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1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeSep 16th 2010
   • (edited Sep 18th 2010)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexIn 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]]).

   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).

2. • CommentRowNumber2.
   • CommentAuthorTobyBartels
   • CommentTimeSep 16th 2010
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   Author: TobyBartels
   Format: MarkdownItexNeat!

   Neat!

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeSep 18th 2010
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   Author: Urs
   Format: MarkdownItexAdded an Idea- and and Applications-section.

   Added an Idea- and and Applications-section.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeSep 18th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItex> but it is at least a Reedy category. You should create a Properties-section in the entry and record that property there.

   but it is at least a Reedy category.

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

5. • CommentRowNumber5.
   • CommentAuthorTim_Porter
   • CommentTimeSep 18th 2010
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexI would second that and ask that a sketch proof be added if it is not too long (and a reference if it is).

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

6. • CommentRowNumber6.
   • CommentAuthorHarry Gindi
   • CommentTimeSep 18th 2010
   • (edited Sep 18th 2010)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexIt'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.

   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.

7. • CommentRowNumber7.
   • CommentAuthorHarry Gindi
   • CommentTimeSep 18th 2010
   • (edited Sep 18th 2010)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexSee: [Google Books](http://books.google.com/books?id=v2TVvLJKCeIC&lpg=PP1&ots=uj6QlPu1DU&dq=dwyer%20hirschhorn%20kan%20smith%20homotopy%20limit%20functor&pg=PA107#v=onepage&q&f=false) Also, the section 22.8: [Google Books](http://books.google.com/books?id=v2TVvLJKCeIC&lpg=PP1&ots=uj6QlPu1DU&dq=dwyer%20hirschhorn%20kan%20smith%20homotopy%20limit%20functor&pg=PA67#v=onepage&q&f=false) 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).

   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).

8. • CommentRowNumber8.
   • CommentAuthorHurkyl
   • CommentTimeAug 30th 2022
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   Author: Hurkyl
   Format: MarkdownItexI'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 $\mathbf{T}^{op}$ is the category of zigzag types, rather than $\mathbf{T}$. The category $\mathbf{T}$ 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 $\mathbf{T}$ is, I presume, meant so that the assignment $t \to \mathbf{C}^t(X, Y)$ from types to "zigzags from $X$ to $Y$ of type $t$" is covariant in $t$. However, I think it's somewhat more idiomatic to have the variance match that of the category of "walking" diagrams and expect $t \to \mathbf{C}^t(X, Y)$ to be a contravariant functor in $t$. Thus my suggestion. <a href="https://ncatlab.org/nlab/revision/diff/zigzag+category/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/zigzag+category/9">v9</a>, <a href="https://ncatlab.org/nlab/show/zigzag+category">current</a>

   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 $\mathbf{T}^{op}$ is the category of zigzag types, rather than $\mathbf{T}$.

   The category $\mathbf{T}$ 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 $\mathbf{T}$ is, I presume, meant so that the assignment $t \to \mathbf{C}^t(X, Y)$ from types to “zigzags from $X$ to $Y$ of type $t$” is covariant in $t$.

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

   diff, v9, current