  We extend the recently introduced setting of coherent differentiation for
taking into account not only differentiation, but also Taylor expansion in
categories which are not necessarily (left)additive.The main idea consists in
extending summability into an infinitary functor which intuitively maps any
object to the object of its countable summable families.This functor is endowed
with a canonical structure of bimonad.In a linear logical categorical setting,
Taylor expansion is then axiomatized as a distributive law between this
summability functor and the resource comonad (aka.~exponential), allowing to
extend the summability functor into a bimonad on the Kleisli category of the
resource comonad: this extended functor computes the Taylor expansion of the
(nonlinear) morphisms of the Kleisli category.We also show how this categorical
axiomatizations of Taylor expansion can be generalized to arbitrary cartesian
categories, leading to a general theory of Taylor expansion formally similar to
that of differential cartesian categories, although it does not require the
underlying cartesian category to be left additive.We provide several examples
of concrete categories which arise in denotational semantics and feature such
analytic structures.
