Re: preprint announcement

[Vincenzo Ciancia <ciancia@di.unipi.it>]

Il giorno lun, 01/06/2009 alle 16.15 +0300, Panagis Karazeris ha
scritto:
> Dear all,
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> I would like to announce that the following preprint is available as
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> http://arxiv.org/abs/0905.4883
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> as well as from my webpage
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> www.math.upatras.gr/~pkarazer
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> Final Coalgebras in Accessible Categories,
> by Panagis Karazeris, Apostolos Matzaris and Jiri Velebil
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> Abstract:
> We give conditions on a finitary endofunctor of a finitely accessible
> category to admit a final coalgebra. Our conditions always apply to the
> case of a finitary endofunctor of a locally finitely presentable (l.f.p=
.)
> category and they bring an explicit construction of the final coalgebra=
 in
> this case. On the other hand, there are interesting examples of final
> coalgebras beyond the realm of l.f.p. categories to which our results
> apply. We rely on ideas developed by Tom Leinster for the study of
> self-similar objects in topology.=20
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> Best regards,
> Panagis Karazeris
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I do not see the following paper in the references; would it be worth to
provide a comparison?

http://www.sciencedirect.com/science?_ob=3DArticleURL&_udi=3DB75H1-4G7MXP=
F-4&_user=3D144492&_rdoc=3D1&_fmt=3D&_orig=3Dsearch&_sort=3Dd&view=3Dc&_a=
cct=3DC000012038&_version=3D1&_urlVersion=3D0&_userid=3D144492&md5=3D5760=
58372d432ade83f476c43b8b466a


Terminal sequences for accessible endofunctors=20
James Worrell

Abstract:

We consider the behaviour of the terminal sequence of an accessible
endofunctor T on a locally presentable category K. The preservation of
monics by T is sufficient to imply convergence, necessarily to a
terminal coalgebra. We can say much more if K is Set, and =CE=BA is =CF=89=
. In
that case it is well known that we do not necessarily get convergence at
=CF=89, however we show that to ensure convergence we don't need to go to=
 a
higher cardinal, just to the next limit ordinal, =CF=89 + =CF=89.

For an =CF=89-accessible endofunctor T on Set the construction of the
terminal coalgebra can thus be seen as a two stage construction, with
each stage being finitary. The first stage obtains the Cauchy completion
of the initial T-algebra as the =CF=89-th object in the terminal sequence=
 A=CF=89.
In the second stage this object is pruned to get the final coalgebra A=CF=
=89
+=CF=89. We give an example where A=CF=89 is the solution of the correspo=
nding
domain equation in the category of complete ultra-metric spaces.



Thanks

Vincenzo




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