nLab
zigzag category

Idea

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

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

Definition

Let $T$ be the category defined as follows:

• Objects are defined by triples of data $t≔(n,t_{+},t_{-})$, which are partition?s of the sets $\{i{\}}_{1\le i\le n}$ for $n\ge 0$ into two disjoint subsets $t_{+}$ and $t_{-}$. (The corresponding zigzag diagram has nodes $t_0,\dots ,t_n$, and the arrow between $t_{i-1}$ and $t_i$ points forward to $t_i$ if $i\in t_{+}$, and backward if $i\in t_{-}$.)

• Morphisms are monotone maps preserving the partitions.

We will call $T$ 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.)

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.