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added the definition to cyclic homology
next the task is to write out the details for how under the identification of the Hochschild complex with functions on the derived loop space, the cyclic complex is the $S^1$-equivariant functions on the derived loop space.
Thanks for the new material. We should, of course, have it under an entry cyclic homology and not under cyclic cohomology, eventually, when both cases get described (in many cases, not only for associative algebras). But later...
Let’s keep both in the same entry. Otherwise there will be lots of duplication. I’ll add the discussion of cohomology later.
OK, for some time I agree. But eventually the details for special cases like for discrete associative algebras (as opposed to say schemes, spectra etc.) should really be in different entries and the basic entry in my understanding could be about the general categorical nonsense of utilizing Connes’ cyclic modules to get cyclic (co)homology like one uses simplicial objects in other situations. The geometric picture about ${S}^1$ equivariant spaces may fit with also quite well with topological cyclic homology.
The general subject under the traditional heading of cyclic homology is much wider subject than the tradiotional subject of Hochschild homology.
There is an interesting approach which you may like, due Cortinas, using crystalline site with infinitesimal thickenings. I do not know if it was ever compared to the picture in Toen and in BenZvi-Nadler.
There is a very interesting thing when one does the cyclic homology for algebras over cyclic operads. Then there are two kinds which fit into a version of Connes exact sequence which agree for associative algebras because the operad for associative algebras is self-dual so the two different versions coincide and the Connes- exact sequence has only Hochschild and cyclic terms. That is in Getzler-Kapranov paper.
I added the definition of the relative cyclic homology into cyclic homology.
New entry epicyclic space redirecting also epicyclic category and epicyclic set.
Some edits at cyclic homology, still under construction. Also changed the title from cyclic cohomology to cyclic homology.
more updates and added a stub at mixed complex
added full publication data for
and made more explicit in the entry that the theorem in section 4 is reviewing a theorem due to Jones.
also Cor. 7.3.1 in:
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 pointer to
and made “string cohomology” a redirect to here (for the moment, maybe it deserves its own page eventually)
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