nLab zigzag category

Contents

Context

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

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

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 \} \,.

Definition

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.)

There is a functor Z:T opRelCatZ : \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 )(n, t_+, t_-), is the relative category on objects {0,1,,n}\{0, 1, \ldots, n \} and generated by arrows (k1)k(k-1) \to k whenever kt +k \in t_+ and (k1)k(k-1) \xleftarrow{\sim} k whenever kt k \in t_-
  • The realization of a morphism τ\tau sends the arrow between (k1)(k-1) and kk to the product of the arrows between (i1)(i-1) and ii for all ii with τ(i)=k\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 tt from XX to YY in a relative category C\mathbf{C} is nothing more than the category of relative functors Z(t)CZ(t) \to \mathbf{C}.

By forgetting the marked arrows, we get a similar realization T opCat\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.

Last revised on August 30, 2022 at 22:10:45. See the history of this page for a list of all contributions to it.