nLab
zigzag

Zigzag of morphisms

In a category $C$ a zigzag of morphisms is a finite collection of morphisms $(f_i)$ in $C$ of the form

$$\begin{array}{cccccccc}& & x_1& & & & x_3& \cdots \\ & {}^{f_1}\swarrow & & {\searrow }^{f_2}& & {}^{f_3}\swarrow & & {\searrow }^{f_4}& \cdots \\ x_0& & & & x_2& & & & x_4& \cdots \end{array}.$$\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 $S\subset \mathrm{Mor}(C)$, then these zig-zags represent morphisms in the simplicial localizaton or localization of $C$ at $S$.