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From: Rob Goldblatt <[email-masked]>
Subject: categories: preprint on behavioural covarieties
Date: Tue, 1 Jun 2004 12:08:22 +1200
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A preprint of a paper entitled

"A comonadic account of behavioural covarieties of coalgebras"

is available for downloading as a pdf file from

www.mcs.vuw.ac.nz/~rob

Rob Goldblatt

ABSTRACT:
A class K of coalgebras for an endofunctor T on the category of sets is=20=

a behavioural covariety if it is closed under disjoint unions and=20
images of bisimulation relations (hence closed under images and domains=20=

of coalgebraic morphisms, including subcoalgebras). K may be thought of=20=

as the class of all coalgebras that satisfy some computationally=20
significant property. In any logical system suitable for specifying=20
properties of state-transition systems in the Hennessy-Milner style,=20
each formula will define a class of models that is a behavioural=20
variety.

  Assume that the forgetful functor on T-coalgebras has a right adjoint,=20=

providing for the construction of cofree coalgebras, and let G^T be the=20=

comonad arising from this adjunction. Then we show that behavioural=20
covarieties K are (isomorphic to) the Eilenberg-Moore categories of=20
coalgebras for certain comonads G^K naturally associated with G^T.=20
These are called pure subcomonads of G^T, and a categorical=20
characterization of them is given, involving a pullback condition on=20
the naturality squares of a transformation from G^K to G^T.

  We show that=A0 there is a bijective correspondence between =
behavioural=20
covarieties of T-coalgebras and isomorphism classes of pure subcomonads=20=

of G^T.

 =20=