nLab finite homotopy type

Definition

Definition

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

• a simplicial set with a finite number of non-degenerate simplices;

• a CW-complex with a finite number of cells.

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.

Related concepts

• π-finite homotopy type, finite type

• finite set, hereditarily finite set

• finite category, finite limit

• finite spectrum

• finite (infinity,1)-category, finite (infinity,1)-limit

• p-local homotopy type, p-complete homotopy type, rational homotopy type

• (infinity,1)-presheaves on $\infty Grpd_{fin}^{op}$ form the classifying (infinity,1)-topos for objects.

References

• Doug Ravenel, The stable homotopy theory of finite complexes, in Ioan Mackenzie James (ed.) Handbook of Algebraic Topology (1995)