Claudio Hermida, (IST, Lisboa)
Coherence via Universality

In this talk we present our papers [1] and [2], trying to give an overall picture while illustrating with the main example taken from the first paper.

[1] From coherent structures to universal properties (to appear in Journal of Pure and Applied Algebra)
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: i) S has the adjoint-pseudo-algebra property. ii) 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 Cat.

[2] Representable multicategories (to appear in Advances in Mathematics)
Abstract: We introduce the notion of representable multicategory, which stands in the same relation to that of monoidal category as fibration does to contravariant pseudofunctor (into Cat). We give an abstract reformulation of multicategories as monads in a suitable Kleisli bicategory of spans. We describe representability in elementary terms via universal arrows. We also give a doctrinal characterisation of representability based on a fundamental monadic adjunction between the 2-category of multicategories and that of strict monoidal categories. The first main result is the coherence theorem for representable multicategories, asserting their equivalence to strict ones, which we establish via a new technique based on the above doctrinal characterisation. The other main result is a 2-equivalence between the 2-category of representable multicategories and that of monoidal categories and strong monoidal functors. This correspondence extends smoothly to one between bicategories and a localised version of representable multicategories.