  It is known that the quantale of sup-preserving maps from a complete lattice
to itself is a Frobenius quantale if and only if the lattice is completely
distributive. Since completely distributive lattices are the nuclear objects in
the autonomous category of complete lattices and sup-preserving maps, we study
the above statement in a categorical setting. We introduce the notion of
Frobenius structure in an arbitrary autonomous category, generalizing that of
Frobenius quantale. We prove that the monoid of endomorphisms of a nuclear
object has a Frobenius structure. If the environment category is
star-autonomous and has epi-mono factorizations, a variant of this theorem
allows to develop an abstract phase semantics and to generalise the previous
statement. Conversely, we argue that, in a star-autonomous category where the
monoidal unit is a dualizing object, if the monoid of endomorphisms of an
object has a Frobenius structure and the monoidal unit embeds into this object
as a retract, then the object is nuclear.
