## ABSTRACT

24/11/2000

**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*)

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: 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.