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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).
Neat!
Added an Idea- and and Applications-section.
but it is at least a Reedy category.
You should create a Properties-section in the entry and record that property there.
I would second that and ask that a sketch proof be added if it is not too long (and a reference if it is).
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.
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).
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.
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