a paper available

[Marek Zawadowski <zawado@mimuw.edu.pl>]

The following paper

Lax Monoidal Fibrations

Abstract: We introduce the notion of a lax monoidal fibration
and we show how it can be conveniently used to deal with various
algebraic structures that play an important role in some definitions
of the opetopic sets (Baez-Dolan, Hermida-Makkai-Power). We present
the 'standard' such structures, the exponential fibrations of basic
fibrations and three areas of applications. First area is related
to the T-categories of A. Burroni. The monoids in the Burroni lax
monoidal fibrations form the fibration of T-categories.
The construction of the relative Burroni fibrations and free
T-categories in this context, allow us to extend the definition
of the set of opetopes given by T. Leinster to the category
of opetopic sets (internally to any Grothendieck topos, if needed).
We also show that fibration of (1-level) multicategories, considered
by Hermida-Makkai-Power, is equivalent to the fibration of
(finitary, cartesian) polynomial monads. This equivalence is induced
by the equivalence of lax monoidal fibrations of amalgamated signatures,
polynomial diagrams, and polynomial (finitary, endo) functors. Finally,
we develop a similar theory for symmetric signatures, analytic diagrams
(a notion introduced here), and (finitary, multivariable) analytic
(endo)functors. Among other things we show that the fibrations of
symmetric multicategories is equivalent to the fibration of analytic
monads. We also give a characterization of such a fibration of analytic
monads. An object of this fibration is a weakly cartesian monad on
a slice of Set whose functor parts is a finitary functors weakly
preserving wide pullbacks. A morphism of this fibration is a weakly
cartesian morphism of monads whose functor part is a pullback functor.

is available at

http://arxiv.org/abs/0912.4464

and at my home-page

http://www.mimuw.edu.pl/~zawado/papers.htm

Comments are welcome.

Happy New Year to all,
Marek Zawadowski


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