* quotients and finitary functors
@ 2010-09-28 11:29 Ondrej Rypacek
2010-09-29 5:32 ` Jiri Velebil
[not found] ` <Pine.LNX.4.64.1009290717210.25898@newton.feld.cvut.cz>
0 siblings, 2 replies; 3+ messages in thread
From: Ondrej Rypacek @ 2010-09-28 11:29 UTC (permalink / raw)
To: categories
Dear All,
It is well known to me that a set functor is finitary iff it is a
quotient of a polynomial functor (for some finitary signature).
Are there any similar results for categories other than Set (toposes) ?
Thanks,
Ondrej
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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* Re: quotients and finitary functors
2010-09-28 11:29 quotients and finitary functors Ondrej Rypacek
@ 2010-09-29 5:32 ` Jiri Velebil
[not found] ` <Pine.LNX.4.64.1009290717210.25898@newton.feld.cvut.cz>
1 sibling, 0 replies; 3+ messages in thread
From: Jiri Velebil @ 2010-09-29 5:32 UTC (permalink / raw)
To: Ondrej Rypacek; +Cc: categories
Dear Ondrej,
The answer is yes, for every category \A for which it makes
good sense to say a ``finitary endofunctor''.
Hence, take any locally finitely presentable category \A
and denote by J : \A_\fp --> \A the full inclusion representing
finitely presentable objects in \A.
Then J exhibits \A as a free cocompletion of \A under filtered
colimits, hence there is an equivalence
(1) finitary endofunctors of \A ~ [\A_\fp,\A]
Now, if you consider another functor, namely
E : |\A_\fp| --> \A (where |\A_\fp| is the underlying discrete
category of \A_\fp), then restriction-along-E provides you
with a monadic functor
U: [\A_\fp,\A] ---> [|\A_\fp|,\A]
By [KP] (reference below), the category [|\A_\fp|,\A] can be
seen as the category of finitary signatures on \A. The left
adjoint F to U assigns a polynomial functor H_\Sigma to
the signature \Sigma.
Monadicity of U then implies that every finitary functor H
(using the equivalence of categories (1) above) can be
expressed as a coequalizer of the form
F(\Gamma) ---> F(\Sigma) ---> H
--->
and this is the ``quotient'' you asked about.
All of the above can be proved slightly more generally
by replacing ``locally finitely presentable'' by ``locally
\lambda-presentable''.
[KP] G.M.Kelly and A.J.Power, Adjunctions whose counits
are coequalizers, and presentations of finitary enriched
monads, Journal of Pure and Applied Algebra
Volume 89, Issues 1-2, 8 October 1993, Pages 163-179,
doi:10.1016/0022-4049(93)90092-8
Hope it helped,
Jirka
On Tue, 28 Sep 2010, Ondrej Rypacek wrote:
> Dear All,
> It is well known to me that a set functor is finitary iff it is a
> quotient of a polynomial functor (for some finitary signature).
>
> Are there any similar results for categories other than Set (toposes) ?
>
>
> Thanks,
> Ondrej
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
^ permalink raw reply [flat|nested] 3+ messages in thread
* Re: quotients and finitary functors
[not found] ` <Pine.LNX.4.64.1009290717210.25898@newton.feld.cvut.cz>
@ 2010-09-29 5:41 ` Jiri Velebil
0 siblings, 0 replies; 3+ messages in thread
From: Jiri Velebil @ 2010-09-29 5:41 UTC (permalink / raw)
To: Ondrej Rypacek; +Cc: categories
Two typos in my previous posting. Apologies, J.
On Wed, 29 Sep 2010, Jiri Velebil wrote:
> Then J exhibits \A as a free cocompletion of \A under filtered
^^^ should be \A_\fp
> E : |\A_\fp| --> \A (where |\A_\fp| is the underlying discrete
^^^^^^^^^^^^^^^^^^^^ should be E : |\A_\fp| --> \A_\fp
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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2010-09-28 11:29 quotients and finitary functors Ondrej Rypacek
2010-09-29 5:32 ` Jiri Velebil
[not found] ` <Pine.LNX.4.64.1009290717210.25898@newton.feld.cvut.cz>
2010-09-29 5:41 ` Jiri Velebil
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