Abstract

There are several kinds of linear type theory in the literature, some with their associated notion of categorical model. Our aim on this paper is to systematize the relationships amongst three of these linear type theories and their models. We point out that mere soundness and completeness of a linear type theory with respect to a class of categorical models is not sufficient to uniquely identify the most appropriate class of models for the theory in question. We recommend instead the use of internal languages. Considering their internal languages we relate the categories of models in the literature via reflections and coreections.

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