Not signed in (Sign In)

Start a new discussion

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 beauty bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality education elliptic-cohomology enriched fibration 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 lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories 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 science set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory 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
    • CommentTimeAug 9th 2011

    stub for totalization

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeAug 9th 2011
    • (edited Aug 9th 2011)

    Is this related to (one of the two) totalization(s) of a double complex in homological algebra and related totalizations in homological algebra ?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeAug 9th 2011

    Good point, the terminology is a bit inconclusive in some places of the literature, I think. I have added the following paragraph to totalization:

    Formally the dual to totalization is geometric realization: where totalization is the end over a powering with Δ\Delta, realization is the coend over the tensoring.

    But various other operations carry names similar to “totalization”. For instance a total chain complex is related under Dold-Kan correspondence to the diagonal of a bisimplicial set – see Eilenberg-Zilber theorem. As discussed at bisimplicial set, this is weakly homotopy equivalent to the operation that is often called TotTot and called the total simplicial set of a bisimplicial set.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeAug 10th 2011

    Whereas the diagonal of a bisimplicial set is actually its geometric realization, considered as a simplicial object in sSet!

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeAug 11th 2011
    • (edited Aug 11th 2011)
    • Paul Bressler, Alexander Gorokhovsky, Ryszard Nest, Boris Tsygan, Deformation quantization of gerbes, Advances in Mathematics 214 (2007) 230–266, pdf

    reviews the constructon of the totalization of a cosimplicial DGLA in Section 3.4 (defined as certain colimit) and “prove that isomorphism classes of descent data of a cosimplicial DGLA are in one-to-one correspondence with isomorphism classes of Maurer–Cartan elements of its totalization”.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeSep 24th 2018
    • (edited Sep 24th 2018)

    added statement and pointer to proof of the fact that the totalization, in the traditional sense, is a model for the homotopy limit over the cosimplicial objects, if the latter is Reedy fibrant.

    diff, v6, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 24th 2018

    also added brief mentioning of Bousfield-Kan spectral sequence and Eilenberg-Moore spectral sequence.

    (phew, that entry had been sitting there all the years lacking most of the homotopy theoretic information…)

    diff, v6, current

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)