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nLab > Latest Changes: Dold-Kan correspondence

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 17th 2009
    • (edited Nov 30th 2012)
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 14th 2010

    added to Dold-Kan correspondence a brief section on how it yields Quillen equivalences of the natural model category structures in the game

    (I thought I had written something along these lines before somewhere, but maybe I didn’t…)

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJul 27th 2010
    • (edited Jul 27th 2010)

    added to Dold-Kan corresponden

    I did this mainly to record Richard Garner’s argument. But in fact I feel that Ronnie Brown’s argument is a little more powerful, as it also factors through the nonabelian version.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 4th 2010
    • (edited Nov 4th 2010)

    tried to brush-up the entry Dold-Kan correspondence. Here are some things I did:

    • rewrote the Idea-section, trying to make it be more forcefully to the point

    • split the Statement-section into subsection “Equivalence of categories” and “Quillen equivalence of model categories”

    • at “Equivalence of categories” I removed a detailed discussion of the normalized chains complex functor itself. This material I instead copied over to Moore complex, where it belongs. In that entry, I split the Definition-seciton into one for general simplicial groups and one for abelian simplicial groups

    • removed all the material in the subsection “Monoidal version” and instead added a pointer there to monoidal Dold-Kan correspondence

    • this means I in particular have deleted the quote from Kathryn Hess there, which it said “should be worked into the entry”. Instead I have taken care that all the information that was in this quote is now at “monoidal Dold-Kan correspondence”. Notably there is a detailed list of literature and unpublished results and attributions of results there.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 4th 2010

    I tried to collect the three subsections on globular and on nonabelian versions and the section describing the connection to strict groupal oo-groupoids

    I wasn’t sure how to do it and ended up deciding to collect them in a subsection titled Statement (general nonabelian case). But momentarily that does not quite live up to its title yet.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeNov 4th 2010

    added to the section Equivalence of categories the gory details of the components of the natural isomorphisms

    Id→NΓ Id \to N \Gamma

    and

    ΓN→Id. \Gamma N \to Id \,.

    Does anyone know a source that checks explicitly that this makes the equivalence an adjoint equivalence ?

    Of course it’s straightforward to check. But tedious.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeNov 4th 2010
    • (edited Nov 4th 2010)

    I have cited Weibel, exercise 8.4.2 now for the statement (at the end of DK – equivalence of categories).

    Then I started a section Applications. So far it contains: construction of Eilenberg-MacLane objects and embedding of abelian sheaf cohomology into nonabelian cohomology

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJun 27th 2011

    have recoded the statement that the Dold-Kan correspondence is compatible with W¯\bar W-delooping of simplicial abelian groups in a new Properties-section Looping and delooping

    • CommentRowNumber9.
    • CommentAuthorjim_stasheff
    • CommentTimeJun 28th 2011
    Moore complex - and no citation of Moore!
    and not even an indication of which Moore!!
    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJun 28th 2011

    and no citation of Moore

    Can you provide it?

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJun 30th 2011

    Hi Jim,

    I have added your Alg-Top findings to Moore complex

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeNov 30th 2012
    • (edited Nov 30th 2012)

    I see that at Dold-Kan correspondence somebody had added a very first sentence to read

    In some sense the Dold-Kan correspondence is the categorification of the statement that ℕ and ℤ are isomorphic in the category of sets in that they have the same cardinality.

    I don’t understand this sentence. I think it needs to come with more explanation.

    Also I think this is unsuited for the very first sentence of the Idea section. It’s more like a Zen koan than an explanation of an idea.

    I have moved the sentence now to [Properties – Relation to categorification](http://ncatlab.org/nlab/show/Dold-Kan correspondence#RelationToCategorification). There sombody please expand on the details alluded to by the sentence.

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeDec 1st 2012

    I don’t suppose you noticed who that was? It doesn’t make any sense to me.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeDec 2nd 2012
    • (edited Dec 2nd 2012)

    I hesitate to check the History-list, because it can be very tedious to find the point where something was added (the quickest general method I can think of is to do a binary search on the length of the history list, last time I had to do this it took me forever…).

    But now I did and in this case I was lucky: it was Stephan Spahn (or somebody signing with his name….), in the latest edit before mine. I’ll check with him now.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeDec 2nd 2012

    I have removed the sentence for the moment. If Stephan or somebody later remembers some story similar to this sentence which is worth recording, we can still do so.

  16. Hi Urs,

    in reply to your comment number 6: what about the following?

    In Kan’s original paper (Functors involving css complexes), Proposition 6.3., he proves (if I managed to untangle the notations correctly), that the usual normalized Moore functor NN is naturally isomorphic to the functor tensor product −⊗ ΔNℤ[Δ •]-\otimes_\Delta N\mathbb{Z}[\Delta^\bullet], where ℤ[Δ •]\mathbb{Z}[\Delta^\bullet] denotes the free simplicial abelian group on the Yoneda embedding.

    Then by abstract nonsense we arrive (Kan arrived, in that paper) at the “hom” expression for the right adjoint Γ\Gamma.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeJun 23rd 2016

    Thanks for getting back to this. I don’t have free resources at the moment to look into this. But if you are sure, please add a corresponding remark to the entry!

  18. I would like to be sure that I untangled Kan’s notations correctly before adding the remark, though…

    At any rate, the article is available here.

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeSep 12th 2020

    added pointer to

    diff, v105, current

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeSep 14th 2020

    added pointer to

    • J. F. Jardine, Lemma 1.5 in: Presheaves of chain complexes, K-theory 30.4 (2003): 365-420 (pdf)

    diff, v108, current

    • CommentRowNumber21.
    • CommentAuthorTim_Porter
    • CommentTimeSep 19th 2020

    Created some links on relevant names.

    diff, v110, current

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeJul 12th 2021

    I forget if I had an argument for this:

    Is the Dold-Kan correspondence N:sAb⇄Ch • +:DK N \colon sAb \rightleftarrows Ch^+_\bullet \colon DK compatible with internal homs, in that the following is a simplicial weak equivalence

    [S,frgt∘DK(V)] • ≃sSet(S×Δ[•],frgt∘DK(V)) ≃Ch • +(N∘ℤ(S×Δ[•]),V) →EZCh • +(N∘ℤ(S)⊗N∘ℤ(Δ[•]),V) ≃Ch • +(N∘ℤ(Δ[•]),[N∘ℤ(S),V]) ≃(frgt∘DK([N∘ℤ(S),V])) • \begin{aligned} \big[ S, \, frgt \circ DK(V) \big]_\bullet & \;\simeq\; sSet \big( S \times \Delta[\bullet], \, frgt \circ DK(V) \big) \\ & \;\simeq\; Ch_\bullet^+ \big( N \circ \mathbb{Z}(S \times \Delta[\bullet]), \, V \big) \\ & \;\xrightarrow{EZ}\; Ch_\bullet^+ \big( N \circ \mathbb{Z}(S) \,\otimes\, N \circ \mathbb{Z}(\Delta[\bullet]), \, V \big) \\ & \;\simeq\; Ch_\bullet^+ \big( N \circ \mathbb{Z}(\Delta[\bullet]), \, [ N \circ \mathbb{Z}(S), \, V ] \big) \\ & \;\simeq\; \Big( frgt \circ DK \big( [ N \circ \mathbb{Z}(S), \, V ] \big) \Big)_\bullet \end{aligned}

    ?

    Here ℤ:sSet⇄sAb:frgt \mathbb{Z} \colon sSet \rightleftarrows sAb \colon frgt denotes the free/forgetful adjunction, and EZ denotes precomposition with the Eilenberg-Zilber map.

    • CommentRowNumber23.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 12th 2021

    Re #22: Yes, because the Eilenberg–Zilber map is a chain homotopy equivalence.

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeJul 13th 2021

    Sure, but why does that make the given map a simplicial equivalence?

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeJul 13th 2021

    Oh, I see, you are saying it’s a strong homotopy, not just a weak homotopy equivalence. Right, I had forgotten about that. Thanks!

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeJul 13th 2021

    To be frank. I am still stuck, though:

    So defining

    [S,frgt∘DK(V)] • ≃sSet(S×Δ[•],frgt∘DK(V)) ≃Ch +(N∘ℤ(S×Δ[•]),V) →EZ SCh +(N∘ℤ(S)⊗N∘ℤ(Δ[•]),V) ≃Ch +(N∘ℤ(Δ[•]),[N∘ℤ(S),V]) ≃(frgt∘DK([N∘ℤ(S),V])) • \begin{aligned} \big[ S, \, frgt \circ DK(V) \big]_\bullet & \;\simeq\; sSet \big( S \times \Delta[\bullet], \, frgt \circ DK(V) \big) \\ & \;\simeq\; Ch_+ \big( N \circ \mathbb{Z}(S \times \Delta[\bullet]), \, V \big) \\ & \;\xrightarrow{EZ_S}\; Ch_+ \big( N \circ \mathbb{Z}(S) \,\otimes\, N \circ \mathbb{Z}(\Delta[\bullet]), \, V \big) \\ & \;\simeq\; Ch_+ \big( N \circ \mathbb{Z}(\Delta[\bullet]), \, [ N \circ \mathbb{Z}(S), \, V ] \big) \\ & \;\simeq\; \Big( frgt \circ DK \big( [ N \circ \mathbb{Z}(S), \, V ] \big) \Big)_\bullet \end{aligned}

    and

    (frgt∘DK([N∘ℤ(S),V])) • ≃Ch +(N∘ℤ(Δ[•]),[N∘ℤ(S),V]) ≃Ch +(N∘ℤ(S)⊗N∘ℤ(Δ[•]),V) →AW SCh +(N∘ℤ(S×Δ[•]),V) ≃sSet(S×Δ[•],frgt∘DK(V)) ≃[S,frgt∘DK(V)] • \begin{aligned} \Big( frgt \circ DK \big( [ N \circ \mathbb{Z}(S), \, V ] \big) \Big)_\bullet & \;\simeq\; Ch_+ \big( N \circ \mathbb{Z}(\Delta[\bullet]), \, [ N \circ \mathbb{Z}(S), \, V ] \big) \\ & \;\simeq\; Ch_+ \big( N \circ \mathbb{Z}(S) \,\otimes\, N \circ \mathbb{Z}(\Delta[\bullet]), \, V \big) \\ & \xrightarrow{AW_S} Ch_+ \big( N \circ \mathbb{Z}(S \times \Delta[\bullet]), \, V \big) \\ & \;\simeq\; sSet \big( S \times \Delta[\bullet], \, frgt \circ DK(V) \big) \\ & \;\simeq\; \big[ S, \, frgt \circ DK(V) \big]_\bullet \end{aligned}

    we have that the simplicial map

    [S,frgt∘DK(V)] •→EZ S(frgt∘DK([N∘ℤ(S),V])) •→AW S[S,frgt∘DK(V)] • \big[ S, \, frgt \circ DK(V) \big]_\bullet \xrightarrow{\;EZ_S\;} \Big( frgt \circ DK \big( [ N \circ \mathbb{Z}(S), \, V ] \big) \Big)_\bullet \xrightarrow{\;AW_S\;} \big[ S, \, frgt \circ DK(V) \big]_\bullet

    is the identity. Fine. We need moreover a simplicial homotopy from

    (frgt∘DK([N∘ℤ(S),V])) •→AW S[S,frgt∘DK(V)] •→EZ S(frgt∘DK([N∘ℤ(S),V])) • \Big( frgt \circ DK \big( [ N \circ \mathbb{Z}(S), \, V ] \big) \Big)_\bullet \xrightarrow{\;AW_S\;} \big[ S, \, frgt \circ DK(V) \big]_\bullet \xrightarrow{\;EZ_S\;} \Big( frgt \circ DK \big( [ N \circ \mathbb{Z}(S), \, V ] \big) \Big)_\bullet

    to the identity. I take it that your implicit suggestion was to use the chain homotopies

    (N∘ℤ(Δ[1]))⊗N∘ℤ(S×Δ[•])→EZ∘AW⇒idN∘ℤ(S×Δ[•]) \Big( N \circ \mathbb{Z}(\Delta[1]) \Big) \otimes N \circ \mathbb{Z}(S \times \Delta[\bullet]) \xrightarrow{ EZ \circ AW \Rightarrow id} N \circ \mathbb{Z}(S \times \Delta[\bullet])

    from the Eilenberg-Zilber theorem and look at their image under Ch +(−,V)Ch_+\big( -, V \big). But it is unclear to me how this gives a simplicial homotopy as needed.

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeJul 13th 2021

    But I think I am fine in the simple case that I am interested in, which is (a) S=Δ[1]/∂Δ[1]S = \Delta[1]/\partial \Delta[1] and (b) VV is concentrated on the integers in some degree. Namely, in that case we know with (a) that the simplicial homotopy groups on both sides are isomorphic (by the discussion at free loop space of a classifying space here), both being, by (b) copies of the integers in two degrees. So now the retraction-part of the EZ theorem says that EZ SEZ_S induces in these degrees a retraction of the integers onto the integers. But that must be an isomorphism.

    • CommentRowNumber28.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 14th 2021

    Re #26: Here is a simpler argument. Since you have already constructed a map, to show that it is a weak equivalence for all S, it suffices to observe that it sends homotopy colimits in S to homotopy limits.

    Indeed, [S,frgt∘DK(V)] and (frgt∘DK([N∘ℤ(S),V])) are fully derived constructions: DK, frgt, N, Z[-] all preserve weak equivalences; frgt∘DK lands in Kan complexes, so [S,frgt∘DK(V)] is derived, and NZ[-] lands in projectively cofibrant chain complexes, so [N∘ℤ(S),V] is derived. Furthermore, frgt preserves homotopy limits, Z[-] preserves homotopy colimits, N and DK preserve both, and internal homs send homotopy colimits in the first argument to homotopy limits.

    Thus, it suffices to verify the claim for S=Δ^0, for which the corresponding map is an isomorphism.

    • CommentRowNumber29.
    • CommentAuthorGuest
    • CommentTimeMay 10th 2022
    The functor called Gamma in this article is the Eilenberg-Maclane space K, and was first introduced in their article in 1954, long before Dold and Kan wrote anything on the subject. They did not know that the normalized chain functor of the Eilenberg-Maclane space of a chain complex A was naturally isomorphic to A - this is the precise statement that is proved by Dold and Kan. They prove this by showing that the normalized chain functor is both a left and a right adjoint to K: equivalently, that the two definitions of the normalized chain functor, as a kernel of face maps or as a cokernel of degeneracy maps, are naturally isomorphic.

    Ezra Getzler

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