nLab homotopy 3-type

Idea

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

Definition

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

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

See the general discussion in homotopy n-type.

Models

There are many useful algebraic models for a homotopy $3$-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 $3$-groupoid ought to qualify, by the homotopy hypothesis.

Related concepts

homotopy level	n-truncation	homotopy theory	higher category theory	higher topos theory	homotopy type theory
h-level 0	(-2)-truncated	contractible space	(-2)-groupoid		true/​unit type/​contractible type
h-level 1	(-1)-truncated	contractible-if-inhabited	(-1)-groupoid/​truth value	(0,1)-sheaf/​ideal	mere proposition/​h-proposition
h-level 2	0-truncated	homotopy 0-type	0-groupoid/​set	sheaf	h-set
h-level 3	1-truncated	homotopy 1-type	1-groupoid/​groupoid	(2,1)-sheaf/​stack	h-groupoid
h-level 4	2-truncated	homotopy 2-type	2-groupoid	(3,1)-sheaf/​2-stack	h-2-groupoid
h-level 5	3-truncated	homotopy 3-type	3-groupoid	(4,1)-sheaf/​3-stack	h-3-groupoid
h-level $n+2$	$n$-truncated	homotopy n-type	n-groupoid	(n+1,1)-sheaf/​n-stack	h-$n$-groupoid
h-level $\infty$	untruncated	homotopy type	∞-groupoid	(∞,1)-sheaf/​∞-stack	h-$\infty$-groupoid