category theory

# Zigzags

## Zigzags of morphisms

In a category $C$ a zigzag of morphisms is a finite collection of morphisms $\left({f}_{i}\right)$ in $C$ of the form

$\begin{array}{cccccccc}& & {x}_{1}& & & & {x}_{3}& \cdots \\ & {}^{{f}_{1}}↙& & {↘}^{{f}_{2}}& & {}^{{f}_{3}}↙& & {↘}^{{f}_{4}}& \cdots \\ {x}_{0}& & & & {x}_{2}& & & & {x}_{4}& \cdots \end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $S\subset \mathrm{Mor}\left(C\right)$, 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)