zigzag category


Category theory

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts





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

{t 0t 1t 2t 3t 4}. \{ t_0 \leftarrow t_1 \leftarrow t_2 \rightarrow t_3 \leftarrow t_4 \cdots \} \,.


Let T\mathbf{T} be the category defined as follows:

  • An object is given by a triple of data t(n,t +,t )t \coloneqq (n,t_+,t_-), which is a partition of the poset {i|1in}\{i \;|\; 1\leq i\leq n\} for some natural number n0n \geq 0 into two subset parts t +t_+ and t t_-. (The corresponding diagram as above has nodes t 0,,t nt_0, \ldots, t_n, and the arrow between t i1t_{i-1} and t it_i points forward to t it_i if it +i \in t_+, and backward if it i \in t_-.)

  • A morphism is a monotone map preserving the partitions.

We will call T\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.)


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.

Revised on August 19, 2013 16:12:04 by Toby Bartels (