The Eckmann–Hilton dual of the Postnikov decomposition of a space is the homology decomposition (or Moore space decomposition) of a space.
A Postnikov decomposition for a simply connected CWcomplex X is a commutative diagram
such that p_{n∗} : π_{r}(X) → π_{r}(P_{n}(X)) is an isomorphism for r ≤ n and π_{r}(P_{n}(X)) = 0 for r > n. Let F_{n} be the homotopy fiber of q_{n}. Then the exact sequence
π_{r+1}(P_{n}X) →^{qn∗} π_{r+1}(P_{n−1}X) → π_{r}(F_{n}) → π_{r}(P_{n}X) →^{qn∗} π_{r}(P_{n−1}X)
shows that F_{n} is an Eilenberg–MacLane space K(π_{n}X, n). Constructing P_{n+1}(X) inductively from P_{n}(X) requires knowing the nth kinvariant, which is a map of the form k_{n} : P_{n}(X) → Y_{n}. The space P_{n+1}(X) is then the homotopy fiber of k_{n}. Thus there is a homotopy fibration sequence
K(π_{n+1}X, n+1) → P_{n+1}(X) → P_{n}(X) → Y_{n}
This means that K(π_{n+1}X, n+1) is homotopy equivalent to the loop space ΩY_{n}. Consequently,
π_{r}(Y_{n}) ≅ π_{r−1}(ΩY_{n}) ≅ π_{r−1}(K(π_{n+1}X, n+1) = π_{n+1}X, r = n+2,
= 0, otherwise.
and we see that Y_{n} is a K(π_{n+1}X, n+2). Thus the nth kinvariant is a map k_{n} : P_{n}(X) → K(π_{n+1}X, n+2)
Note that it induces the zero map on all homotopy groups, but is not necessarily homotopic to the constant map. The original space X is weakly homotopy equivalent to the inverse limit of the P_{n}(X).
Applying the paradigm of Eckmann–Hilton duality, we arrive at the homology decomposition principle from the Postnikov decomposition principle by changing:

 the direction of all arrows
 π_{∗} to H_{∗}
 loops Ω to suspensions S
 fibrations to cofibrations and fibers to cofibers
 Eilenberg–MacLane spaces K(G, n) to Moore spaces M(G, n)
 inverse limits to direct limits
A homology decomposition (or Moore space decomposition) for a simply connected CWcomplex X is a commutative diagram
such that j_{n∗} : H_{r}(X_{≤n}) → H_{r}(X) is an isomorphism for r ≤ n and H_{r}(X_{≤n}) = 0 for
r > n. Let C_{n} be the homotopy cofiber of i_{n}. Then the exact sequence
H_{r}(X_{≤n−1}) →^{in∗} H_{r}(X_{≤n}) → H_{r}(C_{n}) →^{in∗} H_{r−1}(X_{≤n−1}) → H_{r−1}(X_{≤n})
shows that C_{n} is a Moore space M(H_{n}X, n). Constructing X_{≤n+1} inductively from X_{≤n} requires knowing the nth kinvariant, which is a map of the form k_{n} : Y_{n} → X_{≤n}.
The space X_{≤n+1} is then the homotopy cofiber of k_{n}. Thus there is a homotopy cofibration sequence
Y_{n} →^{kn} X_{≤n} →^{in+1} X_{≤n+1} → M(H_{n+1}X, n+1)
This means that M(H_{n+1}X, n+1) is homotopy equivalent to the suspension SY_{n}. Consequently,
H^{˜}_{r}(Y_{n}) ≅ H^{∼}_{r+1}(SY_{n}) ≅ H_{r+1}(M(H_{n+1}X, n+1)) = H_{n+1}X, r = n,
= 0, otherwise
and we see that Y_{n} is an M(H_{n+1}X, n). Thus the nth kinvariant is a map k_{n} : M(H_{n+1}X, n) → X_{≤n}
It induces the zero map on all reduced homology groups, which is a nontrivial statement to make in degree n:
k_{n∗} : H_{n}(M(H_{n+1}X, n)) ∼= H_{n+1}(X) → H_{n}(X) ∼= H_{n}(X_{≤n})
The original space X is homotopy equivalent to the direct limit of the X_{≤n}. The Eckmann–Hilton duality paradigm, while being a very valuable organizational principle, does have its natural limitations. Postnikov approximations possess rather good functorial properties: Let p_{n}(X) : X → P_{n}(X) be a stagen Postnikov approximation for X, that is, p_{n}(X)_{∗} : π_{r}(X) → π_{r}(P_{n}(X)) is an isomorphism for r ≤ n and π_{r}(P_{n}(X)) = 0 for r > n. If Z is a space with π_{r}(Z) = 0 for r > n, then any map g : X → Z factors up to homotopy uniquely through P_{n}(X). In particular, if f : X → Y is any map and p_{n}(Y) : Y → P_{n}(Y) is a stagen Postnikov approximation for Y, then, taking Z = P_{n}(Y) and g = p_{n}(Y) ◦ f, there exists, uniquely up to homotopy, a map p_{n}(f) : P_{n}(X) → P_{n}(Y) such that
homotopy commutes. Let X = S^{2} ∪_{2} e^{3} be a Moore space M(Z/2,2) and let Y = X ∨ S^{3}. If X_{≤2} and Y_{≤2} denote stage2 Moore approximations for X and Y, respectively, then X_{≤2} = X and Y_{≤2} = X. We claim that whatever maps i : X_{≤2} → X and j : Y_{≤2} → Y such that i_{∗} : H_{r}(X_{≤2}) → H_{r}(X) and j_{∗} : H_{r}(Y_{≤2}) → H_{r}(Y) are isomorphisms for r ≤ 2 one takes, there is always a map f : X → Y that cannot be compressed into the stage2 Moore approximations, i.e. there is no map f_{≤2} : X_{≤2} → Y_{≤2} such that
commutes up to homotopy. We shall employ the universal coefficient exact sequence for homotopy groups with coefficients. If G is an abelian group and M(G, n) a Moore space, then there is a short exact sequence
0 → Ext(G, π_{n+1}Y) →^{ι} [M(G, n), Y] →^{η} Hom(G, π_{n}Y) → 0,
where Y is any space and [−,−] denotes pointed homotopy classes of maps. The map η is given by taking the induced homomorphism on π_{n} and using the Hurewicz isomorphism. This universal coefficient sequence is natural in both variables. Hence, the following diagram commutes:
Here we will briefly write E_{2}(−) = Ext(Z/2,−) so that E_{2}(G) = G/2G, and E^{Y} (−) = Ext(−, π_{3}Y). By the Hurewicz theorem, π_{2}(X) ∼= H_{2}(X) ∼= Z/2, π_{2}(Y) ∼= H_{2}(Y) ∼= Z/2, and π_{2}(i) : π_{2}(X_{≤2}) → π_{2}(X), as well as π_{2}(j) : π_{2}(Y_{≤2}) → π_{2}(Y), are isomorphisms, hence the identity. If a homomorphism φ : A → B of abelian groups is onto, then E_{2}(φ) : E_{2}(A) = A/2A → B/2B = E_{2}(B) remains onto. By the Hurewicz theorem, Hur : π_{3}(Y) → H_{3}(Y) = Z is onto. Consequently, the induced map E_{2}(Hur) : E_{2}(π_{3}Y) → E_{2}(H_{3}Y) = E_{2}(Z) = Z/2 is onto. Let ξ ∈ E_{2}(H_{3}Y) be the generator. Choose a preimage x ∈ E_{2}(π_{3}Y), E_{2}(Hur)(x) = ξ and set [f] = ι(x) ∈ [X,Y]. Suppose there existed a homotopy class [f_{≤2}] ∈ [X_{≤2}, Y_{≤2}] such that
j_{∗}[f_{≤2}] = i_{∗}[f].
Then
η_{≤2}[f_{≤2}] = π_{2}(j)_{∗}η_{≤2}[f_{≤2}] = ηj_{∗}[f_{≤2}] = ηi^{∗}[f] = π_{2}(i)^{∗}η[f] = π_{2}(i)^{∗}ηι(x) = 0.
Thus there is an element ε ∈ E_{2}(π_{3}Y_{≤2}) such that ι_{≤2}(ε) = [f_{≤2}]. From ιE_{2}π_{3}(j)(ε) = j_{∗}ι_{≤2}(ε) = j_{∗}[f_{≤2}] = i^{∗}[f] = i^{∗}ι(x) = ιE^{Y} π_{2}(i)(x)
we conclude that E_{2}π_{3}(j)(ε) = x since ι is injective. By naturality of the Hurewicz map, the square
commutes and induces a commutative diagram upon application of E_{2}(−):
It follows that
ξ = E_{2}(Hur)(x) = E_{2}(Hur)E_{2}π_{3}(j)(ε) = E_{2}H_{3}(j)E_{2}(Hur)(ε) = 0,
a contradiction. Therefore, no compression [f_{≤2}] of [f] exists.
Given a cellular map, it is not always possible to adjust the extra structure on the source and on the target of the map so that the map preserves the structures. Thus the category theoretic setup automatically, and in a natural way, singles out those continuous maps that can be compressed into homologically truncated spaces.
[…] of flags of subspaces Vi with dimVi = i. For G = Sp(2n,C), the Borel subgroups B are the subgroups preserving a halfflag of isotropic subspaces and the quotient G/B is the variety of all such flags. Any parabolic subgroup P may be described as the subgroup that preserves some partial flag. Thus (partial) flag manifolds are in ICW. A special case is that of a maximal parabolic subgroup, preserving a single subspace V. The corresponding quotient SL(n, C)/P is a Grassmannian G(k, n) of kdimensional subspaces of Cn. For G = Sp(2n,C), one obtains Lagrangian Grassmannians of isotropic kdimensional subspaces, 1 ≤ k ≤ n. So Grassmannians are objects in ICW. The interleaf category is closed under forming fibrations. […]