category theory

# Contents

## Idea

A tower is a diagram of the shape of the poset of natural numbers

$\cdots \to X_2 \to X_1 \to X_0 \,.$

A limit over a diagram of this form is called a sequential limit/directed limit.

Different contexts lead to different notions of morphism of towers, so it is important to consider what category of towers is appropriate for any given use of these objects. In addition to Postnikov towers, and related uses in decomposing homotopy types, towers also occur as a simple type of pro-object in a category. In that situation the morphisms considered between towers are usually pro-morphisms.

In homotopy theory in the presence of (infinity,1)-limits/(infinity,1)-colimits, every tower is a tower of homotopy fibers.

## Examples

Revised on May 27, 2016 16:52:20 by Urs Schreiber (131.220.184.222)