homotopy 3-type


Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




A homotopy 33-type is a homotopy type where we consider its properties only up to the 33nd homotopy group π 3\pi_3, a homotopy n-type for n=3n = 3


A continuous map XYX \to Y is a homotopy 33-equivalence if it induces isomorphisms on π i\pi_i for 0i30 \leq i \leq 3 at each basepoint. Two spaces share the same homotopy 33-type if they are linked by a zig-zag chain of homotopy 33-equivalences.

For any nice space XX, you can kill its homotopy groups in higher dimensions by attaching cells, thus constructing a new space YY so that the inclusion of XX into YY is a homotopy 33-equivalence; up to (weak) homotopy equivalence, the result is the same for any space with the same homotopy 33-type. Accordingly, a homotopy 33-type may alternatively be defined as a space with trivial π i\pi_i for i>3i \gt 3, or as the unique (weak) homotopy type of such a space, or as its fundamental \infty-groupoid (which should be a 33-groupoid).

See the general discussion in homotopy n-type.


There are many useful algebraic models for a homotopy 33-type. (Assume the homotopy type is connected for simplicity.)

  1. 2-crossed modules
  2. crossed squares
  3. cat-2-groups
  4. Gray-groups: one-object groupoidal Gray-categories

One measure of the usefulness of a given model may be its ease of calculation (e.g., with a generalised van Kampen theorem) and of extraction of topologically significant invariants. In the above a lot more is known, from this viewpoint, about the second and third model than for the first.

Of course, any sufficient weak notion of 33-groupoid ought to qualify, by the homotopy hypothesis.

homotopy leveln-truncationhomotopy theoryhigher category theoryhigher topos theoryhomotopy type theory
h-level 0(-2)-truncatedcontractible space(-2)-groupoidtrue/​unit type/​contractible type
h-level 1(-1)-truncatedcontractible-if-inhabited(-1)-groupoid/​truth value(0,1)-sheaf/​idealmere proposition/​h-proposition
h-level 20-truncatedhomotopy 0-type0-groupoid/​setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/​groupoid(2,1)-sheaf/​stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoid(3,1)-sheaf/​2-stackh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoid(4,1)-sheaf/​3-stackh-3-groupoid
h-level n+2n+2nn-truncatedhomotopy n-typen-groupoid(n+1,1)-sheaf/​n-stackh-nn-groupoid
h-level \inftyuntruncatedhomotopy type∞-groupoid(∞,1)-sheaf/​∞-stackh-\infty-groupoid

Revised on April 25, 2013 21:39:46 by Urs Schreiber (