Fibrations and Yoneda structure for multicategories
by
Claudio Hermida
Mathematics Departmente, IST, Lisbon

We continue the development of the abstract theory of multicategories (relative to a cartesian monad) started in ``Representable Multicategories'' (Adv. in Math. 151, 164-225, 2000) and ``From coherent structures to universal properties'' (JPAA, 165(1), 7-61, 2001), so that our multicategories are monads in Spn_T or Bimod_T. We set up the basics of a theory of (co)fibrations for multicategories. The main difficulty is that, given the nature of 2-cells in Multicat (which detect only linear morphisms) the usual representable notion of fibration in a 2-category is unsuitable in this context. The key to the development of a good theory of fibrations for multicategories is the monadic adjunction between Multicat and MonCat. We give elementary definitions of fibrations for multicategories (in Set) and then proceed to an abstract characterisation (namely the free-monoidal-category functor reflects the property of being a (co)fibration). We give an indexed version in terms of homomorphisms of monoidal bicategories, which we use to construct free cofibrations and a comprehensive factorisation of morhisms of multicategories (initial, discrete cofibration). An interesting example of discrete cofibrations are `algebras for operads'. Another main example of cofibrations are `representable multicategories' (cofibred over the terminal multicategory). Fibrations of multicategories set up a 2-fibration over Multicat.

We also establish a change-of-base result for Yoneda structures (in Street-Walters' sense) along a left (bi)adjoint between 2-categories. We use the (bi)adjunction between Multicat and MonCat to obtain thus a Yoneda structure for multicategories, whose resulting Yoneda embedding yields a full and faithful representation of a multicategory as a multicategory of modules (presheaves) and multilinear morphisms.

Date received: May 15, 2002