Integral Categories and Calculus Categories
J.R.B. Cockett, JS Lemay

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, , 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, , is invertible. We observe that a differential category having antiderivatives, in this sense, is always…
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Taxonomy
TopicsNeural Networks and Applications
