Re: quotients and finitary functors

[Jiri Velebil <velebil@math.feld.cvut.cz>]

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


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