Smallest 3-category not equivalent to a strict 3-category

Question

For modular lattices there is a canonical example of the "smallest" non-modular lattice, $\mathbf N_5$. Is there a similar example for 3-categories which are not equivalent to any strict 3-category? I have read so many times that there do exist such 3-categories but I have never seen an explicit example of one.

Answer 1

Simply connected 3-groupoids $X$ have two homotopy groups $\pi_2(X), \pi_3(X)$, and are classified by this pair of homotopy groups together with a $k$-invariant, namely a cohomology class in $H^4(B^2 \pi_2, \pi_3)$. It turns out that $X$ is strictifiable iff the $k$-invariant vanishes iff $X$ is just the product $B^2 \pi_2 \times B^3 \pi_3$.

The case of the fundamental $3$-groupoid of $S^2$ corresponds to $\pi_2 = \pi_3 = \mathbb{Z}$ with $k$-invariant the element of

$$H^4(B^2 \pi_2, \pi_3) \cong H^4(\mathbb{CP}^{\infty}, \mathbb{Z}) \cong \mathbb{Z}$$

which as a cohomology operation $H^2(-, \mathbb{Z}) \to H^4(-, \mathbb{Z})$ is just the cup square. A "smaller" example is to take $\pi_2 = \pi_3 = \mathbb{Z}_2$ with $k$-invariant the cup square $H^2(-, \mathbb{Z}_2) \to H^4(-, \mathbb{Z}_2)$.

Answer 2

The homotopy $3$-groupoid of $S^2$ is not equivalent to a strict $3$-category. While it may or may not be the strictly smallest weak 3-category with this property, this seems to be the standard example.

This was, as far as I know, first proven by Clemens Berger. For a reference, see Carlos T. Simpson's book "Homotopy theory of higher categories", Corollary I.4.3.3. and section I.4.4. in this draft.