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



Paths and cylinders

Homotopy groups

Basic facts



Zigzags of morphisms

In a category CC a zigzag of morphisms is a finite collection of morphisms (f i)(f_i) in CC of the form

x 1 x 3 f 1 f 2 f 3 f 4 x 0 x 2 x 4 . \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 CC – 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 SMor(C)S \subset Mor(C), then these zig-zags represent morphisms in the simplicial localizaton or localization of CC at SS.

Revised on August 26, 2012 18:33:13 by Urs Schreiber (