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.

]]>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!

]]>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 $N$ is naturally isomorphic to the functor tensor product $-\otimes_\Delta N\mathbb{Z}[\Delta^\bullet]$, where $\mathbb{Z}[\Delta^\bullet]$ 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 $\Gamma$.

]]>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.

]]>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.

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

]]>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.

]]>Hi Jim,

I have added your Alg-Top findings to Moore complex

]]>and no citation of Moore

Can you provide it?

]]>and not even an indication of which Moore!! ]]>

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

]]>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

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

$Id \to N \Gamma$and

$\Gamma N \to Id \,.$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.

]]>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.

]]>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 correspondencethis 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.

added to Dold-Kan corresponden

a paragraph on the cubical version

a paragraph on how the simplical version factors through the globular version by the $\omega$-nerve.

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.

]]>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…)

]]>added reference to dendroidal version of Dold-Kan correspondence

]]>