Dual calculus for functors to spectra

Splitting criteria for homotopy functors of spectra

“…In this paper, we give a clear answer to the problem by introducing a new construction (originally developed in [1]) which relates P n F (respectively, P n F ) to P n−1 F (respectively, P n−1 F ) and Tate obstructions. This construction leads to new splitting criteria which generalize the results in [6], and sheds some light on new obstructions for splitting of both tower and cotower.…”

“…In this section, we will show that the splitting criteria in [6] are special cases of Theorem 2.10 above. As before, we will let F : Spec → Spec be a reduced homotopy functor.…”

“…×n be the diagonal functor, cr n the n th -cross effect, cr n the n th -cocross effect, and t the Tate map (see [6] for details). For any homotopy functor F : Spec → Spec, we define…”

“…An advantage of Johnson-McCarthy's Taylor tower is that it is dualizable. Therefore, in [6], Randy McCarthy constructs the so-called Taylor cotower (see Figure 2) of functors from pointed categories with finite products to the category of spectra, and he uses it to study the splitting phenomena of homotopy functors of spectra. It turns out that, in the case of homotopy functors of spectra, the theories of Taylor tower and cotower are equivalent if all Tate obstructions (i.e., Tate( ⊥ n F ; Σ n ) in Definition 1.3 below) vanish.…”

“…Throughout this paper, we will assume the reader to be familiar with notations, constructions and properties of calculus and dual calculus developed in [5] and [6]. We will also denote the closed model category of spectra (with stably weak equivalences) by Spec.…”