Integral Categories and Calculus Categories

TL;DR

This paper develops an abstract framework for integration in symmetric monoidal categories, introduces calculus categories combining differentiation and integration, and explores their properties and examples.

Contribution

It axiomatizes integration in differential categories, defines calculus categories, and shows how integral transformations relate to antiderivatives and differential structures.

Findings

Integral transformation axiomatized in symmetric monoidal categories.

Calculus categories satisfy the two fundamental theorems of calculus.

Examples illustrate differential, integral, and calculus categories with various modalities.

Abstract

Differential categories are now an established abstract setting for differentiation. However not much attention has been given to the process which is inverse to differentiation: integration. This paper presents the parallel development for integration by axiomatizing an integral transformation, $s_A: !A \to !A \otimes A$, in a symmetric monoidal category with a coalgebra modality. When integration is combined with differentiation, the two fundamental theorems of calculus are expected to hold (in a suitable sense): a differential category with integration which satisfies these two theorem is called a {\em calculus category\/}. Modifying an approach to antiderivatives by T. Ehrhard, we define having antiderivatives as the demand that a certain natural transformation, $K: !A \to !A$, is invertible. We observe that a differential category having antiderivatives, in this sense, is always a calculus category. When the coalgebra modality is monoidal, it is natural to demand an extra coherence between integration and the coalgebra modality. In the presence of this extra coherence we show that a calculus category with a monoidal coalgebra modality has its integral transformation given by antiderivatives and, thus, that the integral structure is uniquely determined by the differential structure. The paper finishes by providing a suite of separating examples. Examples of differential categories, integral categories, and calculus categories based on both monoidal and (mere) coalgebra modalities are presented. In addition, differential categories which are not integral categories are discussed and vice versa.