The category of simply-connected spaces is blessed with certain features that make homotopy theory tractable. In the first place, there is the Whitehead Theorem (Theorem 4.5) that tells us when a mapping of spaces of the homotopy type of CW-complexes is a homotopy equivalence—the necessary condition that the mapping induces an isomorphism of integral homology groups is also sufficient. Secondly, the Postnikov tower of a simply-connected space is a tower of principal fibrations pulled back via the κ-invariants of the space (Theorem 8bis.37). This makes cohomological obstruction theory accessible, if not computable ([Brown, E57], [Schön90], [Sergeraert94]). Furthermore, the system of local coefficients that arises in the description of the E2-term of the Leray-Serre spectral sequence is simple when the base space of a fibration is simply-connected, and the cohomology Eilenberg-Moore spectral sequence converges strongly for a fibration pulled back from such a fibration.

A defect of the category of simply-connected spaces is the fact that certain constructions do not stay in the category. The dishearteningly simple example is the based loop space functor—if (X, x0) is simply-connected, Ω(X, x0) need not be. Furthermore, the graded group-valued functor, the homotopy groups of a space, does not always distinguish distinct homotopy types of spaces that are not simply-connected. A classic example is the pair of spaces X1 = ℝP2m × S2n and X2 = S2m × ℝP2n; the homotopy groups in each degree κ are abstractly isomorphic, πκ (X1) ≅ πκ (X2).