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

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Zigzags
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 $S$ .

Revised on August 26, 2012 18:33:13
by

Urs Schreiber
(89.204.137.239)