preprint announcement

preprint announcement

[Panagis Karazeris]

Dear all,

I would like to announce that the following preprint is available as

http://arxiv.org/abs/0905.4883

as well as from my webpage

www.math.upatras.gr/~pkarazer

Final Coalgebras in Accessible Categories,
by Panagis Karazeris, Apostolos Matzaris and Jiri Velebil

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. 

Best regards,
Panagis Karazeris



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Re: preprint announcement

[Vincenzo Ciancia]

Il giorno lun, 01/06/2009 alle 16.15 +0300, Panagis Karazeris ha
scritto:
> Dear all,
>=20
> I would like to announce that the following preprint is available as
>=20
> http://arxiv.org/abs/0905.4883
>=20
> as well as from my webpage
>=20
> www.math.upatras.gr/~pkarazer
>=20
> Final Coalgebras in Accessible Categories,
> by Panagis Karazeris, Apostolos Matzaris and Jiri Velebil
>=20
> 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
>=20
> Best regards,
> Panagis Karazeris
>=20

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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