Module eilenberg_maclane 

The general Eilenberg–MacLane construction K(A, n) — one engine, every rung of the tower.

two_type built K(A, 2) by hand. Every space in the Postnikov tower is glued from K(πₙ, n)’s, so the honest way to “keep climbing” is to build the general K(A, n) for any n, not a fresh module per dimension. This is the Dold–Kan model Γ(A[n]): an m-simplex is an A-labeling of the order-preserving surjections [m] ↠ [n] (the chain complex is concentrated in degree n, so a face that fails to stay surjective dies — C_{n−1} = C_{n+1} = 0). It is a simplicial abelian group, hence a Kan complex, and we verify by enumeration, for n = 2, 3:

• πₙ = A: the inner horn at degree n has |A| fillers — the n-cells.
• Kan: every horn fills (the face-tuple map is onto the compatible horns).
• exactly n-truncated: inner horns at degree n + 2 fill uniquely — no π_{n+1}.

n = 3 is the first genuine 3-type object the framework possesses (K(A, 2) was the first 2-type).

Functions

homology_size

The k-th homology |H_k(K(A,n))| via the normalized (Moore) chain complex: N_k = ⋂_{i≥1} ker dᵢ with differential d_0. Computed independently of the horn-filling homotopy extractor, so that their agreement is the genuine Hurewicz statement, not a tautology.

homotopy_group_size

Extract the homotopy group π_k of K(A, n) directly from the Kan complex, by horn-filling. This is the homotopy hypothesis realized computationally — given the ∞-groupoid object, read its homotopy back out. Returns |π_k|: |A| at k = n, trivial (1) everywhere else. (K(A, n≥2) is 1-connected, so π_0 = π_1 = 1 by construction.)

is_omega_spectrum

Whether the Eilenberg–MacLane spaces {K(A, n)} form an Ω-spectrum: each is the loop space of the next, so the homotopy is invariant under the simultaneous shift n ↦ n+1, k ↦ k+1 (π_k(K(A,n)) = π_{k+1}(K(A,n+1))). That deloopability is what makes the sequence a spectrum.

stable_homotopy

The stable homotopy of the Eilenberg–MacLane spectrum HA: π_k(HA) = colim_n π_{n+k}(K(A,n)). For HA it is concentrated in degree 0 (|A| at k = 0, trivial elsewhere) — exactly what makes HA represent ordinary cohomology Hⁿ(−; A). Read off the stable range (N = 4).