Module postnikov 

The Postnikov k-invariant — the gluing data of a 2-type, and the obstruction to breaking it into a product.

two_type built K(A,2) and kan_complex built K(G,1). A general 2-truncated ∞-groupoid is not just a π₁ and a π₂ sitting side by side: by Mac Lane–Whitehead / Sinh’s theorem it is classified by (π₁ = G, π₂ = A, the G-action, k ∈ H³(G; A)). The class k — the Postnikov invariant — is the associator of the corresponding 2-group, and it says how the two levels are twisted:

• k = 0 ⟺ the 2-type is the product K(G,1) × K(A,2) — the levels decouple.
• k ≠ 0 ⟺ a genuinely entangled 2-type, provably not a product.

So k is “the single object carrying all πₙ with their interactions” reduced to one checkable cohomology class. And it is, once more, a symmetry-breaking obstruction: splitting the 2-type into a product is the ultimate symmetry break (decoupling π₁ from π₂), and k ≠ 0 is exactly the obstruction to it — the higher sibling of the deadlock β₁. We compute group cohomology directly (A = Z/modulus, trivial action) and crush the canonical case H³(Z/2; Z/2) = Z/2: the cup-cube cocycle is a 3-cocycle that is not a coboundary, so its 2-type is no product.

Functions

bockstein

The Bockstein β : Hⁿ(−; Z/2) → Hⁿ⁺¹(−; Z/2) — the connecting homomorphism of the short exact sequence 0 → Z/2 →(×2) Z/4 → Z/2 → 0. By a theorem of Steenrod, β = Sq¹. Computed honestly: lift the Z/2-cochain to Z/4, take δ over Z/4 (which is even on a Z/2-cocycle), divide by 2, and reduce mod 2. The lowest genuine Steenrod square, reached as a connecting map rather than a cup-square.

coboundary

The group-cohomology coboundary δⁿ : Cⁿ(G; A) → Cⁿ⁺¹(G; A), with A = Z/modulus and trivial G-action. f is a length-order^n table; the result is length order^{n+1}.

cohomology_size

|Hⁿ(G; A)| — cocycles modulo coboundaries (both are subgroups of Cⁿ, so the order is the index).

cup

The cup product Cᵖ × Cᵠ → Cᵖ⁺ᵠ (Alexander–Whitney; Z/modulus coefficients, trivial action, no signs): (f ∪ h)(a₁,…,a_{p+q}) = f(a₁,…,a_p) · h(a_{p+1},…,a_{p+q}). It makes the obstruction groups a graded ring H*(G; A) — the multiplicative algebra binding all the k-invariant levels.

cup_cube_z2

The canonical nontrivial 3-cocycle (the cup-cube x³), generator of H³(Z/2; Z/2) = Z/2.

cup_power_z2

The cup-power xⁿ of Z/2 with Z/2 coefficients: α(g_1,…,g_n) = g_1·g_2·⋯·g_n (1 iff every argument is 1). As the n-fold cup product of the nonzero 1-cocycle, it is an n-cocycle and the generator of Hⁿ(Z/2; Z/2) = Z/2 — the explicit nonzero obstruction at level n.

cup_square

The cup-square cohomology operation x ↦ x ∪ x : Hⁿ(−; A) → H²ⁿ(−; A). Over Z/2 it is the TOP Steenrod square Sqⁿ — the first genuine cohomology operation (a natural transformation of the functor Hⁿ(−;A)), the secondary structure carried by the cohomology the Eilenberg–MacLane spectrum represents.

is_coboundary

Is f an n-coboundary? (f = δβ for some (n-1)-cochain β)

is_cocycle

Is f an n-cocycle? (δf = 0)