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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This is an unpublished note, but on request I have now uploaded it to the nLab
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.
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 $k-1$.
Another approach to cyclic sets via discrete approximation of a circle is in
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.
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.
How Moerdijk’s does construction generalize to all skew-simplicial sets ?
Added a recent paper by Connes
Do the applications to cohomology suggest a Quillen model structure on cyclic sets?
Yes, a model structure was given by Spalinski in 95. I have added pointers to the entry here.
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.
Hm, but that model structure is supposed to model S^1-equivariant homotopy types.
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 ?
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.
I have added a link to some slides by Spalinski, discussing cyclic, dihedral and quaternionic sets and their model category structures.
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).
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.
Is it actually an oo-topos as suggested at cohomology?
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.
Since there seems to be no electronic copy of the original
(?)
I have added pointer to
and then to
Jean-Louis Loday, Cyclic Spaces and $S^1$-Equivariant Homology (doi:10.1007/978-3-662-21739-9_7)
Chapter 7 in: Cyclic Homology, Grundlehren 301, Springer 1992 (doi:10.1007/978-3-662-21739-9)
Added a PDF link:
Thanks! Have copied this over to the other entries, too.
It doesn’t seem right, because for example there are infinitely many endofunctors on the free category $B\mathbb{N}$ on the loop on $0$.
The article says
… the cycle category $\Lambda$, is the full subcategory of Cat whose objects are the categories $[n]_\Lambda$ which are freely generated by the graph $0\to 1\to 2\to\ldots\to n\to 0$ …
I believe this is wrong? Rather, I think that the cycle category is the subcategory of Cat on functors between cycles $[n]_\Lambda$ that preserve the “winding number”. Am I misunderstanding something?
Yes, it’s wrong. My memory is that this has been brought up before. Let me have a look.
I got rid of the entire paragraph where that glitch appears, because I have my doubts that the category of shapes here can be described as a full subcategory of $Cat$ in any way. But the description at cycle category is correct.
Added MR777584 Zbl0534.18009 reviews links for Connes CRAS article.
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