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).

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