  Recently, there has been growing interest in bicategorical models of
programming languages, which are "proof-relevant" in the sense that they keep
distinct account of execution traces leading to the same observable outcomes,
while assigning a formal meaning to reduction paths as isomorphisms.
  In this paper we introduce a new model, a bicategory called thin spans of
groupoids. Conceptually it is close to Fiore et al.'s generalized species of
structures and to Melli\`es' homotopy template games, but fundamentally differs
as to how replication of resources and the resulting symmetries are treated.
Where those models are saturated -- the interpretation is inflated by the fact
that semantic individuals may carry arbitrary symmetries -- our model is thin,
drawing inspiration from thin concurrent games: the interpretation of terms
carries no symmetries, but semantic individuals satisfy a subtle invariant
defined via biorthogonality, which guarantees their invariance under symmetry.
  We first build the bicategory $\mathbf{Thin}$ of thin spans of groupoids. Its
objects are certain groupoids with additional structure, its morphisms are
spans composed via plain pullback with identities the identity spans, and its
$2$-cells are span morphisms making the induced triangles commute only up to
natural isomorphism. We then equip $\mathbf{Thin}$ with a pseudocomonad $!$,
and finally show that the Kleisli bicategory $\mathbf{Thin}_{!}$ is cartesian
closed.
