Have uploaded the file now to the nLab server (here).

]]>Updated URL for notes of Malkiewich

Neil Strickland

]]>Added MR777584 Zbl0534.18009 reviews links for Connes CRAS article.

]]>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.

]]>Yes, it’s wrong. My memory is that this has been brought up before. Let me have a look.

]]>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?

]]>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 cycle category Λ, is the full subcategory of Cat whose objects are the categories [n]Λ which are freely generated by the graph 0→1→2→…→n→0." ]]>

Thanks! Have copied this over to the other entries, too.

]]>Added a PDF link:

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

269 should have been 296

]]>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

(?)

I have added pointer to

- 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

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)

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.

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

]]>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.

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

]]>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.

]]>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 ?

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

]]>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.

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

]]>