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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