  We introduce a bicategorical model of linear logic which is a novel variation
of the bicategory of groupoids, profunctors, and natural transformations. Our
model is obtained by endowing groupoids with additional structure, called a
kit, to stabilize the profunctors by controlling the freeness of the groupoid
action on profunctor elements. The theory of generalized species of structures,
based on profunctors, is refined to a new theory of \emph{stable species} of
structures between groupoids with Boolean kits. Generalized species are in
correspondence with analytic functors between presheaf categories; in our
refined model, stable species are shown to be in correspondence with
restrictions of analytic functors, which we characterize as being stable, to
full subcategories of stabilized presheaves. Our motivating example is the
class of finitary polynomial functors between categories of indexed sets, also
known as normal functors, that arises from kits enforcing free actions. We show
that the bicategory of groupoids with Boolean kits, stable species, and natural
transformations is cartesian closed. This makes essential use of the logical
structure of Boolean kits and explains the well-known failure of cartesian
closure for the bicategory of finitary polynomial functors between categories
of set-indexed families and cartesian natural transformations. The paper
additionally develops the model of classical linear logic underlying the
cartesian closed structure and clarifies the connection to stable domain
theory.
