An Idempotent Completion Functor in Homotopy Theory

Abstract

We observe that Artin-Mazur style $R$-completions ($R$ is a commutative ring with identity) induce analogous idempotent completions on the weak prohomotopy category pro-Ho(Top). Because Ho(Top) is a subcategory of pro-Ho(Top) and pro-Ho(Top) is closely related to the topologized homotopy category of J. F. Adams and D. Sullivan, our construction represents the Sullivan completions as homotopy limits of idempotent functors. In addition, we show that the Sullivan completion is idempotent on those spaces (in analogy with the Bousfield and Kan ${R_\infty }$-completion on $R$-good spaces) for which its cohomology with coefficients in $R$ agrees with that of our Artin-Mazur style completion. Finally, we rigidify the Artin-Mazur completion to obtain an idempotent Artin-Mazur completion on a category of generalized prospaces which preserves fibration and suitably defined cofibration sequences. (Our previous results on idempotency and factorization lift to the rigid completion.) Our results answer questions of Adams, Sullivan, and, later, A. Deleanu.