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.

翻译：我们扩展了最近引入的相干微分框架，使其不仅能处理微分，还能处理未必是（左）可加的范畴中的泰勒展开。主要思路是将可和性扩展为一个无穷函子，该函子直观地将任何对象映射到其可数可和族对象。这个函子具有双幺半群的标准结构。在线性逻辑范畴环境中，泰勒展开被公理化为该可和性函子与资源余幺半群（即指数）之间的分配律，从而允许将可和性函子扩展为资源余幺半群的克莱斯利范畴上的双幺半群：这个扩展函子计算克莱斯利范畴中（非线性）态射的泰勒展开。我们还展示了泰勒展开的这种范畴论公理化如何推广到任意笛卡尔范畴，从而产生一种形式上类似于微分笛卡尔范畴的泰勒展开通用理论，尽管它不要求底层笛卡尔范畴是左可加的。我们提供了几个具体范畴的例子，这些例子出现在指称语义中并具有这种解析结构。