Abstract. We construct a Taylor tower for functors from pointed categories to abelian categories via cotriples associated to cross effect functors. The tower was inspired by Goodwillie's Taylor tower for functors of spaces, and is related to Dold and Puppe's stable derived functors and Mac Lane's Q-construction. We study the layers, DnF = fiber(PnF → P n−1 F ), and the limit of the tower. For the latter we determine a condition on the cross effects that guarantees convergence. We define differentials for functors, and establish chain and product rules for them. We conclude by studying exponential functors in this setting and describing their Taylor towers.

In this paper we show that every semisimple Banach algebra over R or C has the uniqueness of norm property, that is we show that if 31 is a Banach algebra with each of the norms || ||, || ||' then these norms define the same topology. This result is deduced from a maximum property of the norm in a primitive Banach algebra (Theorem 1).In the following F is a field which may be taken throughout as R, the real field, or C, the complex field. If 36 is a normed space then (B(36) will denote the space of bounded linear operators on 36. LEMMA 1. Let F, G be closed sub spaces of the Banach space E such that F+G = E. Then there exists L>0 such that if xÇzE then there is an ƒ G F with(0 11/ 11 SL|W|.(ii) x-fEG. M\'*M\\4M'for all aG3l, £G$, where || -|| is the norm in 3Ï and || -||' the norm in 36.The theorem asserts that the natural map 3I-»(B(ï) is continuous. It is a much stronger version of [4, Theorem 2.2.7] but applicable only to primitive algebras. It would be interesting to know how far it can be generalized. is continuous then the map a-*ab -tab!;, being a composition of continuous maps, is continuous. Since 36 is strictly irreducible, if £=^0 we can, by a suitable choice of &, make &£ any particular vector in 36 and so if a->a% is continuous for one nonzero J it is continuous for all £ in 36. We shall deduce a contradic-537

Abstract. In this paper, we consider abelian functor calculus, the calculus of functors of abelian categories established by the second author and McCarthy. We carefully construct a category of abelian categories and suitably homotopically defined functors, and show that this category, equipped with the directional derivative, is a cartesian differential category in the sense of Blute, Cockett, and Seely. This provides an abstract framework that makes certain analogies between classical and functor calculus explicit. Inspired by Huang, Marcantognini, and Young's chain rule for higher order directional derivatives of functions, we define a higher order directional derivative for functors of abelian categories. We show that our higher order directional derivative is related to the iterated partial directional derivatives of the second author and McCarthy by a Faà di Bruno style formula. We obtain a higher order chain rule for our directional derivatives using a feature of the cartesian differential category structure, and with this provide a formulation for the nth layers of the Taylor tower of a composition of functors F • G in terms of the derivatives and directional derivatives of F and G, reminiscent of similar formulations for functors of spaces or spectra by Arone and Ching. Throughout, we provide explicit chain homotopy equivalences that tighten previously established quasi-isomorphisms for properties of abelian functor calculus.

We construct a functor of spaces, M" , and show that its multilinearization is equivalent to the nth layer of the Taylor tower of the identity functor of spaces. This allows us to identify the derivatives of the identity functor and determine their homotopy type.

We study functors F : C f → D where C and D are simplicial model categories and C f is the category consisting of objects that factor a fixed morphism f : A → B in C. We define the analogs of Eilenberg and Mac Lane's cross effect functors in this context, and identify explicit adjoint pairs of functors whose associated cotriples are the diagonals of the cross effects. With this, we generalize the cotriple Taylor tower construction of Deriving calculus with cotriples (by the second and third authors) from the setting of functors from pointed categories to abelian categories to that of functors from C f to S, a suitable category of spectra, to produce a tower of functors · · · → Γ n+1