preprint: Representable Multicategories

preprint: Representable Multicategories

[Claudio Hermida]

The preprint `Representable multicategories' is available at


http://www.maths.usyd.edu.au:8000/u/hermida

 

     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. 

-- 
Claudio Hermida				  

School of Mathematics and Statistics F07,
University of Sydney,
Sydney, NSW 2006,
Australia