nLab
tower of homotopy fibers

Contents

Context

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

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Definition

Given a homotopy theory, i.e. an (infinity,1)-category, then a tower of homotopy fibers or tower of fibrations or similar is a tower diagram of the form

hofib(f 2) X 2 f 2 A 2 hofib(f 1) X 1 f 1 A 1 hofib(f 0) X f 0 A 0 \array{ \vdots \\ {}^{\mathllap{hofib(f_2)}}\downarrow \\ X_2 &\stackrel{f_2}{\longrightarrow}& A_2 \\ {}^{\mathllap{hofib(f_1)}}\downarrow \\ X_1 &\stackrel{f_1}{\longrightarrow}& A_1 \\ {}^{\mathllap{hofib(f_0)}}\downarrow \\ X &\stackrel{f_0}{\longrightarrow}& A_0 }

where each hook is a homotopy fiber sequence.

Properties

The long exact sequences of homotopy groups for each of the hooks in the tower combine to yield an exact couple. The corresponding spectral sequence of an exact couple is a means to (approximately) computethe homotopy groups of the base object XX of the tower

Examples

References

Last revised on March 1, 2016 at 13:32:28. See the history of this page for a list of all contributions to it.