[Paper Review] Differential restriction categories

J.R.B. Cockett, G. S. H. Cruttwell+1 more|arXiv (Cornell University)|Aug 20, 2012

Homotopy and Cohomology in Algebraic TopologyMathematics17 references21 citations

TL;DR

This paper introduces differential restriction categories, an algebraic framework that unifies cartesian differential categories and restriction categories to model partial smooth maps on open subsets of R^n. It establishes that differential structure lifts through both join completion and classical completion, showing that even germs of functions at points—like those defined only at a single point—can carry well-behaved differentials, thereby extending differential calculus to partial and local maps in a purely algebraic way.

ABSTRACT

We combine two recent ideas: cartesian differential categories, and restriction categories. The result is a new structure which axiomatizes the category of smooth maps defined on open subsets of $\R^n$ in a way that is completely algebraic. We also give other models for the resulting structure, discuss what it means for a partial map to be additive or linear, and show that differential restriction structure can be lifted through various completion operations.

Motivation & Objective

• To formalize a category-theoretic framework for partial smooth maps, such as those defined on open subsets of R^n, using algebraic axioms.
• To unify cartesian differential categories and restriction categories into a new structure that captures the algebraic essence of differentiation and partiality.
• To define and characterize additive, linear, and strongly additive maps in the context of partial maps.
• To demonstrate that differential structure is preserved under completion operations, particularly join and classical completion.
• To show that even local or pointwise-defined maps (germs) can be meaningfully differentiated within this framework.

Proposed method

• Introduces differential restriction categories as a combination of cartesian differential categories and restriction categories, with axioms for differentiation, restriction, and their interaction.
• Defines left additive restriction categories and uses joins of compatible maps to model partiality and compatibility.
• Applies the join completion construction to freely add joins of compatible partial maps, proving that differential structure lifts to the completed category.
• Applies the classical completion construction to allow classical reasoning about restriction categories, showing differential structure lifts via germs of functions.
• Uses the fractional monad and rational functions to construct models of the theory, particularly in the context of weak rigs to handle non-distributive fraction constructions.
• Proves that the classical completion of a differential restriction category remains a differential restriction category by showing that germs of functions at points inherit well-defined derivatives.

Experimental results

Research questions

• RQ1 Can differential calculus be algebraically axiomatized for partial maps defined on open subsets of R^n, such as smooth functions with isolated singularities?
• RQ2 How can the concepts of additivity and linearity be meaningfully extended to partial maps in a categorical setting?
• RQ3 Does differential structure on a restriction category lift through the join completion, which freely adds joins of compatible partial maps?
• RQ4 Can differential structure be preserved under the classical completion of a restriction category, which allows reasoning about maps defined on closed sets or germs?
• RQ5 What is the categorical and algebraic nature of differentials for functions defined only at a point or on a closed set, and how does this relate to germs?

Key findings

• The classical completion of a differential restriction category is again a differential restriction category, showing that differential structure is compatible with classical reasoning about partial maps.
• Differential structure lifts through the join completion, preserving joins of compatible partial maps and ensuring that the derivative commutes with join operations.
• Maps defined only at a point—such as f(x) = 2x at x = 5—can be interpreted as germs, and their differentials are well-defined within the framework.
• The construction of rational functions via the fractional monad provides a model of the theory, though it requires working with weak rigs due to non-distributivity of the fraction construction.
• The unit functor from a differential restriction category to its classical completion preserves additivity, strong additivity, and linearity, ensuring structural consistency across completions.
• The theory provides a semantic foundation for the simply typed differential lambda calculus in settings with partiality and resource-sensitive computation.