2012/09/11 by Rosona Eldred, Eldred, Rosona
Mathematics · #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1209.2384
openalex publication_date 2012/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we show that for reduced homotopy endofunctors of spaces, F, and for all n ≥ 1 there are adjoint functors Rn, Ln with Tn F ≃ Rn F Ln, where Pn F is the n-excisive approximation to F, constructed by taking the homotopy colimit over iterations of Tn F. This then endows Tn of the identity with the structure of a monad and the Tn F's are the functor version of bimodules over that monad. It follows that each Tn F (and PnF) takes values in spaces of symmetric Lusternik-Schnirelman cocategory n, as defined by Hopkins. This also recovers recent results of Chorny-Scherer. The spaces Tn F(X) are in fact classically nilpotent (in the sense of Berstein-Ganea) but not nilpotent in the sense of Biedermann and Dwyer. We extend the original constructions of dual calculus to our setting, establishing the n-co-excisive approximation for a functor, and dualize our constructions to obtain analogous results concerning constructions Tn, Pn,and LS category.