  In this paper, we build on a recent bicategorical model called thin spans of
groupoids, introduced by Clairambault and Forest. Notably, thin spans feature a
decomposition of symmetry into two sub-groupoids of polarized -- positive and
negative -- symmetries. We first construct a variation of the original
exponential of thin spans, based on sequences rather than families. Then we
give a syntactic characterisation of the interpretation of simply-typed
lambda-terms in thin spans, in terms of rigid intersection types and rigid
resource terms. Finally, we formally relate thin spans with the weighted
relational model and generalized species of structure. This allows us to show
how some quantities in those models reflect polarized symmetries: in particular
we show that the weighted relational model counts witnesses from generalized
species of structure, divided by the cardinal of a group of positive
symmetries.
