Abstract. For accessible set-valued functors it is well known that weak preservation of limits is equivalent to representability, and weak preservation of connected limits to familial representability. In contrast, preservation of weak wide pullbacks is equivalent to being a coproduct of quotients of hom-functors modulo groups of automorphisms. For finitary functors this was proved by André Joyal who called these functors analytic. We introduce a generalization of Joyal’s concept from endofunctors of Set to endofunctors of a symmetric monoidal category. 1.

We describe a sufficient condition for the process of left Kan extension to be a conservative functor. This is useful in the study of graphic Fourier transforms and quantum categories and groupoids. 1