From coherent structures to universal properties

Abstract Given a 2-category K admitting a calculus of bimodules, and a 2-monad T on it compatible with such calculus, we construct a 2-category L with a 2-monad S on it such that • S has the adjoint-pseudo-algebra property. • The 2-categories of pseudo-algebras of S and T are equivalent. Thus, coherent structures (pseudo- T -algebras) are transformed into universally characterised ones (adjoint-pseudo- S -algebras). The 2-category L consists of lax algebras for the pseudo-monad induced by T on the bicategory of bimodules of K . We give an intrinsic characterisation of pseudo- S -algebras in terms of representability. Two major consequences of the above transformation are the classifications of lax and strong morphisms, with the attendant coherence result for pseudo-algebras. We apply the theory in the context of internal categories and examine monoidal and monoidal globular categories (including their monoid classifiers) as well as pseudo-functors into C at .