is stated more or less explicitly. That’s on page 85 of Duskin’s “Simplicial methods”. Does anyone know of any other reference that does? I would like to list them withj page and verse on this statement in the nLab etry.

4. • CommentRowNumber4.
   • CommentAuthorJohn Baez
   • CommentTimeJun 30th 2010
   • PermaLink
   Author: John Baez
   Format: MarkdownItexHave you tried Tim Porter's enormous electronic tome on simplicial methods? I remember him talking a lot about decalage in his talks at the Barcelona 2-group conference. (It's probably more efficient to ask Tim than to look through that book... but sorry, I'm too lazy to do this.)

   Have you tried Tim Porter’s enormous electronic tome on simplicial methods? I remember him talking a lot about decalage in his talks at the Barcelona 2-group conference.

   (It’s probably more efficient to ask Tim than to look through that book… but sorry, I’m too lazy to do this.)

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJun 30th 2010
   • (edited Jun 30th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> Have you tried Tim Porter's enormous electronic tome on simplicial methods? In [[timporter:crossed menagerie]] the fact that "twisted Cartesian products" are nothig but pullbacks of a universal twisted product is made explicit, but this fact -- maybe with less emphasis or more between the lines -- has been in the literature for long. The literature that I am aware of I listed at <a href="http://ncatlab.org/nlab/show/simplicial+group#SimplicialBundleReferences">simplicial group -- delooping and bundles -- references </a>. But the fact that I am after, that everythig is much simpler even and all the "twisted" structures nothing but explicit components for the abstractly and easily defined homotopy fiber I don't see in the _menagerie_ (though maybe it's there, I haven't read the full thing). The only somewhat intermediate step to this statement in the literature that I am aware of is Duskin's remark on p. 85 of "Simplicial methods..." where he observes that the "universal twisted product" is just the [[decalage]] of $\mathbf{B}G$, which unfortunately is written $\overline{W}G$ in the literature.

   Have you tried Tim Porter’s enormous electronic tome on simplicial methods?

   In crossed menagerie (timporter) the fact that “twisted Cartesian products” are nothig but pullbacks of a universal twisted product is made explicit, but this fact – maybe with less emphasis or more between the lines – has been in the literature for long. The literature that I am aware of I listed at simplicial group – delooping and bundles – references .

   But the fact that I am after, that everythig is much simpler even and all the “twisted” structures nothing but explicit components for the abstractly and easily defined homotopy fiber I don’t see in the menagerie (though maybe it’s there, I haven’t read the full thing).

   The only somewhat intermediate step to this statement in the literature that I am aware of is Duskin’s remark on p. 85 of “Simplicial methods…” where he observes that the “universal twisted product” is just the decalage of $\mathbf{B}G$, which unfortunately is written $\overline{W}G$ in the literature.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJun 30th 2010
   • (edited Jun 30th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItex<a href="http://ncatlab.org/timporter/files/menagerie10.pdf#page=239">page 239</a> mentions the [[fiber sequence]] $G \to W G \to \overline{W}G$...

   page 239 mentions the fiber sequence $G \to W G \to \overline{W}G$…

7. • CommentRowNumber7.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 3rd 2010
   • (edited Jul 3rd 2010)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexUrs, I owe you an apology. It seems there's something a little fishy about the claim after all. (And it teaches me that you have to be careful in the simplicial world, because things happen there that you wouldn't expect from experience with $Top$. I forget this at times.) I discovered the fishiness while following up on my claim that your argument can be simplified. Both sides of the asserted isomorphism $$Dec(Y)(-) \coloneqq Y(1 + -) \cong Y^{\Delta_1} \times_Y Y_0$$ has a left adjoint. (Here $Y_0$ is the discrete simplicial set on the set of 0-simplices.) The left adjoint of decalage is the left Kan extension along yoneda $\Delta \to Set^{\Delta^{op}}$ of $$\Delta \stackrel{1 + -}{\to} \Delta \stackrel{yoneda}{\to} Set^{\Delta^{op}},$$ and the left adjoint of the other side is given by pushout of $$\array{ Y & \stackrel{\{0\} \times 1}{\to} & \Delta_1 \times Y \\ \pi \downarrow & & \\ \pi_0(Y) & & }$$ (because $\Delta_1 \times -$ is left adjoint to $(-)^{\Delta_1}$, and $\pi(-)$ is left adjoint to $(-)_0)$. Since left adjoints preserve colimits and every simplicial set $Y$ is a colimit of representables, we'd just have to show that the left adjoints are isomorphic when restricted to representables $Y = \Delta_{n} = \Delta(-, [n+1])$ (naturally in $n$). Since $\pi_0(\Delta_n) = 1$, we'd just have to show the following is a pushout (and naturally so in $n$). $$\array{ \Delta_n & \stackrel{0 \times id}{\to} & I \times \Delta_n \\ \downarrow & & \downarrow \\ 1 & \to & \Delta_{1+n} }$$ The only problem is that this is false (!). For this to be a pushout, we have to check that it is a pushout objectwise, i.e., that we have a pushout of sets $$\array{ \Delta([m], [1+n]) & \stackrel{0 \times id}{\to} & \Delta([m], [2]) \times \Delta([m], [1+n]) \\ \downarrow & & \downarrow \\ 1 & \to & \Delta([m], [2+n]) }$$ Take $n=1$, $m=2$. If you take cardinalities of sets, you would get a false pushout: $$\array{ 3 & \hookrightarrow & 9 \\ \downarrow & & \downarrow \\ 1 & \to & 6 }$$ because the lower right corner of the pushout ought be $7$, not $6$. I can explain in more geometric language what is happening. In dimension $n=1$, we are taking a simplicial square $I \times I$ and then contracting the edge $0 \times I$ to a point. So this contracts the 1-simplex $(0, 0) \to (0, 1)$ to a point. But the quotient does not identify the 1-simplex $(0, 1) \to (1, 1)$ with the diagonal simplex $(0, 0) \to (1, 1)$. Thus there are four nondegenerate 1-simplices in the pushout, whereas we expect just three in the cone $\Delta_2$. Similarly, there are two nondegenerate 2-simplices in the pushout, but only one in $\Delta_2$. I was pretty shocked when it began to dawn on me that something was wrong, because it upsets something that I have assumed as obviously true *for years*. But I'm pretty sure that the above is error-free.

   Urs, I owe you an apology. It seems there’s something a little fishy about the claim after all. (And it teaches me that you have to be careful in the simplicial world, because things happen there that you wouldn’t expect from experience with $Top$. I forget this at times.) I discovered the fishiness while following up on my claim that your argument can be simplified.

   Both sides of the asserted isomorphism

   $$Dec(Y)(-) \coloneqq Y(1 + -) \cong Y^{\Delta_1} \times_Y Y_0$$

   has a left adjoint. (Here $Y_0$ is the discrete simplicial set on the set of 0-simplices.) The left adjoint of decalage is the left Kan extension along yoneda $\Delta \to Set^{\Delta^{op}}$ of

   $$\Delta \stackrel{1 + -}{\to} \Delta \stackrel{yoneda}{\to} Set^{\Delta^{op}},$$

   and the left adjoint of the other side is given by pushout of

   $$\array{ Y & \stackrel{\{0\} \times 1}{\to} & \Delta_1 \times Y \\ \pi \downarrow & & \\ \pi_0(Y) & & }$$

   (because $\Delta_1 \times -$ is left adjoint to $(-)^{\Delta_1}$, and $\pi(-)$ is left adjoint to $(-)_0)$. Since left adjoints preserve colimits and every simplicial set $Y$ is a colimit of representables, we’d just have to show that the left adjoints are isomorphic when restricted to representables $Y = \Delta_{n} = \Delta(-, [n+1])$ (naturally in $n$). Since $\pi_0(\Delta_n) = 1$, we’d just have to show the following is a pushout (and naturally so in $n$).

   $$\array{ \Delta_n & \stackrel{0 \times id}{\to} & I \times \Delta_n \\ \downarrow & & \downarrow \\ 1 & \to & \Delta_{1+n} }$$

   The only problem is that this is false (!). For this to be a pushout, we have to check that it is a pushout objectwise, i.e., that we have a pushout of sets

   $$\array{ \Delta([m], [1+n]) & \stackrel{0 \times id}{\to} & \Delta([m], [2]) \times \Delta([m], [1+n]) \\ \downarrow & & \downarrow \\ 1 & \to & \Delta([m], [2+n]) }$$

   Take $n=1$, $m=2$. If you take cardinalities of sets, you would get a false pushout:

   $$\array{ 3 & \hookrightarrow & 9 \\ \downarrow & & \downarrow \\ 1 & \to & 6 }$$

   because the lower right corner of the pushout ought be $7$, not $6$.

   I can explain in more geometric language what is happening. In dimension $n=1$, we are taking a simplicial square $I \times I$ and then contracting the edge $0 \times I$ to a point. So this contracts the 1-simplex $(0, 0) \to (0, 1)$ to a point. But the quotient does not identify the 1-simplex $(0, 1) \to (1, 1)$ with the diagonal simplex $(0, 0) \to (1, 1)$. Thus there are four nondegenerate 1-simplices in the pushout, whereas we expect just three in the cone $\Delta_2$. Similarly, there are two nondegenerate 2-simplices in the pushout, but only one in $\Delta_2$.

   I was pretty shocked when it began to dawn on me that something was wrong, because it upsets something that I have assumed as obviously true for years. But I’m pretty sure that the above is error-free.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJul 4th 2010
   • (edited Jul 4th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks, Todd, of course you are right. Silly me. I'll fix entry as soon as I have a minute...

   Thanks, Todd, of course you are right. Silly me. I’ll fix entry as soon as I have a minute…

9. • CommentRowNumber9.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 4th 2010
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexHa, you think *you* feel silly? I wonder if I'll be able to track down all the places I've been spreading the lie myself! (And my erroneous belief was so deeply ingrained that it took me quite some time to be convinced it really was a mistake.) Don't feel silly. I think in fact it's a very easy trap to fall into. (Not long ago there was an MO thread on falling into mathematical traps...) While I'm in a confessional mood, I'll mention that last night I fixed a mistake I had perpetrated quite a while ago at [[connected space]]. I had long suspected there was a mistake there, and was just about to break down and ask MO for a counterexample, when I finally managed to find one.

   Ha, you think you feel silly? I wonder if I’ll be able to track down all the places I’ve been spreading the lie myself! (And my erroneous belief was so deeply ingrained that it took me quite some time to be convinced it really was a mistake.)

   Don’t feel silly. I think in fact it’s a very easy trap to fall into. (Not long ago there was an MO thread on falling into mathematical traps…)

   While I’m in a confessional mood, I’ll mention that last night I fixed a mistake I had perpetrated quite a while ago at connected space. I had long suspected there was a mistake there, and was just about to break down and ask MO for a counterexample, when I finally managed to find one.

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJul 4th 2010
    • PermaLink
    Author: Urs
    Format: MarkdownItexThanks, Todd, but I feel I should have known better. But also I feel, most mistakes are there so that we learn somehting. Here is still something to be learned I think: after all, the main point I found noteworthy was 1. decalage is a fibration replacemnt of the morphism $X_0 \to X$ for fibrant $X$; 1. there is a canonical such fibration replacement, namely $X^I \times_X X_0$. Then I concluded erroneously that both replacments are in fact isomorphic. But as we have seen now, in fact decalage gives a _smaller_ replacement. For the moment I just rolled back the entry. Eventually I would like to restorre a corrected version of my last expanded version with a discussion of ho decalage factorizes the standard fibration replacement.

    Thanks, Todd, but I feel I should have known better.

    But also I feel, most mistakes are there so that we learn somehting. Here is still something to be learned I think:

    after all, the main point I found noteworthy was

    1. decalage is a fibration replacemnt of the morphism $X_0 \to X$ for fibrant $X$;

    2. there is a canonical such fibration replacement, namely $X^I \times_X X_0$.

    Then I concluded erroneously that both replacments are in fact isomorphic. But as we have seen now, in fact decalage gives a smaller replacement.

    For the moment I just rolled back the entry. Eventually I would like to restorre a corrected version of my last expanded version with a discussion of ho decalage factorizes the standard fibration replacement.

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJul 20th 2010
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded to [[simplicial group]] the statement that for a principal action of a simplicial group $G$ on a Kan complex P, the quotient map $$ P \to X := P/G $$ is always a Kan fibration.

    added to simplicial group the statement that for a principal action of a simplicial group $G$ on a Kan complex P, the quotient map

    $$P \to X := P/G$$

    is always a Kan fibration.

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeAug 10th 2011
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have added statement and proof of the observation that for any simplicial group $G$ there is a fiber sequence of Kan complexes $$ G_0 \to G \to B \Omega G/G_0 $$ in [simplicial group -- properties -- fiber sequences](http://ncatlab.org/nlab/show/simplicial+group#FiberSequences).

    I have added statement and proof of the observation that for any simplicial group $G$ there is a fiber sequence of Kan complexes

    $$G_0 \to G \to B \Omega G/G_0$$

    in simplicial group – properties – fiber sequences.

13. • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeSep 14th 2011
    • PermaLink
    Author: Urs
    Format: MarkdownIn the References-section at [[simplicial group]] I have added two more items for John Moore with the original proof that simplicial groups are Kan, using the information that is being thrown around currently on the CatTheory mailing list

    In the References-section at simplicial group I have added two more items for John Moore with the original proof that simplicial groups are Kan, using the information that is being thrown around currently on the CatTheory mailing list

14. • CommentRowNumber14.
    • CommentAuthorzskoda
    • CommentTimeSep 14th 2011
    • PermaLink
    Author: zskoda
    Format: MarkdownItexSo 1954 is the year of the theorem...

    So 1954 is the year of the theorem…

15. • CommentRowNumber15.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 14th 2011
    • PermaLink
    Author: jim_stasheff
    Format: Text@urs: one point in the literature where the (evident) observation that for G a simplicial group the thing called WG is nothing but decalage of the thing called W¯G is stated more or less explicitly and the following discussion - if I only knew what decalage meant! I know - I can follow the links had I but world enough and time

    @urs: one point in the literature where the (evident) observation that for G a simplicial group the thing called
    WG is nothing but decalage of the thing called
    W¯G is stated more or less explicitly
    
    and the following discussion - if I only knew what decalage meant! I know - I can follow the links
    had I but world enough and time
16. • CommentRowNumber16.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 15th 2011
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexThe decalage is the simplicial version of the path space. So $WG = Dec \overline{W}G$ is morally like $PBG$, the based path space of the classifying space. Except that $Dec \overline{W}G$ is a simplicial group (well, not explicitly - it is iso to the underlying simplicial set of a simplicial group), with $G$ as a subgroup and $WG/G = \overline{W}G$.

    The decalage is the simplicial version of the path space. So $WG = Dec \overline{W}G$ is morally like $PBG$, the based path space of the classifying space. Except that $Dec \overline{W}G$ is a simplicial group (well, not explicitly - it is iso to the underlying simplicial set of a simplicial group), with $G$ as a subgroup and $WG/G = \overline{W}G$.

17. • CommentRowNumber17.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 15th 2011
    • PermaLink
    Author: jim_stasheff
    Format: TextThe example was clear but I had to find out what (mathematical) decalage meant by following the link will now consult a French dictionary for the street meaning \bar WG is no more unfortunate than any other historical notations cf BA only the full two sided notation B(M,A,N) is unambiguous jim

    The example was clear but I had to find out what (mathematical) decalage meant by following the link
    will now consult a French dictionary for the street meaning
    
    \bar WG is no more unfortunate than any other historical notations
    cf BA only the full two sided notation B(M,A,N) is unambiguous
    
    jim
18. • CommentRowNumber18.
    • CommentAuthordomenico_fiorenza
    • CommentTimeFeb 24th 2014
    • PermaLink
    Author: domenico_fiorenza
    Format: MarkdownItexAt [[simplicial group]] I can't find what I thought was the basic example, i.e., the [[singular simplicial complex]] of a [[topological group]]. so maybe I'm wrong and that is actually not an example? or have I missed that example from the page on simplicial groups?

    At simplicial group I can’t find what I thought was the basic example, i.e., the singular simplicial complex of a topological group. so maybe I’m wrong and that is actually not an example? or have I missed that example from the page on simplicial groups?

19. • CommentRowNumber19.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 24th 2014
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexIt should be an example because singularization is a right adjoint and therefore preserves the finite product diagrams that express something as a topological group.

    It should be an example because singularization is a right adjoint and therefore preserves the finite product diagrams that express something as a topological group.

20. • CommentRowNumber20.
    • CommentAuthorMike Shulman
    • CommentTimeJan 2nd 2015
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexDoes anyone know to what extent the simplicial theory of things like $W G$ and $\overline{W} G$ can be extended from simplicial groups to grouplike simplicial monoids?

    Does anyone know to what extent the simplicial theory of things like $W G$ and $\overline{W} G$ can be extended from simplicial groups to grouplike simplicial monoids?

21. • CommentRowNumber21.
    • CommentAuthorMike Shulman
    • CommentTimeJan 2nd 2015
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexIn particular, I'm wondering about a simplicial version of May's theorem 7.6 from *Classifying spaces and fibrations* that if $G$ is a grouplike topological monoid, then $E G\times_{B G} X = B(*,G,X) \to B G$ is a quasifibration.

    In particular, I’m wondering about a simplicial version of May’s theorem 7.6 from Classifying spaces and fibrations that if $G$ is a grouplike topological monoid, then $E G\times_{B G} X = B(*,G,X) \to B G$ is a quasifibration.

22. • CommentRowNumber22.
    • CommentAuthorZhen Lin
    • CommentTimeJan 2nd 2015
    • PermaLink
    Author: Zhen Lin
    Format: MarkdownItexThe proof of Quillen's theorem B uses a lemma (cf Lemma 5.7 in [Goerss and Jardine, Ch. IV]), that essentially says that $B (*, \mathcal{C}, X) \to N \mathcal{C}$ is a quasifibration for any category $\mathcal{C}$ and any _homotopically constant_ diagram $X : \mathcal{C} \to sSet$. It seems to me that this generalises to simplicially enriched categories $\mathcal{C}$ as well.

    The proof of Quillen’s theorem B uses a lemma (cf Lemma 5.7 in [Goerss and Jardine, Ch. IV]), that essentially says that $B (*, \mathcal{C}, X) \to N \mathcal{C}$ is a quasifibration for any category $\mathcal{C}$ and any homotopically constant diagram $X : \mathcal{C} \to sSet$. It seems to me that this generalises to simplicially enriched categories $\mathcal{C}$ as well.

23. • CommentRowNumber23.
    • CommentAuthorMike Shulman
    • CommentTimeJan 2nd 2015
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexAnd, of course, any action by a *grouplike* monoid is homotopically constant. Thanks!

    And, of course, any action by a grouplike monoid is homotopically constant. Thanks!

24. • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeJul 4th 2021
    • PermaLink
    Author: Urs
    Format: MarkdownItexWhere Moore's theorem is stated ([here](https://ncatlab.org/nlab/show/simplicial+group#EverySimplicialGroupIsAKanComplex)), I added more pointers to references, such as to * [[Edward B. Curtis]], Section 3 of: _Simplicial homotopy theory_, Advances in Mathematics 6 (1971) 107–209 (<a href="https://doi.org/10.1016/0001-8708(71)90015-6">doi:10.1016/0001-8708(71)90015-6</a>, [MR279808](http://www.ams.org/mathscinet-getitem?mr=279808)) and also pointer to where in Weibel's book this is stated and proven. <a href="https://ncatlab.org/nlab/revision/diff/simplicial+group/64">diff</a>, <a href="https://ncatlab.org/nlab/revision/simplicial+group/64">v64</a>, <a href="https://ncatlab.org/nlab/show/simplicial+group">current</a>

    Where Moore’s theorem is stated (here), I added more pointers to references, such as to

    • Edward B. Curtis, Section 3 of: Simplicial homotopy theory, Advances in Mathematics 6 (1971) 107–209 (doi:10.1016/0001-8708(71)90015-6, MR279808)

    and also pointer to where in Weibel’s book this is stated and proven.

    diff, v64, current

25. • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeMay 14th 2023
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to * [[Daniel Quillen]], II §3.8 of: *[[Homotopical Algebra]]*, Lecture Notes in Mathematics **43**, Springer (1967) &lbrack;[doi:10.1007/BFb0097438](https://doi.org/10.1007/BFb0097438)&rbrack; for another account of Moore's theorem <a href="https://ncatlab.org/nlab/revision/diff/simplicial+group/68">diff</a>, <a href="https://ncatlab.org/nlab/revision/simplicial+group/68">v68</a>, <a href="https://ncatlab.org/nlab/show/simplicial+group">current</a>

    added pointer to

    • Daniel Quillen, II §3.8 of: Homotopical Algebra, Lecture Notes in Mathematics 43, Springer (1967) [doi:10.1007/BFb0097438]

    for another account of Moore’s theorem

    diff, v68, current

26. • CommentRowNumber26.
    • CommentAuthorTim_Porter
    • CommentTimeJul 24th 2023
    • PermaLink
    Author: Tim_Porter
    Format: MarkdownItexChanged some slightly awkward wording. <a href="https://ncatlab.org/nlab/revision/diff/simplicial+group/73">diff</a>, <a href="https://ncatlab.org/nlab/revision/simplicial+group/73">v73</a>, <a href="https://ncatlab.org/nlab/show/simplicial+group">current</a>

    Changed some slightly awkward wording.

    diff, v73, current