Updated reference to Helmstutler article to published version (which is superior to the arxiv version currently linked)

Helmstutler

]]>Minor format.

]]>Ficed two dead links

G. Nan Tie

]]>Fixed two dead links

]]>added a brief paragraph (here) making explicit that there is a generalization to abelian categories and further to additive categories in which all retractions come from direct sums.

]]>Thanks for bringing this up. I had no idea that there was such a wild mis-redirect at *Omega-group* all along. Have removed it (announced there).

In the second paragraph of the Idea there is a mention/link to $\infty$-groups, the link in source being `[[∞-group]]`

It currently opens Omega-group

but I think the intended target would be infinity-group

At Omega-group there is a yellow warning:

`Note: infinity-group and Omega-group both redirect for "∞-group".`

Maybe the ∞-group and plural redirects Omega group should be deleted (?).

]]>Don’t remember what happened there. I have removed these words now, but anyone versed in the history of the subject should feel free to further adjust/expand.

Also fixed a few minor glitches, such as the typesetting of “Słomińska”.

]]>Corrected a broken link to the Dold–Puppe paper.

This article currently claims

This remarkable article, which appeared shortly after the work by Dold and Puppe but was apparently not influenced by that …

where “this remarkable article” refers to the 1958 paper by Kan and “the work by Dold and Puppe” refers to the 1961 paper by Dold–Puppe.

How exactly does a 1958 paper “appear shortly after” a 1961 paper?

]]>Ezra Getzler ]]>

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.

]]>But I think I am fine in the simple case that I am interested in, which is (a) $S = \Delta[1]/\partial \Delta[1]$ and (b) $V$ 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 $EZ_S$ induces in these degrees a retraction of the integers onto the integers. But that must be an isomorphism.

To be frank. I am still stuck, though:

So defining

$\begin{aligned} \big[ S, \, frgt \circ DK(V) \big]_\bullet & \;\simeq\; sSet \big( S \times \Delta[\bullet], \, frgt \circ DK(V) \big) \\ & \;\simeq\; Ch_+ \big( N \circ \mathbb{Z}(S \times \Delta[\bullet]), \, V \big) \\ & \;\xrightarrow{EZ_S}\; Ch_+ \big( N \circ \mathbb{Z}(S) \,\otimes\, N \circ \mathbb{Z}(\Delta[\bullet]), \, V \big) \\ & \;\simeq\; Ch_+ \big( N \circ \mathbb{Z}(\Delta[\bullet]), \, [ N \circ \mathbb{Z}(S), \, V ] \big) \\ & \;\simeq\; \Big( frgt \circ DK \big( [ N \circ \mathbb{Z}(S), \, V ] \big) \Big)_\bullet \end{aligned}$and

$\begin{aligned} \Big( frgt \circ DK \big( [ N \circ \mathbb{Z}(S), \, V ] \big) \Big)_\bullet & \;\simeq\; Ch_+ \big( N \circ \mathbb{Z}(\Delta[\bullet]), \, [ N \circ \mathbb{Z}(S), \, V ] \big) \\ & \;\simeq\; Ch_+ \big( N \circ \mathbb{Z}(S) \,\otimes\, N \circ \mathbb{Z}(\Delta[\bullet]), \, V \big) \\ & \xrightarrow{AW_S} Ch_+ \big( N \circ \mathbb{Z}(S \times \Delta[\bullet]), \, V \big) \\ & \;\simeq\; sSet \big( S \times \Delta[\bullet], \, frgt \circ DK(V) \big) \\ & \;\simeq\; \big[ S, \, frgt \circ DK(V) \big]_\bullet \end{aligned}$we have that the simplicial map

$\big[ S, \, frgt \circ DK(V) \big]_\bullet \xrightarrow{\;EZ_S\;} \Big( frgt \circ DK \big( [ N \circ \mathbb{Z}(S), \, V ] \big) \Big)_\bullet \xrightarrow{\;AW_S\;} \big[ S, \, frgt \circ DK(V) \big]_\bullet$is the identity. Fine. We need moreover a simplicial homotopy from

$\Big( frgt \circ DK \big( [ N \circ \mathbb{Z}(S), \, V ] \big) \Big)_\bullet \xrightarrow{\;AW_S\;} \big[ S, \, frgt \circ DK(V) \big]_\bullet \xrightarrow{\;EZ_S\;} \Big( frgt \circ DK \big( [ N \circ \mathbb{Z}(S), \, V ] \big) \Big)_\bullet$to the identity. I take it that your implicit suggestion was to use the chain homotopies

$\Big( N \circ \mathbb{Z}(\Delta[1]) \Big) \otimes N \circ \mathbb{Z}(S \times \Delta[\bullet]) \xrightarrow{ EZ \circ AW \Rightarrow id} N \circ \mathbb{Z}(S \times \Delta[\bullet])$from the Eilenberg-Zilber theorem and look at their image under $Ch_+\big( -, V \big)$. But it is unclear to me how this gives a simplicial homotopy as needed.

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

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

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

]]>I forget if I had an argument for this:

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

$\begin{aligned} \big[ S, \, frgt \circ DK(V) \big]_\bullet & \;\simeq\; sSet \big( S \times \Delta[\bullet], \, frgt \circ DK(V) \big) \\ & \;\simeq\; Ch_\bullet^+ \big( N \circ \mathbb{Z}(S \times \Delta[\bullet]), \, V \big) \\ & \;\xrightarrow{EZ}\; Ch_\bullet^+ \big( N \circ \mathbb{Z}(S) \,\otimes\, N \circ \mathbb{Z}(\Delta[\bullet]), \, V \big) \\ & \;\simeq\; Ch_\bullet^+ \big( N \circ \mathbb{Z}(\Delta[\bullet]), \, [ N \circ \mathbb{Z}(S), \, V ] \big) \\ & \;\simeq\; \Big( frgt \circ DK \big( [ N \circ \mathbb{Z}(S), \, V ] \big) \Big)_\bullet \end{aligned}$?

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

]]>Created some links on relevant names.

]]>added pointer to

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

added pointer to

- Daniel Quillen, Section II.4 item 5 in:
*Homotopical Algebra*, Lecture Notes in Mathematics 43, Springer 1967(doi:10.1007/BFb0097438)

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.

]]>