# Contents

## 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.

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)-truncated(-1)-groupoid/truth valuemere proposition, h-proposition
h-level 20-truncateddiscrete space0-groupoid/setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/groupoid(2,1)-sheaf/stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoidh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoidh-3-groupoid
h-level $n+2$$n$-truncatedhomotopy n-typen-groupoidh-$n$-groupoid
h-level $\infty$untruncatedhomotopy type∞-groupoid(∞,1)-sheaf/∞-stackh-$\infty$-groupoid

Revised on April 25, 2013 21:39:46 by Urs Schreiber (82.169.65.155)