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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMar 31st 2014
• (edited Mar 31st 2014)

Have added to cyclic set a pointer to notes from 1996 by Ieke Moerdijk where the theory classified by the topos of cyclic sets is identified (abstract circles).

This is an unpublished note, but on request I have now uploaded it to the nLab

• Ieke Moerdijk, Cyclic sets as a classifying topos, 1996 (pdf)

I have also added a corresponding brief section to classifying topos.

By the way, there is an old query box with an exchange between Mike and Zoran at cyclic set. It seems to me that this has been resolved and the query box could be removed (to make the entry read more smoothly). Maybe Mike and/or Zoran could briefly look into this.

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeMar 31st 2014
• (edited Mar 31st 2014)

Thanks for the MOerdijk’s notes, I wanted them from 2007 and could not get to them. The point of the discussion with Mike was NOT (only) to resolve what should stand in the $n$Lab but to reason why it is the way it is, and for this the discussion is still worthy, including to myself. So I copy the record here with backpointer in the entry.

Mike: I copied (and attempted to clarify) the above from symmetric set, but I don’t think I believe it. If you invert the composite $0\to 1\to 2$ in $[2]$, then the objects $0$ and $2$ become isomorphic and are both a retract of $1$. This localization has exactly one nondegenerate, nonidentity self-map, which exchanges $0$ and $2$. But shouldn’t the object “$2$” in $\Lambda$ have a $\mathbb{Z}/3$ worth of self-maps?

Zoran Škoda: Thanks, Mike, I corrected the cyclic part, the symmetric was OK before. But even $[0]$ has an object with infinity worth of self-maps. If the new map $n\to 0$ is taken into account, then all $n+1$ objects of cyclic $[n+1]_\Lambda$ will be on the same footing: from point $k$ one has identity, going forward one step, 2 steps, 3 steps, and so on, and one is allowed to cross the boundary $k+n-k$, doing more than $n-k$ steps, even $n-1$ step coming all through to your predecessor $k-1$.

• CommentRowNumber3.
• CommentAuthorzskoda
• CommentTimeMar 31st 2014
• (edited Mar 31st 2014)

Another approach to cyclic sets via discrete approximation of a circle is in

• Vladimir Drinfeld, On the notion of geometric realization, math.CT/0304064
• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeMar 31st 2014

Thanks for the pointer to Drinfeld’s result. Ieke mentioned this fact to me (though I am not sure if in relation to Drinfeld), but I forgot to add it. Have done so now.

• CommentRowNumber5.
• CommentAuthorzskoda
• CommentTimeMar 31st 2014

I have added the link to skew-simplicial set (due Krasauskas, and a bit later, independently, Loday and Fiedorowicz) which is the natural common notion comprising examples like symmetric sets, dihedral sets, cyclic sets and simplicial sets. They are presheaves on a skew-simplicial group (or crossed simplicial group, in the language of Loday and Fiedorowicz); Kapranov had few days ago revisited the subject in a long paper related to geometry.

• Tobias Dyckerhoff, Mikhail Kapranov, Crossed simplicial groups and structured surfaces, arxiv/1403.5799

How Moerdijk’s does construction generalize to all skew-simplicial sets ?

• CommentRowNumber6.
• CommentAuthorspitters
• CommentTimeApr 1st 2014

Added a recent paper by Connes

• CommentRowNumber7.
• CommentAuthorspitters
• CommentTimeApr 8th 2014

Do the applications to cohomology suggest a Quillen model structure on cyclic sets?

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeApr 8th 2014

Yes, a model structure was given by Spalinski in 95. I have added pointers to the entry here.

• CommentRowNumber9.
• CommentAuthorzskoda
• CommentTimeApr 8th 2014

The category of cycles is a test category in the sense of Grothendieck, isn’t it ? If so, then Grothendieck knew how to do the homotopy for cyclic sets before Spalinski’s recipe.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeApr 8th 2014

Hm, but that model structure is supposed to model S^1-equivariant homotopy types.

• CommentRowNumber11.
• CommentAuthorzskoda
• CommentTimeApr 8th 2014
• (edited Apr 8th 2014)

The result that cyclic sets are alteratively about $S^1$-equivariant homotopy is earlier (late 1980s) is a result, and is not the artificial by definition but by the nature of the category of cyclic sets (and its relation to simplicial sets) isn’t it ? It may as well that other approaches do the same, including Grothendieck’s applied to this case ?

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeApr 8th 2014

Spalinski might be the first to give a model structure on cyclic sets presenting the homotopy theory of $S^1$-equivariant homotopy types. But I don’t have time to chase references further. I guess if you care it is easy to do.

• CommentRowNumber13.
• CommentAuthorTim_Porter
• CommentTimeApr 9th 2014

I have added a link to some slides by Spalinski, discussing cyclic, dihedral and quaternionic sets and their model category structures.

• CommentRowNumber14.
• CommentAuthorzskoda
• CommentTimeApr 9th 2014
• (edited Apr 9th 2014)

12: it is not about the references (what makes you think that need references) – the question is purely scientific: if the Grothendieck approach, if applicable, gives the same homotopy theory or not. We can do that at later time of course. My remark about earlier references on equivariant structure is just to say that the idea that cyclic sets are about $S^1$-equivariant case is not at all specific to Spalinski’s model category approach, so it is conceivable (but still uninformed case) that it may be the case with Grothendieck’s approach if it is applicable (just a guess).

• CommentRowNumber15.
• CommentAuthorTim_Porter
• CommentTimeApr 9th 2014

I merely noted that Spalinski did not have a page so looked and found those slides! I could not find a home page for him.. strange.

In fact there was some comment about there being a difference and it was the Blumberg structure that gave better results, (but this is from memory.) The S^1-equivariant theory seems to be the orbit based version. I did look at this years ago, and should revisit those ideas.

• CommentRowNumber16.
• CommentAuthorspitters
• CommentTimeApr 10th 2014

Is it actually an oo-topos as suggested at cohomology?

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeApr 6th 2018

in the first lines of the Idea-section I added missing cross-link with cyclic object and mentioning of Hochschild and cyclic (co-)homology.

This entry may deserve cleaning up and harmonization with a bunch of closely related entries.

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeJun 27th 2021

Since there seems to be no electronic copy of the original

• Alain Connes, Cohomologie cyclique et foncteurs $Ext^n$, C.R.A.S. 269 (1983), Série I, 953-958

(?)

• Pierre Cartier, Section 1.6 of: Homologie cyclique : rapport sur des travaux récents de Connes, Karoubi, Loday, Quillen…, Séminaire Bourbaki: volume 1983/84, exposés 615-632, Astérisque, no. 121-122 (1985), Exposé no. 621 (numdam:SB_1983-1984__26__123_0)

and then to

• CommentRowNumber19.
• CommentAuthorTim_Porter
• CommentTimeJun 28th 2021

269 should have been 296

• CommentRowNumber20.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 28th 2021

• Alain Connes, Cohomologie cyclique et foncteurs $Ext^n$, C.R.A.S. 296 (1983), Série I, 953-958. PDF.