# Cartesian Integral Categories and Contextual Integral Categories

*J. Robin B. Cockett and Jean−Simon Pacaud Lemay*

### Abstract

The notion of a Cartesian integral category is introduced and motivated. Morally, this notion should be the coKleisli category of a (tensor) integral category. However, unfortunately, integral categories, as introduced, do not in general have enough structure for this to be true. The objective of this paper is then to find an appropriate (tensor) integral structure on a coalgebra modality which actually produces as its coKleisli category a Cartesian integral category. The main theorem of this paper is that any contextual integral category with a pseudo-linearizing transformation has its coKlleisli category a Cartesian integral category. The result seems to be quite general as, in particular, it covers all example in which the linear maps arise from a differential.