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The zigzag category is the category whose objects are diagrams that look like
Let $\mathbf{T}$ be the category defined as follows:
An object is given by a triple of data $t \coloneqq (n,t_+,t_-)$, which is a partition of the poset $\{i \;|\; 1\leq i\leq n\}$ for some natural number $n \geq 0$ into two subset parts $t_+$ and $t_-$. (The corresponding diagram as above has nodes $t_0, \ldots, t_n$, and the arrow between $t_{i-1}$ and $t_i$ points forward to $t_i$ if $i \in t_+$, and backward if $i \in t_-$.)
A morphism is a monotone map preserving the partitions.
We will call $\mathbf{T}$ the zigzag category, or the category of zigzag types.
Also, notice that we have cleverly hidden the empty set among the objects. We pat ourselves on the backs for doing this. (Here the zigzag type consists of a single node.)
There is a functor $Z : \mathbf{T}^{op} \to \mathbf{RelCat}$ that realizes each zigzag as a relative category. It is given by:
In particular, in this realization, the morphisms of zigzag types are precisely the relative functors that are also nondecreasing on object labels and preserve bottom and top objects.
Then, the category of zigzags of type $t$ from $X$ to $Y$ in a relative category $\mathbf{C}$ is nothing more than the category of relative functors $Z(t) \to \mathbf{C}$.
By forgetting the marked arrows, we get a similar realization $\mathbf{T}^{op} \to \mathbf{Cat}$ for talking about zigzags in ordinary categories.
Such zig-zag diagrams serve to model morphisms in an (∞,1)-category in terms of a presentation by a category with weak equivalences. See simplicial localization of a homotopical category.
Last revised on August 30, 2022 at 22:10:45. See the history of this page for a list of all contributions to it.