(This is related to my earlier question on Kan's simplicial formula as Curtis mentions the link with the Hopf map, which has a very pretty formula that links well with the Samelson / Whitehead shuffle product mentioned there.)

In his 1950 paper on a Certain Exact Sequence, J. H. C. Whitehead defined a set of groups $\Gamma_n(K)$ for a pointed CW-complex. Briefly in the long exact sequence of the pair $(K^{(n)},K^{(n-1)})$ where $K^{(n)}$ is the $n$-skeleton o f$K$, $\Gamma_n(K)$ is the image of $\pi_n(K^{(n-1)})\to \pi_n(K^{(n)})$. He showed that $\Gamma_3(K)\cong \Gamma(\pi_2(K))$ where this $\Gamma$ is the `universal quadratic function' functor (which is introduced in that same paper).

What is known for general $K$, about $\Gamma_n(K)$? For instance, what universal properties might it have?

Noting that it is the vanishing of the $\Gamma_n$ that corresponds to the homotopy type of $K$ being faithfully modelled by the crossed complex (in the sense of Brown and Higgins) of the space $K$ filtered by its skeleta. As a crossed complex corresponds to a strict $\infty$-groupoid I presume these $\Gamma_n(K)$ measure the obstructions to `strictness' of the $\infty$-groupoid corresponding to $K$. (This is fairly clear in low dimensions if one looks at Conduché's 2-crossed modules as the obstruction / Peiffer lifting is clearly the interchange law of 2-category theory viewed from the crossed module / crossed complex perspective.)

What are the categorical obstructions to varying degrees of strictness of the homotopy type of a (weak) infinity groupoid / simplicial group(oid) and how do they interpret in terms of the $\Gamma_n$s?


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