[Paper Review] A Tangent Category Alternative to the Faà di Bruno Construction

Jean-Simon Pacaud Lemay|arXiv (Cornell University)|May 4, 2018

Homotopy and Cohomology in Algebraic Topology4 citations

TL;DR

This paper presents a new construction of cofree Cartesian differential categories using tangent category theory, replacing the complex Faà di Bruno formula with a simpler, functorially based chain rule. The key contribution is an equivalent cofree Cartesian differential category structure defined via D-sequences, which simplifies composition and differential combinator operations by leveraging the tangent functor, offering a more intuitive and less combinatorial alternative to the original Faà di Bruno construction.

ABSTRACT

The Faà di Bruno construction, introduced by Cockett and Seely, constructs a comonad $\mathsf{Fa{\grave{a}}}$ whose coalgebras are precisely Cartesian differential categories. In other words, for a Cartesian left additive category $\mathbb{X}$, $\mathsf{Fa{\grave{a}}}(\mathbb{X})$ is the cofree Cartesian differential category over $\mathbb{X}$. Composition in these cofree Cartesian differential categories is based on the Faà di Bruno formula, and corresponds to composition of differential forms. This composition, however, is somewhat complex and difficult to work with. In this paper we provide an alternative construction of cofree Cartesian differential categories inspired by tangent categories. In particular, composition defined here is based on the fact that the chain rule for Cartesian differential categories can be expressed using the tangent functor, which simplifies the formulation of the higher order chain rule.

Motivation & Objective

• To address the complexity and combinatorial burden of the Faà di Bruno construction in defining cofree Cartesian differential categories.
• To develop an alternative, more intuitive construction of cofree Cartesian differential categories using the framework of tangent categories.
• To simplify the formulation of higher-order chain rules by expressing them through the tangent functor rather than symmetric trees and multinomial coefficients.
• To generalize the construction beyond additive categories by introducing pre-D-sequences for arbitrary categories with finite products.
• To establish equivalence between the new construction and the original Faà di Bruno comonad, ensuring the new structure retains the universal cofree property.

Proposed method

• Introduce pre-D-sequences as generalized sequences of morphisms in categories with finite products, forming the foundation for the new construction.
• Define D-sequences as a special subclass of pre-D-sequences that satisfy the differential combinator axioms of Cartesian differential categories.
• Construct composition of D-sequences using the tangent functor's action, where the chain rule is expressed via the natural transformation T(fg) = T(f)T(g), simplifying higher-order composition.
• Define the differential combinator D on D-sequences as a shift operation that maps fn to fn+1, aligning with the standard differential combinator in Cartesian differential categories.
• Use the tangent functor's naturality and product preservation to define the differential combinator and composition in a way that avoids symmetric tree notation.
• Prove that the resulting category of D-sequences forms a Cartesian differential category and is equivalent to the Faà di Bruno construction via a comonad equivalence.

Experimental results

Research questions

• RQ1 Can a cofree Cartesian differential category be constructed without relying on the Faà di Bruno formula's complex combinatorics?
• RQ2 Can the higher-order chain rule in Cartesian differential categories be reformulated using the tangent functor instead of symmetric trees?
• RQ3 Is there a natural generalization of D-sequences to categories without additive structure, enabling differentiation in broader categorical contexts?
• RQ4 Does the new construction preserve the universal property of cofreeness, i.e., is it equivalent to the Faà di Bruno comonad?
• RQ5 Can the differential combinator and composition be defined more intuitively using tangent category structures rather than multinomial coefficients?

Key findings

• The new construction defines a cofree Cartesian differential category via D-sequences, which are equivalent to the Faà di Bruno construction in the sense of comonad coalgebras.
• Composition in the new construction is defined via the tangent functor's functoriality, avoiding the need for symmetric trees and multinomial coefficients.
• The differential combinator is given by a simple shift operation, and linearity of D-sequences is characterized by f• = i•·f0.
• The construction generalizes to arbitrary categories with finite products through pre-D-sequences, enabling differentiation in non-additive settings.
• The category of D-sequences forms a Cartesian differential category, and the resulting comonad is equivalent to the Faà di Bruno comonad, preserving the universal cofree property.
• The paper demonstrates that the tangent functor provides a cleaner, more structural formulation of the higher-order chain rule, replacing the combinatorial Faà di Bruno formula.