nLab tower of homotopy fibers

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

$$\array{ \vdots \\ {}^{\mathllap{hofib(f_2)}}\downarrow \\ X_2 &\stackrel{f_2}{\longrightarrow}& A_2 \\ {}^{\mathllap{hofib(f_1)}}\downarrow \\ X_1 &\stackrel{f_1}{\longrightarrow}& A_1 \\ {}^{\mathllap{hofib(f_0)}}\downarrow \\ X &\stackrel{f_0}{\longrightarrow}& A_0 }$$

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) compute the homotopy groups of the base object $X$ of the tower

Examples

• Postnikov tower

• Adams tower

Related concepts

• filtered object in an (infinity,1)-category

  • filtered chain complex

  • filtered spectrum

• spectral sequence of a tower of fibrations

References

• Aldridge Bousfield, Daniel Kan, chapter IX of Homotopy limits, completions and localization, Springer 1972

• Paul Goerss, Rick Jardine, chapter VI of Simplicial homotopy theory, 1996