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Fibration theorems and the Taylor tower of the identity for spectral operadic algebras

Schonsheck, Nikolas
http://rave.ohiolink.edu/etdc/view?acc_num=osu161839958092492

Abstract Details

2021, Doctor of Philosophy, Ohio State University, Mathematics.
We work in the setting of algebras over a reduced operad in spectra. Our first main result establishes a spectral algebra analog of the Bousfield-Kan “fibration lemma” under appropriate conditions. In particular, we show that completion with respect to topological Quillen homology (or TQ-completion, for short) preserves homotopy fibration sequences provided that the base and total O-algebras are connected. Our argument essentially boils down to proving that the natural map from the homotopy fiber to its TQ-completion tower induces a pro-isomorphism on homotopy groups. More generally, we also show that similar results remain true if we replace “homotopy fibration sequence” with “homotopy pullback square.” Our second main result concerns the convergence of the Taylor tower of the identity functor in O-algebras. Specifically, we show that if A is a (−1)-connected O-algebra with 0-connected TQ-homology spectrum TQ(A), then there is a natural weak equivalence between the limit of the Taylor tower of the identity functor evaluated on A and the TQ-completion of A. Since, in this context, the identity functor is only known to be 0-analytic, this result extends knowledge of the Taylor tower of the identity beyond its “radius of convergence.”
John Harper (Advisor)
Niles Johnson (Committee Member)
Crichton Ogle (Committee Member)
50 p.
Mathematics
Algebraic topology; homotopy theory; completion; localization; functor calculus; Quillen homology

Recommended Citations

Citations

  • Schonsheck, N. (2021). Fibration theorems and the Taylor tower of the identity for spectral operadic algebras [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu161839958092492

    APA Style (7th edition)

  • Schonsheck, Nikolas. Fibration theorems and the Taylor tower of the identity for spectral operadic algebras. 2021. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu161839958092492.

    MLA Style (8th edition)

  • Schonsheck, Nikolas. "Fibration theorems and the Taylor tower of the identity for spectral operadic algebras." Doctoral dissertation, Ohio State University, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=osu161839958092492

    Chicago Manual of Style (17th edition)

osu161839958092492
188
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