[Paper Review] A convenient differential category

Richard Blute, Thomas Ehrhard+1 more|arXiv (Cornell University)|Jun 16, 2010

Homotopy and Cohomology in Algebraic Topology4 citations

TL;DR

This paper establishes that the category of Mackey-complete, separated, topological convex bornological vector spaces—known as convenient vector spaces—forms a differential category, providing a categorical model for differential linear logic. By leveraging bornological structures and the characterization of smooth maps via smooth curves, the authors construct a codereliction map using the derivative at zero, proving the category satisfies the axioms of a differential category with a well-defined notion of differentiation.

ABSTRACT

In this paper, we show that the category of Mackey-complete, separated, topological convex bornological vector spaces and bornological linear maps is a differential category. Such spaces were introduced by Frölicher and Kriegl, where they were called convenient vector spaces. While much of the structure necessary to demonstrate this observation is already contained in Frölicher and Kriegl's book, we here give a new interpretation of the category of convenient vector spaces as a model of the differential linear logic of Ehrhard and Regnier. Rather than base our proof on the abstract categorical structure presented by Frölicher and Kriegl, we prefer to focus on the bornological structure of convenient vector spaces. We believe bornological structures will ultimately yield a wide variety of models of differential logics.

Motivation & Objective

• To establish a categorical model of differential linear logic using convenient vector spaces.
• To demonstrate that the category of convenient vector spaces supports a well-defined notion of differentiation compatible with differential category axioms.
• To reinterpret the structure of convenient vector spaces through bornological linear maps, emphasizing their suitability for modeling higher-order smooth functionals.
• To connect the abstract framework of differential categories with concrete analytical structures in infinite-dimensional analysis.
• To lay the foundation for future work on integral logic and holomorphic function categories in similar settings.

Proposed method

• The authors define smooth maps between convenient vector spaces as those preserving smooth curves, based on Boman's theorem and the monoid of smooth real maps.
• They construct the comonad !E using the bornological tensor product and completion, with !E isomorphic to the space of smooth curves from R to E.
• The codereliction map is defined as the derivative at zero of the inclusion map ι: E → !E, given by coder(v) = lim_{t→0} (δ_{tv} - δ_0)/t.
• The differential category structure is verified by checking the axioms, including the Leibniz rule and the codereliction identity, using componentwise limits in the tensor product.
• The bialgebra structure on !E is defined via diagonal, counit, sum, and zero maps, with Δ(δ_x) = δ_x ⊗ δ_x and ε(δ_x) = 1.
• The proof relies on the fact that the derivative in this category corresponds to the standard Fréchet derivative, ensuring consistency with classical analysis.

Experimental results

Research questions

• RQ1 Can the category of convenient vector spaces serve as a model for differential linear logic?
• RQ2 How can the notion of smoothness in infinite-dimensional spaces be characterized without relying on norms?
• RQ3 What is the categorical structure of the coKleisli category of convenient vector spaces, and does it support a differential category structure?
• RQ4 Is the codereliction map in this category naturally derived from the derivative of a canonical inclusion?
• RQ5 Can this framework be extended to holomorphic or real-analytic functions, and what logical structure would emerge?

Key findings

• The category Con of convenient vector spaces equipped with bornological linear maps is a differential category, satisfying all required axioms for differential linear logic.
• The codereliction map is explicitly constructed as the derivative at zero of the inclusion ι: E → !E, given by coder(v) = lim_{t→0} (δ_{tv} - δ_0)/t.
• The Leibniz rule is verified by showing that the derivative of Δ(δ_{tv}) matches the tensor product of the codereliction map with itself and the identity, using componentwise limits.
• The identity coder;ε = id is proven by showing that the evaluation of the codereliction map on v yields v through the limit lim_{t→0} (tv)/t = v.
• The category Con is shown to be a model of intuitionistic linear logic, with finite biproducts and a well-defined bialgebra structure on the !-comonad.
• The results suggest that convenient vector spaces are well-suited for modeling integration and holomorphic functionals, with potential for future logical extensions.