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 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 sheaves 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.
    • CommentAuthorHarry Gindi
    • CommentTimeSep 16th 2010
    • (edited Sep 18th 2010)

    In keeping with the convention set up for simplex category and globe category, I have created zigzag category. I don’t know if this is a useful shape category aside from its use in localization (in that I’m not sure whether or not it is a test category, but it is at least a Reedy category).

    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeSep 16th 2010

    Neat!

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 18th 2010

    Added an Idea- and and Applications-section.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeSep 18th 2010

    but it is at least a Reedy category.

    You should create a Properties-section in the entry and record that property there.

    • CommentRowNumber5.
    • CommentAuthorTim_Porter
    • CommentTimeSep 18th 2010

    I would second that and ask that a sketch proof be added if it is not too long (and a reference if it is).

    • CommentRowNumber6.
    • CommentAuthorHarry Gindi
    • CommentTimeSep 18th 2010
    • (edited Sep 18th 2010)

    It’s asserted (I think without proof) in Dwyer-Hirschhorn-Kan-Smith section 35, which proves that the (nerve of the) Grothendieck and Hammock localizations are Dwyer-Kan weakly equivalent.

    • CommentRowNumber7.
    • CommentAuthorHarry Gindi
    • CommentTimeSep 18th 2010
    • (edited Sep 18th 2010)

    See: Google Books

    Also, the section 22.8: Google Books

    So no proof is given, although It doesn’t seem like a proof would be by any means difficult.

    What I’d be really interested in seeing is a better description of its “categorical realization” (see the page on simplicial localization of a homotopical category). It seems like prime material for a description of each zigzag as a double category (or maybe even a 3-fold category) where the horizontal arrows are zigzags forwards, the vertical arrows are the backwards arrows (and the third bunch of arrows are the order-preserving ones?). That way, we could look at double-functor categories into the canonical double category we get from “crossing a category with its opposite” (not the product, just a description of something I can’t describe better).

    • CommentRowNumber8.
    • CommentAuthorHurkyl
    • CommentTimeAug 30th 2022

    I’ve added the construction of the “walking zigzag” associated to a zigzag type.

    I also want to suggest that we adjust the definition so that T op\mathbf{T}^{op} is the category of zigzag types, rather than T\mathbf{T}.

    The category T\mathbf{T} of zigzag types is defined in Homotopy Limit Functors on Model Categories and Homotopical Categories, which I presume this entry is based on. The variance on T\mathbf{T} is, I presume, meant so that the assignment tC t(X,Y)t \to \mathbf{C}^t(X, Y) from types to “zigzags from XX to YY of type tt” is covariant in tt.

    However, I think it’s somewhat more idiomatic to have the variance match that of the category of “walking” diagrams and expect tC t(X,Y)t \to \mathbf{C}^t(X, Y) to be a contravariant functor in tt. Thus my suggestion.

    diff, v9, current