# Homotopy functor calculus vs functor calculus in additive categories

Consider the Goodwillie calculus of a homotopy functor $$F : \mathrm{Sp} \to \mathrm{Sp}$$, where $$\mathrm{Sp}$$ denotes an appropriate model for spectra (... orthogonal spectra for instance).

Then Goodwillie (and others) produced a Taylor tower of approximations to $$F$$, where the levels of the tower are spectra with a $$\Sigma_n$$-action.

Calculus of functors was also developed by Johnson and McCarthy, for additive categories, which produces an analogous tower associated to functors between additive categories.

I am aware that Johnson and McCarthy consider when the two towers are the same, or more precisely when the functors in their additive sense, fit into a Goodwillie tower. My question concerns the opposite direction:

When we consider the homotopy category of spectra, it is a triangulated category, hence additive. Has anyone ever considered how the Goodwillie and Johnson-McCarthy towers relate in this instance?

That is, given a homotopy functor $$F: \mathrm{Sp} \to \mathrm{Sp}$$ it will produce a tower in the sense of Goodwillie, but we can consider $$F$$ as a functor between homotopy cateogories, $$F:\mathrm{Ho}(\mathrm{Sp}) \to \mathrm{Ho}(\mathrm{Sp})$$, and we get another tower in the sense of Johnson-McCarthy. As such my question may be rephrased as follows:

When does the Goodwillie tower and the Johnson-McCarthy tower coincide for a homotopy functor $$F: \mathrm{Sp} \to \mathrm{Sp}$$?

• If I am not mistaken, the construction of Johnson-McCarthy requires the existence of certain colimits (e.g., quotients). I am not sure you can apply it to a functor between homotopy categories. – Gregory Arone Oct 19 '18 at 5:33