+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Category theory +-- {: .hide} [[!include category theory - contents]] =-- #### Homotopy theory +--{: .hide} [[!include homotopy - contents]] =-- =-- =-- # Zigzags * table of contents {: toc} ## Zigzags of morphisms In a [[category]] $C$ a **zigzag of morphisms** is a finite collection of [[morphism]]s $(f_i)$ in $C$ of the form $$ \array{ && x_1 &&&& x_3 & \cdots \\ & {}^{\mathllap{f_1}}\swarrow && \searrow^{\mathrlap{f_2}} && {}^{\mathllap{f_3}}\swarrow && \searrow^{\mathrlap{f_4}} & \cdots \\ x_0 &&&& x_2 &&&& x_4 & \cdots } \,. $$ A zigzag consisting just out of two morphisms is a _roof_ or [[span]]. General such zig-zags of morphisms represent ordinary morphisms in the _groupoidification_ of $C$ -- the [[Kan fibrant replacement]] of its [[nerve]], its [[simplicial localization]] or its 1-categorical [[localization]] at all its morphisms. More generally, if in these zig-zags the left-pointing morphisms are restricted to be in a class $S \subset Mor(C)$, then these zig-zags represent morphisms in the simplicial localizaton or localization of $C$ at $S$. ## Related concepts * [[resolution]], [[derived functor]] * [[span]], [[correspondence]] * [[Burnside category]] * [[zigzag persistence module]] [[!redirects zigzag]] [[!redirects zigzags]] [[!redirects zig-zag]] [[!redirects zig-zags]] [[!redirects zig zag]] [[!redirects zig zags]]