Idempotent functors that preserve cofiber sequences and split suspensions

Abstract

We show that an $f$-localization functor $L_f$ commutes with cofiber sequences of $(N-1)$-connected finite complexes if and only if its restriction to the collection of $(N-1)$-connected finite complexes is $R$-localization for some unital subring $R\sseq\mathbb{Q}$. This leads to a homotopy-theoretical characterization of the rationalization functor: the restriction of $L_f$ to simply-connected spaces (not just the finite complexes) is rationalization if and only if $L_f(S^2)$ is nontrivial and simply-connected, $L_f$ preserves cofiber sequences of simply-connected finite complexes, and for each simply-connected finite complex $K$, $\s^k L_f(K)$ splits as a wedge of copies of $L_f(S^n)$ for large enough $k$ and various values of $n$.