Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 24th 2010

    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 1S^1-equivariant functions on the derived loop space.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeNov 24th 2010

    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...

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeNov 24th 2010

    Let’s keep both in the same entry. Otherwise there will be lots of duplication. I’ll add the discussion of cohomology later.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeNov 24th 2010
    • (edited Nov 24th 2010)

    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{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.

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeNov 24th 2010
    • (edited Nov 24th 2010)

    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.

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeDec 2nd 2010

    I added the definition of the relative cyclic homology into cyclic homology.

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeSep 5th 2013

    New entry epicyclic space redirecting also epicyclic category and epicyclic set.

    • CommentRowNumber8.
    • CommentAuthoradeelkh
    • CommentTimeJul 8th 2014

    Some edits at cyclic homology, still under construction. Also changed the title from cyclic cohomology to cyclic homology.

    • CommentRowNumber9.
    • CommentAuthoradeelkh
    • CommentTimeJul 8th 2014

    more updates and added a stub at mixed complex

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJun 27th 2021

    added full publication data for

    • Jean-Louis Loday, Section 4 of: Free loop space and homology, Chapter 4 in” Janko Latchev, Alexandru Oancea (eds.): Free Loop Spaces in Geometry and Topology, IRMA Lectures in Mathematics and Theoretical Physics 24, EMS 2015 (arXiv:1110.0405, ISBN:978-3-03719-153-8)

    and made more explicit in the entry that the theorem in section 4 is reviewing a theorem due to Jones.

    diff, v25, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJun 27th 2021

    also Cor. 7.3.1 in:

    diff, v25, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeAug 18th 2021

    added pointer to

    and made “string cohomology” a redirect to here (for the moment, maybe it deserves its own page eventually)

    diff, v28, current