Contents

# Contents

## Idea

The concept of a homotopy type (homotopy n-type) all of whose homotopy groups are finite groups does not have an established name. Sometimes it is called $\pi$-finiteness. In the context of groupoid cardinality “tameness” is used. In homological algebra “of finite type” is used, which in homotopy theory however badly clashes with the concept of finite homotopy type which is crucially different from homotopy type with finite homotopy groups.

## Properties

### (Co-)Limits indexed by homotopy types with finite homotopy groups

see at K(n)-local stable homotopy theory (…)

### Relation to coherent objects

###### Proposition

∞Grpd is a coherent (∞,1)-topos and a locally coherent (∞,1)-topos. An object $X$, hence an ∞-groupoid, is an n-coherent object precisely if all its homotopy groups in degree $k \leq n$ are finite. Hence the fully coherent objects here are the homotopy types with finite homotopy groups.

## References

• G. J. Ellis, Spaces with finitely many non-trivial homotopy groups all of which are finite, Topology, 36, (1997), 501–504, ISSN 0040-9383.

• Jean-Louis Loday, Spaces with finitely many non-trivial homotopy groups (pdf)

Last revised on August 15, 2017 at 10:28:15. See the history of this page for a list of all contributions to it.