Coherent Taylor expansion as a bimonad

TL;DR

This paper develops a categorical framework for Taylor expansion using bimonads, extending coherent differentiation to non-additive categories and providing a general theory applicable to various semantic models.

Contribution

It introduces a bimonad-based approach to categorical Taylor expansion, generalizing previous coherent differentiation frameworks to broader categorical settings.

Findings

Defines a summability functor as a bimonad for Taylor expansion

Axiomatizes Taylor expansion via a distributive law in linear logic categories

Provides examples in denotational semantics with analytic structures

Abstract

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.