This paper is about a categorical approach to model a very simple
Semantically Linear lambda calculus, named Sll-calculus. This is a core
calculus underlying the programming language SlPCF. In particular, in this
work, we introduce the notion of Sll-Category, which is able to describe a very
large class of sound models of Sll-calculus. Sll-Category extends in the
natural way Benton, Bierman, Hyland and de Paiva's Linear Category, in order to
soundly interpret all the constructs of Sll-calculus.