nLab zigzag category

Idea

The zigzag category is the category whose objects are diagrams that look like

$$\{ t_0 \leftarrow t_1 \leftarrow t_2 \rightarrow t_3 \leftarrow t_4 \cdots \} \,.$$

Definition

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:

• The realization of a type $(n, t_+, t_-)$, is the relative category on objects $\{0, 1, \ldots, n \}$ and generated by arrows $(k-1) \to k$ whenever $k \in t_+$ and $(k-1) \xleftarrow{\sim} k$ whenever $k \in t_-$
• The realization of a morphism $\tau$ sends the arrow between $(k-1)$ and $k$ to the product of the arrows between $(i-1)$ and $i$ for all $i$ with $\tau(i) = k$.

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.

Applications

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.