structure on the set $[X,{{P_{{n}}}}]^A_B$ of homotopy classes of diagonals, which only exists on a “stable” part ${{n}}\leq 2{{d}}$ and, of course, only if this set is non-empty. It comes from an “up to homotopy” (fibrewise) abelian group structure on Moore–Postnikov stages (of a certain pullback of $Y'{\rightarrow}B$) which we construct algorithmically – this is the heart of the present paper. We remark that the abelian group structure on $[X,{{P_{{n}}}}]^A_B$ was already observed in [@McClendon]; however, this paper did not deal with algorithmic aspects.

In the stable part of the Moore–Postnikov tower, the natural map $[X,{{P_{{n}}}}]^A_B\to[X,{{P_{{{n}}-1}}}]^A_B$ is a group homomorphism and the above mentioned computation of the preimages of a given homotopy class $[\ell]$ may be reduced to a *finite* set of generators of the image; the computation is conveniently summarized in a long exact sequence . This finishes the rough description of our inductive computation.

New tools {#new-tools .unnumbered}
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In the process of building the Moore–Postnikov tower and also later, it is important to work with infinite simplicial sets, such as the Moore–Postnikov stages $P_n$, in an algorithmic way. This is handled by the so-called *equivariant* effective algebraic topology and effective homological algebra. The relevant non-equivariant results are 