Contents

# Contents

## Definition

###### Definition

A homotopy type is called finite if it is presented (via the discussion at homotopy hypothesis) by either

###### Remark

Similarly a spectrum (stable homotopy type) given by a sequence of finite homotopy types is called a finite spectrum.

###### Remark

Beware that a finite homotopy type in general does not have finite and finitely many homotopy groups (see e.g. at homotopy groups of spheres). Homotopy types with finite and finitely many homotopy groups have alternatively been called $\pi$-finite, or tame, or (adapted from homological algebra) “of finite type” (which needs to be carefully distinguishes, therefore, from “finite homotopy type”). See at homotopy type with finite homotopy groups.

## Properties

### Relation to compact homotopy type

The compact objects in ∞Grpd are the retracts of finite homotopy types. Not every such retract is itself a finite homotopy type; the vanishing of Wall's finiteness obstruction is a necessary and sufficient condition for this to happen.

## References

Last revised on January 19, 2016 at 15:23:28. See the history of this page for a list of all contributions to it.