Functor derivatives - a question and a result

Functor derivatives - a question and a result

[Jiri Adamek]

Andre Joyal defined derivatives of analytic functors
in his 1986 paper. Recently I heard the more general definition
of a derivative F' of an endofunctor F defined via a universal
sub-cartesian transformation from F'xId into F. Who is the author
of this definition? The following result seems to indicate that
outside of the realm of analytic functors derivatives may not
be really useful:

Theorem. Every non-faithful functor F:Set -> Set has the derivative
         F' = 0 (the constant functor to the empty set).

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Re: Functor derivatives - a question and a result

[Marcelo Fiore]

On a related matter to the message below by Jiri, let me point out the
following paper:

   M. Fiore.  Differential structure in models of multiplicative
   biadditive intuitionistic linear logic.  In Typed Lambda Calculi
   and Applications (TLCA 2007), LNCS 4583, pp. 163-177, 2007.
   [Available from <http://www.cl.cam.ac.uk/~mpf23/latest.html>]

presenting a categorical framework for differentiation, directly
synthetised from the differential calculus of generalised species
of structures.  Though, as it transpired in conversation with Anders
Kock, the setting is also applicable to convenient vector spaces
and some models of SDG.


On Fri, 2 Nov 2007, Jiri Adamek wrote:
>
> Andre Joyal defined derivatives of analytic functors
> in his 1986 paper. Recently I heard the more general definition
> of a derivative F' of an endofunctor F defined via a universal
> sub-cartesian transformation from F'xId into F. Who is the author
> of this definition? The following result seems to indicate that
> outside of the realm of analytic functors derivatives may not
> be really useful:
>
> Theorem. Every non-faithful functor F:Set -> Set has the derivative
>         F' = 0 (the constant functor to the empty set).
>
> xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> alternative e-mail address (in case reply key does not work):
> J.Adamek@tu-bs.de
> xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
>