Exponentiation for unitary structures

Motivated by the observation that both pretopologies and preapproach limits can be characterized as those convergence relations which have a unit for a suitable composition, we introduce the category Alg u ( T ; V ) of reflexive and unitary lax algebras, for a symmetric monoidal closed lattice V and a Set-monad T = ( T , e , m ) . For T = U the ultrafilter monad, we characterize exponentiable morphisms in Alg u ( U ; V ) . Further, we give a sufficient condition for an object to be exponentiable in the category Alg ( U ; V ) of reflexive and transitive lax algebras. This specializes to known and new results for pretopological, preapproach and approach spaces.