This is related to Transversality in the proof of the Blakers-Massey Theorem. Is it necessary?, and other questions, such as the intuition behind the Freudenthal suspension theorem.

The answers and comments to the question give references to nice proofs of the connectivity result. But of course Blakers and Massey proved an algebraic result for the critical (i.e. first non-vanishing) triad group $\pi_n(X;A,B)$ as a tensor product in the case $A \cap B$ simply connected.

The result was generalised to the $m$-ad and non simply connected case by Ellis and Steiner in "Higher-dimensional crossed modules and the homotopy groups of $(n+1)$-ads" *J. Pure Appl. Algebra* {46} (1987) 117--136, see Theorem 3.7, using in an essential way the van Kampen theorem for $n$-cubes of spaces proved in the paper by Brown and Loday Topology 26 (1987) 311-334 (available here).

Now the proof of the latter theorem involves sophisticated tools in algebraic topology, of which surely Loday's cat$^n$-groups are essential.

Question 1. Is it possible to algebraicise the connectivity proof or proofs so that they give at least some part of the results on the critical group, including the non simply connected case?

Question 2. Is it possible to give a more geometric proof, based on the techniques pioneered by Puppe, to give a new proof of the theorem Loday and I proved, but instead by a direct verification of the universal property of the proposed colimit?

Note that of course the determination given of the critical group implies by induction the connectivity result. The proof in the paper in Topology goes by induction using the algebraic result in one dimension to prove connectivity in the next dimension.

The point of Question 1 is that if one has clever and nice proofs of a connectivity result, can the arguments be given an algebraic model so that by similar arguments one gets an algebraic result? Is it currently like trying to use arguments as in the proof of the usual Seifert-van Kampen Theorem, but without having a fundamental group or groupoid?