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A monadic approach to polycategories

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Juergen Koslowski

In the quest for an elegant formulation of the notion of
``polycategory'' we develop a more symmetric counterpart to
Burroni's notion of ``T- category'', where T is a cartesian
monad on a category X with pullbacks. Our approach involves two
such monads, S and T, that are linked by a suitable
generalization of a distributive law in the sense of Beck. This
takes the form of a span omega : TS <--> ST in the functor
category [X,X] and guarantees essential associativity for a
canonical pullback-induced composition of S-T-spans over X,
identifying them as the 1-cells of a bicategory, whose (internal)
monoids then qualify as ``omega-categories''. In case that
S and T both are the free monoid monad on set, we construct
an omega utilizing an apparently new classical distributive law
linking the free semigroup monad with itself. Our construction then
gives rise to so-called ``planar polycategories'', which nowadays
seem to be of more intrinsic interest than Szabo's original
polycategories. Weakly cartesian monads on X may be accommodated
as well by first quotienting the bicategory of X-spans.

Keywords:
cartesian monad, S-T-span, (cartesian) distributive law, multicategory,
(planar) polycategory, fc-polycategory, associative double semigroup

2000 MSC:
18C15, 18D05

*Theory and Applications of Categories,*
Vol. 14, 2005,
No. 7, pp 125-156.

http://www.tac.mta.ca/tac/volumes/14/7/14-07.dvi

http://www.tac.mta.ca/tac/volumes/14/7/14-07.ps

http://www.tac.mta.ca/tac/volumes/14/7/14-07.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/14/7/14-07.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/14/7/14-07.ps

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