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The aim of this paper is twofold. On the one hand, we show that the kernel $\overline{C(A)}$ of the Bousfield periodization functor $P_A$ is cellularly generated by a space $B$, i.e., we construct a space $B$ such that the smallest closed class $C(B)$ containing $B$ is exactly $\overline{C(A)}$. On the other hand, we show that the partial order $(Spaces,\gg)$ is a complete lattice, where $B\gg A$ if $B\in C(A)$. Finally, as a corollary we obtain Bousfield's theorem, which states that $(Spaces,>)$ is a complete lattice, where $B>A$ if $B\in\overline{C(A)}$. [ABSTRACT FROM AUTHOR]

Let M be a simplicial model category and J : M ? M a simplicial coaugmented functor. Given an object X, the assignment n? J n+1 X defines a cofacial resolution (an augmented cosimplicial space without its codegeneracy maps). Following Bousfield and Kan we define J s X = tot s([ n] ? J n+1 X). An object X is called J-injective if it is a retract of JX in Ho(M) via the natural map. We show that certain homotopy limits of J-injective objects are J s-injective. Our method is to use the notion of pro-weak equivalences which was first introduced in a different language and context by David Edwards and Harold Hastings. The key observation is that a cofacial resolution X (-1) ? X which admits a left contraction gives rise to a pro-weak equivalence of towers { X(-1)} s=0?{tot S X} s = 0. [ABSTRACT FROM AUTHOR]